Introduction
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Convergence of a Sequence, Monotone Sequences
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Cauchy Criterion, Bolzano - Weierstrass Theorem
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Disproving Limits of Sequences
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{"Free":0,"Sample":1,"Paid":2}

[{"Name":"Introduction","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"What is a Sequence","Duration":"9m 42s","ChapterTopicVideoID":26295,"CourseChapterTopicPlaylistID":254619,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26295.jpeg","UploadDate":"2021-09-12T14:02:18.6630000","DurationForVideoObject":"PT9M42S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.725","Text":"In this clip, we\u0027re starting a new topic, sequences."},{"Start":"00:04.725 ","End":"00:09.945","Text":"A sequence is an ordered list of numbers and those numbers are called its terms."},{"Start":"00:09.945 ","End":"00:14.550","Text":"But in some places they\u0027re called members or elements of"},{"Start":"00:14.550 ","End":"00:19.500","Text":"the list and the list could be finite or infinite."},{"Start":"00:19.500 ","End":"00:24.645","Text":"But in this course, we\u0027re only going to be concerned with infinite sequences."},{"Start":"00:24.645 ","End":"00:28.370","Text":"Anyway, let\u0027s look at some examples."},{"Start":"00:28.370 ","End":"00:30.095","Text":"Here\u0027s the 1st one."},{"Start":"00:30.095 ","End":"00:33.605","Text":"The first term is 1 over 1,"},{"Start":"00:33.605 ","End":"00:35.870","Text":"the second term is 1.5,"},{"Start":"00:35.870 ","End":"00:41.410","Text":"then 1/4, 1/4 and you can guess the pattern and so on."},{"Start":"00:41.410 ","End":"00:43.935","Text":"In the second example,"},{"Start":"00:43.935 ","End":"00:47.190","Text":"the first term is 1 over 2,"},{"Start":"00:47.190 ","End":"00:49.160","Text":"2^11, 1 over 2 squared,"},{"Start":"00:49.160 ","End":"00:50.270","Text":"1 over 2 cubed,"},{"Start":"00:50.270 ","End":"00:52.520","Text":"1 over 2^4, and again,"},{"Start":"00:52.520 ","End":"00:53.870","Text":"you can probably guess the pattern,"},{"Start":"00:53.870 ","End":"00:58.740","Text":"the next element term would be 1 over 2 to the fifth."},{"Start":"00:59.170 ","End":"01:03.530","Text":"Third example, just alternating minus 1,"},{"Start":"01:03.530 ","End":"01:05.525","Text":"1, minus 1, 1,"},{"Start":"01:05.525 ","End":"01:08.600","Text":"minus 1, 1, and so on."},{"Start":"01:08.600 ","End":"01:11.555","Text":"Our last example here,"},{"Start":"01:11.555 ","End":"01:17.015","Text":"this is what it looks like and you can probably guess the pattern."},{"Start":"01:17.015 ","End":"01:23.450","Text":"For example, the next one in the sequence would be root 5 over 6."},{"Start":"01:23.450 ","End":"01:25.610","Text":"I think you get the idea."},{"Start":"01:25.610 ","End":"01:29.990","Text":"Now there\u0027s a notation for sequences on the terms of the sequence."},{"Start":"01:29.990 ","End":"01:33.890","Text":"I just want to remind you though that terms sometimes called members or"},{"Start":"01:33.890 ","End":"01:38.480","Text":"elements and we denote them as follows."},{"Start":"01:38.480 ","End":"01:42.350","Text":"The first term of the sequence is usually called a_1,"},{"Start":"01:42.350 ","End":"01:44.435","Text":"the second term a_2,"},{"Start":"01:44.435 ","End":"01:48.410","Text":"the third term a_3, and so on."},{"Start":"01:48.410 ","End":"01:51.185","Text":"But it doesn\u0027t have to be the letter a."},{"Start":"01:51.185 ","End":"01:54.650","Text":"You could have b_1 for the first term,"},{"Start":"01:54.650 ","End":"01:56.240","Text":"b_2, for the second term,"},{"Start":"01:56.240 ","End":"01:59.045","Text":"b_3, and so on."},{"Start":"01:59.045 ","End":"02:07.099","Text":"Some examples, and actually these are the same examples that we had above."},{"Start":"02:07.099 ","End":"02:10.040","Text":"I just added the notation so,"},{"Start":"02:10.040 ","End":"02:11.375","Text":"in the first sequence,"},{"Start":"02:11.375 ","End":"02:13.080","Text":"the first term is a_1,"},{"Start":"02:13.080 ","End":"02:14.340","Text":"next one is a_2,"},{"Start":"02:14.340 ","End":"02:16.395","Text":"a_3, a_4, and so on."},{"Start":"02:16.395 ","End":"02:18.620","Text":"If we have several sequences at once,"},{"Start":"02:18.620 ","End":"02:20.030","Text":"we don\u0027t want to mix the letters up,"},{"Start":"02:20.030 ","End":"02:21.470","Text":"so I\u0027ll give each one a different letter."},{"Start":"02:21.470 ","End":"02:24.969","Text":"This would be say, b_1, b_2, b_3, b_4,"},{"Start":"02:24.969 ","End":"02:27.965","Text":"and then here I used the letter c,"},{"Start":"02:27.965 ","End":"02:34.770","Text":"and here the letter d and the subscript indicates what the position is in the sequence."},{"Start":"02:35.000 ","End":"02:42.020","Text":"Next, I\u0027ll be talking about something called the general term of a sequence."},{"Start":"02:42.020 ","End":"02:44.270","Text":"Now we need to talk about a general term"},{"Start":"02:44.270 ","End":"02:46.220","Text":"because we don\u0027t if we just want to talk about the 1st,"},{"Start":"02:46.220 ","End":"02:47.945","Text":"2nd, 3rd, 4th terms."},{"Start":"02:47.945 ","End":"02:52.490","Text":"I want to talk about the 17th term, the 119th term."},{"Start":"02:52.490 ","End":"02:54.740","Text":"In general, for a number n,"},{"Start":"02:54.740 ","End":"03:00.605","Text":"I want to know what is the element or term in the nth place."},{"Start":"03:00.605 ","End":"03:02.420","Text":"Perhaps to have consistency,"},{"Start":"03:02.420 ","End":"03:10.605","Text":"I should just stick with the same term which is term member element term yeah."},{"Start":"03:10.605 ","End":"03:12.540","Text":"In the nth place,"},{"Start":"03:12.540 ","End":"03:15.110","Text":"and it\u0027s the nth element,"},{"Start":"03:15.110 ","End":"03:20.680","Text":"and it\u0027s usually denoted as a_n."},{"Start":"03:20.680 ","End":"03:25.850","Text":"In the nth place I have a and then a subscript n. Of course,"},{"Start":"03:25.850 ","End":"03:28.100","Text":"if I\u0027m not using a in this sequence,"},{"Start":"03:28.100 ","End":"03:34.190","Text":"so general term would be b_n or c_n or d_n, and so on."},{"Start":"03:34.190 ","End":"03:39.500","Text":"But the subscript denotes what place in the sequence we are."},{"Start":"03:39.500 ","End":"03:44.110","Text":"Now, I\u0027ll give some examples which are actually these examples."},{"Start":"03:44.110 ","End":"03:46.575","Text":"In this example here,"},{"Start":"03:46.575 ","End":"03:49.815","Text":"for example, the 4th term is 1 over 4."},{"Start":"03:49.815 ","End":"03:51.320","Text":"If you look at the pattern,"},{"Start":"03:51.320 ","End":"04:00.030","Text":"the nth term will be 1 over n. So a_n is 1 over n. Then the 2nd example here,"},{"Start":"04:00.030 ","End":"04:07.870","Text":"which you only went up to 4, you can see that in place n in the nth term is 1 over 2^n."},{"Start":"04:07.870 ","End":"04:13.895","Text":"Now given the other couple of examples which correspond to these two."},{"Start":"04:13.895 ","End":"04:17.420","Text":"For this sequence, which is an alternating sequence,"},{"Start":"04:17.420 ","End":"04:20.015","Text":"meaning it goes minus plus minus plus."},{"Start":"04:20.015 ","End":"04:26.240","Text":"The general term would be minus 1^n."},{"Start":"04:26.240 ","End":"04:30.140","Text":"Now, note that if n is odd,"},{"Start":"04:30.140 ","End":"04:35.260","Text":"then minus 1^n is minus 1 and if it\u0027s even,"},{"Start":"04:35.260 ","End":"04:40.480","Text":"it\u0027s plus 1 and luckily turned out right that the 1st place was minus,"},{"Start":"04:40.480 ","End":"04:42.010","Text":"the 2nd place was plus."},{"Start":"04:42.010 ","End":"04:46.430","Text":"On a remark, if you wanted to take it the other way around, 1,"},{"Start":"04:46.430 ","End":"04:51.440","Text":"minus 1,1 minus 1, et cetera."},{"Start":"04:51.440 ","End":"04:57.775","Text":"Then the way you do it is you would say that the general term is minus 1^n."},{"Start":"04:57.775 ","End":"05:03.450","Text":"You would add another minus and then it would reverse the odd and the even."},{"Start":"05:03.450 ","End":"05:06.830","Text":"The odd places would get the plus 1,"},{"Start":"05:06.830 ","End":"05:09.670","Text":"and the even places will get the minus 1."},{"Start":"05:09.670 ","End":"05:12.810","Text":"This happens a lot so often you want to adjust"},{"Start":"05:12.810 ","End":"05:17.895","Text":"minus 1^n to be minus 1^n plus 1 and reverse the sign."},{"Start":"05:17.895 ","End":"05:19.970","Text":"In this example also,"},{"Start":"05:19.970 ","End":"05:23.150","Text":"we can see the pattern under the square root."},{"Start":"05:23.150 ","End":"05:25.730","Text":"We see that we have 1, 2, 3, 4,"},{"Start":"05:25.730 ","End":"05:30.060","Text":"so in place n it would be n. On the denominator,"},{"Start":"05:30.060 ","End":"05:32.820","Text":"in place 1 we have 2, 2nd term is 3,"},{"Start":"05:32.820 ","End":"05:34.615","Text":"3rd term is 4, 4th term is 5,"},{"Start":"05:34.615 ","End":"05:36.995","Text":"so the nth term would be an n plus 1."},{"Start":"05:36.995 ","End":"05:44.630","Text":"This would be the nth term or the general term of this sequence."},{"Start":"05:44.630 ","End":"05:46.520","Text":"N is a convenient letter,"},{"Start":"05:46.520 ","End":"05:51.920","Text":"but sometimes it\u0027s convenient to use i or j or m,"},{"Start":"05:51.920 ","End":"05:53.900","Text":"unless we say otherwise we choose n,"},{"Start":"05:53.900 ","End":"05:58.825","Text":"and usually if a is free we\u0027ll use letter a."},{"Start":"05:58.825 ","End":"06:05.240","Text":"Next, I\u0027ll show you how to use the general term to describe the sequence."},{"Start":"06:05.240 ","End":"06:10.430","Text":"How do we describe a sequence by means of its general element?"},{"Start":"06:10.430 ","End":"06:12.230","Text":"In the previous examples,"},{"Start":"06:12.230 ","End":"06:18.070","Text":"we found the formula for the general term a_n in terms of n and from it,"},{"Start":"06:18.070 ","End":"06:21.020","Text":"you can compute a_1, a_2, a_3,"},{"Start":"06:21.020 ","End":"06:24.590","Text":"or any particular a by just letting n equal 1,"},{"Start":"06:24.590 ","End":"06:26.620","Text":"2, 3, or whatever."},{"Start":"06:26.620 ","End":"06:29.840","Text":"If I just give you the formula or describe"},{"Start":"06:29.840 ","End":"06:34.715","Text":"the general element a_n then I can figure out the whole sequence."},{"Start":"06:34.715 ","End":"06:38.250","Text":"For example, and this was our first example,"},{"Start":"06:38.250 ","End":"06:39.960","Text":"instead of writing 1,"},{"Start":"06:39.960 ","End":"06:42.185","Text":"1/2, 1/3, 1/4 or 1/5, and so on."},{"Start":"06:42.185 ","End":"06:44.750","Text":"All you have to do is say that a_n is"},{"Start":"06:44.750 ","End":"06:47.780","Text":"1 over n. Because then you can figure out all of these."},{"Start":"06:47.780 ","End":"06:51.590","Text":"You would say, the 1st element, I put an equals 1,"},{"Start":"06:51.590 ","End":"06:53.390","Text":"1 over 1, 2nd element,"},{"Start":"06:53.390 ","End":"06:55.535","Text":"n is 2,1 over 2, and so on."},{"Start":"06:55.535 ","End":"06:58.295","Text":"This is much more compact."},{"Start":"06:58.295 ","End":"07:04.525","Text":"Our 2nd example which was this is written more simply as,"},{"Start":"07:04.525 ","End":"07:07.170","Text":"I\u0027ll use a different letter than a is b."},{"Start":"07:07.170 ","End":"07:12.435","Text":"So b_n, the nth element is 1 over 2^n."},{"Start":"07:12.435 ","End":"07:16.530","Text":"Like the 4th element is 1 over 2^4 and so on."},{"Start":"07:16.530 ","End":"07:19.620","Text":"In our 3rd example is minus 1,1,"},{"Start":"07:19.620 ","End":"07:20.960","Text":"minus 1,1, and so on."},{"Start":"07:20.960 ","End":"07:23.059","Text":"We found that the general element,"},{"Start":"07:23.059 ","End":"07:27.415","Text":"general term c_n is minus1^n."},{"Start":"07:27.415 ","End":"07:34.580","Text":"If I just say this is more of a shorthand way in writing this whole thing with a dot,"},{"Start":"07:34.580 ","End":"07:36.575","Text":"dot, dot at the end."},{"Start":"07:36.575 ","End":"07:43.715","Text":"Now there are some other notations that we can use instead of writing a_n is 1 over n,"},{"Start":"07:43.715 ","End":"07:45.980","Text":"there are other possibilities,"},{"Start":"07:45.980 ","End":"07:50.390","Text":"and I\u0027ll illustrate the notations on the series here,"},{"Start":"07:50.390 ","End":"07:53.335","Text":"the 1/5, 1/3, and so on."},{"Start":"07:53.335 ","End":"07:58.235","Text":"What we wrote simply as a_n equals 1 over n,"},{"Start":"07:58.235 ","End":"08:01.775","Text":"can be written in some minor variations."},{"Start":"08:01.775 ","End":"08:06.170","Text":"The most similar one to this is this,"},{"Start":"08:06.170 ","End":"08:12.080","Text":"which is just putting this in a bracket to emphasize that we\u0027re talking about"},{"Start":"08:12.080 ","End":"08:18.720","Text":"the whole sequence and not about the particular general term, the nth term."},{"Start":"08:19.000 ","End":"08:25.310","Text":"This one is a bit more precise because we assume"},{"Start":"08:25.310 ","End":"08:31.130","Text":"always that we start from the first element and keep going to infinity,"},{"Start":"08:31.130 ","End":"08:33.905","Text":"1, 2, 3, 4, and so on."},{"Start":"08:33.905 ","End":"08:40.609","Text":"This just makes it precise that we do start at element 1 and keep going to infinity."},{"Start":"08:40.609 ","End":"08:43.880","Text":"Sometimes we don\u0027t start at number 1,"},{"Start":"08:43.880 ","End":"08:48.390","Text":"sometimes we start at 0 or at any other number."},{"Start":"08:48.390 ","End":"08:52.600","Text":"The 1st one, we could call it a_3, a_4, a_5."},{"Start":"08:53.030 ","End":"08:55.670","Text":"If we want to be really precise,"},{"Start":"08:55.670 ","End":"08:58.390","Text":"then this will tell us where we start and where we end,"},{"Start":"08:58.390 ","End":"09:01.920","Text":"and if up its infinity it means never-ending."},{"Start":"09:02.620 ","End":"09:06.005","Text":"This is the same thing as this one,"},{"Start":"09:06.005 ","End":"09:11.190","Text":"except that here I use curly brackets instead of round brackets."},{"Start":"09:12.350 ","End":"09:18.350","Text":"This is another variation of this one,"},{"Start":"09:18.350 ","End":"09:21.305","Text":"where instead of saying n goes from 1 to infinity,"},{"Start":"09:21.305 ","End":"09:24.605","Text":"I say, n bigger or equal to 1."},{"Start":"09:24.605 ","End":"09:28.550","Text":"We always assume that n is a whole number,"},{"Start":"09:28.550 ","End":"09:32.165","Text":"but it tells us it\u0027s all the whole numbers from 1 onwards."},{"Start":"09:32.165 ","End":"09:35.390","Text":"These are all variations you might see."},{"Start":"09:35.390 ","End":"09:43.080","Text":"In fact, I\u0027ll be using some of these in the rest of this topic."}],"ID":27202},{"Watched":false,"Name":"Examples of Sequences","Duration":"6m 19s","ChapterTopicVideoID":26296,"CourseChapterTopicPlaylistID":254619,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26296.jpeg","UploadDate":"2021-09-12T14:03:52.1030000","DurationForVideoObject":"PT6M19S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.300","Text":"In this clip, we\u0027ll be solving some example exercises and here\u0027s the first 1."},{"Start":"00:06.300 ","End":"00:11.010","Text":"We have to write the first 5 terms of the sequence, which is this."},{"Start":"00:11.010 ","End":"00:13.349","Text":"Notice I\u0027ve used the curly brace notation,"},{"Start":"00:13.349 ","End":"00:15.525","Text":"that\u0027s 1 of the several notations."},{"Start":"00:15.525 ","End":"00:21.345","Text":"Basically what it says is that the nth term is n plus 4 over n squared."},{"Start":"00:21.345 ","End":"00:25.350","Text":"What I want to do is plug in successively n equals 1, 2,"},{"Start":"00:25.350 ","End":"00:29.085","Text":"3, 4, 5 and here we are."},{"Start":"00:29.085 ","End":"00:31.980","Text":"Instead of n here and here,"},{"Start":"00:31.980 ","End":"00:35.250","Text":"I put 1 or 2 or 3 or 4 or 5,"},{"Start":"00:35.250 ","End":"00:37.620","Text":"then this is what I get for a_1,"},{"Start":"00:37.620 ","End":"00:40.410","Text":"a_2, a_3, a_4, a_5, first 5 terms."},{"Start":"00:40.410 ","End":"00:42.705","Text":"If I just simplify them,"},{"Start":"00:42.705 ","End":"00:47.025","Text":"then here\u0027s what we get like 1 plus 4 is 5 over 1,"},{"Start":"00:47.025 ","End":"00:50.100","Text":"2 plus 4, 6 over 4,"},{"Start":"00:50.100 ","End":"00:52.655","Text":"7 over 9, and so on."},{"Start":"00:52.655 ","End":"00:55.609","Text":"Now let\u0027s go on to the next example."},{"Start":"00:55.609 ","End":"00:57.590","Text":"In this next example,"},{"Start":"00:57.590 ","End":"01:00.560","Text":"we have to add the first 5 terms of this sequence."},{"Start":"01:00.560 ","End":"01:02.120","Text":"It\u0027s not a_n, it\u0027s b_n."},{"Start":"01:02.120 ","End":"01:08.005","Text":"That\u0027s okay and it\u0027s written with the notation n goes from 1 to infinity."},{"Start":"01:08.005 ","End":"01:10.470","Text":"We just want from 1 to 5,"},{"Start":"01:10.470 ","End":"01:14.240","Text":"so we just take this expression and plug in successively 1,"},{"Start":"01:14.240 ","End":"01:17.440","Text":"2, 3, 4, and 5. Here\u0027s what we get."},{"Start":"01:17.440 ","End":"01:19.430","Text":"Instead of the n, which is here and here,"},{"Start":"01:19.430 ","End":"01:21.395","Text":"I put 1 and then 2,"},{"Start":"01:21.395 ","End":"01:23.350","Text":"3, and so on."},{"Start":"01:23.350 ","End":"01:25.685","Text":"Now we just simplify a little bit."},{"Start":"01:25.685 ","End":"01:31.910","Text":"Note that minus 1 to an even power is plus 1."},{"Start":"01:31.910 ","End":"01:33.305","Text":"So we\u0027ll start with plus,"},{"Start":"01:33.305 ","End":"01:35.870","Text":"and minus 1 to an odd power will be minus."},{"Start":"01:35.870 ","End":"01:37.765","Text":"It will alternate."},{"Start":"01:37.765 ","End":"01:39.705","Text":"This is what we get."},{"Start":"01:39.705 ","End":"01:41.540","Text":"Of course the plus is unnecessary,"},{"Start":"01:41.540 ","End":"01:43.550","Text":"but it\u0027s just for emphasis that we have plus,"},{"Start":"01:43.550 ","End":"01:46.805","Text":"minus, plus, minus, plus."},{"Start":"01:46.805 ","End":"01:55.920","Text":"In this example, we\u0027re given a sequence c_n n bigger or equal to 1,"},{"Start":"01:55.920 ","End":"01:58.410","Text":"meaning n goes 1, 2, 3, 4, 5,"},{"Start":"01:58.410 ","End":"02:02.955","Text":"and so on and we\u0027re given c_n descriptively."},{"Start":"02:02.955 ","End":"02:05.760","Text":"We\u0027re told that c_n is the nth prime number."},{"Start":"02:05.760 ","End":"02:07.830","Text":"It\u0027s not a formula, but it\u0027s"},{"Start":"02:07.830 ","End":"02:11.180","Text":"definitely well defined because we know what the prime numbers are,"},{"Start":"02:11.180 ","End":"02:13.380","Text":"at least the first few."},{"Start":"02:13.570 ","End":"02:16.885","Text":"Assuming you know your primes,"},{"Start":"02:16.885 ","End":"02:20.265","Text":"c_1 is the first prime number is 2."},{"Start":"02:20.265 ","End":"02:22.640","Text":"I know some people think that 1 is a prime,"},{"Start":"02:22.640 ","End":"02:24.650","Text":"but it\u0027s not considered to be."},{"Start":"02:24.650 ","End":"02:26.360","Text":"Next prime number is 3,"},{"Start":"02:26.360 ","End":"02:28.825","Text":"then 5, then 7, then 11."},{"Start":"02:28.825 ","End":"02:31.490","Text":"If they ask for the 6th member,"},{"Start":"02:31.490 ","End":"02:35.180","Text":"for example, I could say c_6 is 13,"},{"Start":"02:35.180 ","End":"02:37.630","Text":"c_7 is the 7th prime,"},{"Start":"02:37.630 ","End":"02:42.975","Text":"it is 17, and so on but we\u0027re just asked for 5 so that\u0027ll do."},{"Start":"02:42.975 ","End":"02:47.420","Text":"This exercise is a little bit different to the previous ones."},{"Start":"02:47.420 ","End":"02:51.080","Text":"We also have a sequence and we want to write the first 5 terms."},{"Start":"02:51.080 ","End":"02:55.250","Text":"But this time, the sequence is defined recursively."},{"Start":"02:55.250 ","End":"02:57.725","Text":"I\u0027ll just highlight that term."},{"Start":"02:57.725 ","End":"03:02.840","Text":"What it means is that we don\u0027t have an explicit formula for"},{"Start":"03:02.840 ","End":"03:07.860","Text":"each n. What we do have is 1 explicit 1,"},{"Start":"03:07.860 ","End":"03:11.400","Text":"usually the 1st 1 is given explicitly a_1."},{"Start":"03:11.400 ","End":"03:17.120","Text":"But subsequent a_n\u0027s are given in terms of previous a_n\u0027s,"},{"Start":"03:17.120 ","End":"03:23.325","Text":"so we have to build up the sequence 1 by 1 and you\u0027ll see in a moment."},{"Start":"03:23.325 ","End":"03:26.220","Text":"Okay. As I said, the 1st 1 is just given to us,"},{"Start":"03:26.220 ","End":"03:27.960","Text":"so we just copy that."},{"Start":"03:27.960 ","End":"03:30.460","Text":"Now, for the next 1,"},{"Start":"03:30.460 ","End":"03:32.910","Text":"how do we find a_2?"},{"Start":"03:32.910 ","End":"03:39.525","Text":"We can get a_2 if we put n equals 1 in this formula, and we get."},{"Start":"03:39.525 ","End":"03:43.370","Text":"Notice that everywhere there\u0027s n I\u0027m putting 1 and it\u0027s colored in blue."},{"Start":"03:43.370 ","End":"03:45.785","Text":"a_1 plus 1, which is a_2,"},{"Start":"03:45.785 ","End":"03:51.450","Text":"will be given from a_1 plus 2 times 1."},{"Start":"03:51.450 ","End":"03:55.780","Text":"Now, we know a_1 because it\u0027s written here."},{"Start":"03:55.940 ","End":"04:00.780","Text":"So a_1 is 1 and 2 times 1 is 2,"},{"Start":"04:00.780 ","End":"04:03.825","Text":"so we have 1 plus 2 is 3."},{"Start":"04:03.825 ","End":"04:06.790","Text":"So a_2 is 3."},{"Start":"04:06.790 ","End":"04:08.360","Text":"But we didn\u0027t find it directly."},{"Start":"04:08.360 ","End":"04:12.739","Text":"We found it in terms of the previous n. In fact, this,"},{"Start":"04:12.739 ","End":"04:15.540","Text":"and I\u0027ll highlight it,"},{"Start":"04:16.090 ","End":"04:20.795","Text":"is a special case of a recursive formula."},{"Start":"04:20.795 ","End":"04:25.220","Text":"Sometimes we say recursive formula,"},{"Start":"04:25.220 ","End":"04:29.165","Text":"but sometimes we say recursive rule."},{"Start":"04:29.165 ","End":"04:30.800","Text":"Anyway, don\u0027t worry about the name,"},{"Start":"04:30.800 ","End":"04:33.665","Text":"that\u0027s just learn how to use such a thing."},{"Start":"04:33.665 ","End":"04:35.150","Text":"Let\u0027s go on to the next 1."},{"Start":"04:35.150 ","End":"04:37.415","Text":"If I want a_3,"},{"Start":"04:37.415 ","End":"04:43.025","Text":"what I have to do is let n equals 2 because then 2 plus 1 is 3."},{"Start":"04:43.025 ","End":"04:47.059","Text":"So we get, if n is 2 in the recursive formula,"},{"Start":"04:47.059 ","End":"04:52.595","Text":"a_2 plus 1 equals a_2 plus twice 2."},{"Start":"04:52.595 ","End":"04:56.075","Text":"Now a_2 we found already is 3,"},{"Start":"04:56.075 ","End":"04:57.350","Text":"so we put that here,"},{"Start":"04:57.350 ","End":"04:59.300","Text":"and twice 2 is 4 is 7."},{"Start":"04:59.300 ","End":"05:03.740","Text":"We got a_3 from previous value a_2."},{"Start":"05:03.740 ","End":"05:07.190","Text":"We still have to do 2 more of these to get a_4 and a_5."},{"Start":"05:07.190 ","End":"05:09.590","Text":"To get a_4, I think you get the idea,"},{"Start":"05:09.590 ","End":"05:11.965","Text":"we let n equals 3."},{"Start":"05:11.965 ","End":"05:14.490","Text":"So n equals 3,"},{"Start":"05:14.490 ","End":"05:19.095","Text":"a_3 plus 1 is a_3 plus twice 3."},{"Start":"05:19.095 ","End":"05:22.050","Text":"We have a_3 because it\u0027s here, it\u0027s 7."},{"Start":"05:22.050 ","End":"05:30.280","Text":"I should have really been indicating that I get the 7 from here like a_2 was 3."},{"Start":"05:30.280 ","End":"05:37.890","Text":"That was what gave me this and the 1st 1 is what gave me this here."},{"Start":"05:37.890 ","End":"05:43.200","Text":"Okay. So a_4 is 13,"},{"Start":"05:43.200 ","End":"05:47.410","Text":"here is 7 plus 6, and we have 1 more."},{"Start":"05:47.810 ","End":"05:50.910","Text":"This time we put n equals 4."},{"Start":"05:50.910 ","End":"05:52.605","Text":"If we want to get a_5,"},{"Start":"05:52.605 ","End":"05:56.175","Text":"so a_4 plus 1 is a_n, which is a_4,"},{"Start":"05:56.175 ","End":"06:01.500","Text":"plus twice 4 and a_ 4 is 13,"},{"Start":"06:01.500 ","End":"06:03.240","Text":"which I got from here."},{"Start":"06:03.240 ","End":"06:07.050","Text":"Twice 4 is 8, 13 and 8 is 21."},{"Start":"06:07.050 ","End":"06:12.120","Text":"Now we have all the first 5 elements,"},{"Start":"06:12.120 ","End":"06:16.485","Text":"terms, members of the sequence."},{"Start":"06:16.485 ","End":"06:20.020","Text":"That\u0027s it for this exercise."}],"ID":27203},{"Watched":false,"Name":"Defining a Sequence as a Function","Duration":"3m 21s","ChapterTopicVideoID":26297,"CourseChapterTopicPlaylistID":254619,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26297.jpeg","UploadDate":"2021-09-12T14:04:37.9070000","DurationForVideoObject":"PT3M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.750","Text":"Up to now, we\u0027ve been using a sequence as an ordered list of numbers."},{"Start":"00:07.750 ","End":"00:11.485","Text":"There is another mathematical way of defining a sequence."},{"Start":"00:11.485 ","End":"00:16.010","Text":"A sequence can be considered as a function of sorts."},{"Start":"00:16.310 ","End":"00:19.030","Text":"We\u0027ll illustrate it on an example."},{"Start":"00:19.030 ","End":"00:27.120","Text":"Let\u0027s suppose we have this sequence where the nth term a_n is n plus 4 over n squared."},{"Start":"00:27.120 ","End":"00:31.989","Text":"We can actually view it as a function from the natural numbers."},{"Start":"00:31.989 ","End":"00:33.820","Text":"N is a natural number."},{"Start":"00:33.820 ","End":"00:35.760","Text":"After I\u0027ve made the computation,"},{"Start":"00:35.760 ","End":"00:38.280","Text":"I get a real number, so I have a function."},{"Start":"00:38.280 ","End":"00:40.495","Text":"The domain is natural numbers,"},{"Start":"00:40.495 ","End":"00:42.430","Text":"the range is real numbers,"},{"Start":"00:42.430 ","End":"00:46.985","Text":"and we can write it as f of n equals n plus 4 over n squared."},{"Start":"00:46.985 ","End":"00:54.450","Text":"We define instead of a_n as the nth member of the sequence we, in this notation,"},{"Start":"00:54.450 ","End":"00:57.225","Text":"would say f of n."},{"Start":"00:57.225 ","End":"01:03.065","Text":"This interpretation of a sequence as a function will be useful later on."},{"Start":"01:03.065 ","End":"01:07.460","Text":"For example, it will help us to determine if a sequence is"},{"Start":"01:07.460 ","End":"01:12.965","Text":"increasing or decreasing based on the derivative of this function."},{"Start":"01:12.965 ","End":"01:17.059","Text":"Just point out that it\u0027s not necessarily the natural numbers"},{"Start":"01:17.059 ","End":"01:22.445","Text":"because this notation doesn\u0027t tell us where a runs from."},{"Start":"01:22.445 ","End":"01:24.365","Text":"If I said a_n."},{"Start":"01:24.365 ","End":"01:27.955","Text":"N goes from 1 to infinity,"},{"Start":"01:27.955 ","End":"01:30.200","Text":"then it would be the natural numbers."},{"Start":"01:30.200 ","End":"01:31.579","Text":"But as I mentioned,"},{"Start":"01:31.579 ","End":"01:34.850","Text":"the series doesn\u0027t necessarily start at 1 even if it\u0027s infinite,"},{"Start":"01:34.850 ","End":"01:36.469","Text":"it could start at 0,"},{"Start":"01:36.469 ","End":"01:40.560","Text":"or at 3, or something else in which case we just modify it"},{"Start":"01:40.560 ","End":"01:42.720","Text":"slightly and say not the natural numbers but"},{"Start":"01:42.720 ","End":"01:45.390","Text":"the natural numbers from 3 onwards or something."},{"Start":"01:45.390 ","End":"01:46.710","Text":"I don\u0027t want to get too technical,"},{"Start":"01:46.710 ","End":"01:49.620","Text":"we\u0027ll just stick to the natural numbers to the real numbers,"},{"Start":"01:49.620 ","End":"01:56.090","Text":"and I want to move on and talk about the sketch or graph of such a function."},{"Start":"01:56.090 ","End":"01:58.585","Text":"I\u0027ll just show it to you right away."},{"Start":"01:58.585 ","End":"02:01.940","Text":"We see it\u0027s just a set of isolated points."},{"Start":"02:01.940 ","End":"02:04.385","Text":"That\u0027s what the graph of a sequence looks like."},{"Start":"02:04.385 ","End":"02:10.710","Text":"We only have values for n which is a whole number 1, 2, 3, 4."},{"Start":"02:10.710 ","End":"02:12.410","Text":"We don\u0027t have anything in between,"},{"Start":"02:12.410 ","End":"02:16.600","Text":"so we don\u0027t join the points with a curve."},{"Start":"02:16.600 ","End":"02:19.200","Text":"The way we get it, it\u0027s just regular."},{"Start":"02:19.200 ","End":"02:21.180","Text":"We plug in values."},{"Start":"02:21.180 ","End":"02:23.170","Text":"Let\u0027s say when n is 1,"},{"Start":"02:23.170 ","End":"02:27.455","Text":"then we\u0027d get 1 plus 4 over 1 squared is 5."},{"Start":"02:27.455 ","End":"02:30.080","Text":"I\u0027ll just write the y value is 5."},{"Start":"02:30.080 ","End":"02:31.985","Text":"Of course, the point is 1, 5."},{"Start":"02:31.985 ","End":"02:36.155","Text":"I want to plug in 2. 2 plus 6 over 4 that\u0027s 1/2."},{"Start":"02:36.155 ","End":"02:39.950","Text":"Plug in 3 I would get 7/9,"},{"Start":"02:39.950 ","End":"02:43.020","Text":"and so on. That\u0027s what it looks like."},{"Start":"02:43.020 ","End":"02:49.690","Text":"However, if you did join it with a curve that would give you something else."},{"Start":"02:49.690 ","End":"02:53.165","Text":"That would give you a function from the real numbers to the real numbers."},{"Start":"02:53.165 ","End":"02:56.940","Text":"f of x is x plus 4 over x squared,"},{"Start":"02:56.940 ","End":"02:59.705","Text":"and that\u0027s something totally different."},{"Start":"02:59.705 ","End":"03:05.930","Text":"But they are related and we will be using the properties of the derivative,"},{"Start":"03:05.930 ","End":"03:12.965","Text":"for example, to show that a sequence is decreasing if the function is decreasing."},{"Start":"03:12.965 ","End":"03:16.340","Text":"Anyway, you\u0027ll see more about that later."},{"Start":"03:16.340 ","End":"03:19.250","Text":"That\u0027s assuming you\u0027ve studied derivatives."},{"Start":"03:19.250 ","End":"03:22.320","Text":"I\u0027m just going to end the clip here."}],"ID":27204},{"Watched":false,"Name":"Limit of a Sequence","Duration":"3m 5s","ChapterTopicVideoID":26298,"CourseChapterTopicPlaylistID":254619,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26298.jpeg","UploadDate":"2021-09-12T14:05:27.1900000","DurationForVideoObject":"PT3M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.440","Text":"In this clip, we\u0027ll be talking about the limit of a sequence and it\u0027s"},{"Start":"00:04.440 ","End":"00:08.430","Text":"got some similarity with limits of functions, as we shall see."},{"Start":"00:08.430 ","End":"00:10.665","Text":"Anyway, we\u0027ll start with an example."},{"Start":"00:10.665 ","End":"00:16.440","Text":"The familiar 1 over n sequence in the nth place,"},{"Start":"00:16.440 ","End":"00:18.300","Text":"we\u0027ll write a few terms,"},{"Start":"00:18.300 ","End":"00:21.330","Text":"starts off 1, 1/2, 1/3 and so on,"},{"Start":"00:21.330 ","End":"00:23.370","Text":"and very quickly we get to small numbers,"},{"Start":"00:23.370 ","End":"00:25.950","Text":"1 over 400 ,1 over 10,000,"},{"Start":"00:25.950 ","End":"00:28.215","Text":"1 over a billion."},{"Start":"00:28.215 ","End":"00:30.180","Text":"If we keep writing them,"},{"Start":"00:30.180 ","End":"00:31.230","Text":"keep adding more terms."},{"Start":"00:31.230 ","End":"00:38.340","Text":"I think it\u0027s intuitively obvious anyway that they approach 0,"},{"Start":"00:38.340 ","End":"00:41.230","Text":"get closer and closer to 0."},{"Start":"00:42.110 ","End":"00:46.730","Text":"We say that the limit of this sequence is 0,"},{"Start":"00:46.730 ","End":"00:51.725","Text":"and we use a notation similar to functions."},{"Start":"00:51.725 ","End":"00:57.880","Text":"We say that the limit as n goes to infinity of 1 over n is 0,"},{"Start":"00:57.880 ","End":"01:01.880","Text":"and there\u0027s an alternative notation."},{"Start":"01:01.880 ","End":"01:06.050","Text":"Sorry, this arrow doesn\u0027t have to be quite so long."},{"Start":"01:06.050 ","End":"01:11.555","Text":"1 over n tends to 0 as n tends to infinity."},{"Start":"01:11.555 ","End":"01:14.940","Text":"This notation, you don\u0027t need the word lim."},{"Start":"01:15.330 ","End":"01:19.940","Text":"I want to introduce another term, converges."},{"Start":"01:19.940 ","End":"01:23.570","Text":"We say that the sequence converges to 0."},{"Start":"01:23.570 ","End":"01:26.540","Text":"If it has a limit of 0, then it converges to 0."},{"Start":"01:26.540 ","End":"01:28.520","Text":"Nothing special about 0."},{"Start":"01:28.520 ","End":"01:30.200","Text":"We could say if the limit was 3,"},{"Start":"01:30.200 ","End":"01:32.030","Text":"then it would converge to 3."},{"Start":"01:32.030 ","End":"01:34.850","Text":"But if we don\u0027t care what it converges to,"},{"Start":"01:34.850 ","End":"01:38.705","Text":"we just say that the sequence is convergent."},{"Start":"01:38.705 ","End":"01:41.040","Text":"It has a limit."},{"Start":"01:41.160 ","End":"01:44.710","Text":"Now, not every sequence has a limit,"},{"Start":"01:44.710 ","End":"01:48.920","Text":"this 1 did, but there are examples which don\u0027t."},{"Start":"01:48.920 ","End":"01:52.450","Text":"For example, look at the following sequence,"},{"Start":"01:52.450 ","End":"01:54.430","Text":"the 1 that alternates 1 minus 1,"},{"Start":"01:54.430 ","End":"01:56.965","Text":"1 minus 1, and so on."},{"Start":"01:56.965 ","End":"02:00.835","Text":"I claim that it does not have a limit,"},{"Start":"02:00.835 ","End":"02:02.920","Text":"and why it doesn\u0027t it have a limit, although,"},{"Start":"02:02.920 ","End":"02:05.875","Text":"I think it\u0027s fairly intuitive?"},{"Start":"02:05.875 ","End":"02:09.640","Text":"Well, it keeps alternating between 1 and minus 1,"},{"Start":"02:09.640 ","End":"02:13.420","Text":"so it doesn\u0027t approach any 1 single number."},{"Start":"02:13.420 ","End":"02:21.070","Text":"It approaches 2 different numbers so it doesn\u0027t have a limit."},{"Start":"02:21.070 ","End":"02:23.980","Text":"If it doesn\u0027t have a limit, It\u0027s not convergent."},{"Start":"02:23.980 ","End":"02:26.360","Text":"There\u0027s a name for not convergent."},{"Start":"02:26.360 ","End":"02:29.600","Text":"Instead of that, we say divergent."},{"Start":"02:29.600 ","End":"02:32.240","Text":"If it\u0027s not convergent, it\u0027s divergent,"},{"Start":"02:32.240 ","End":"02:36.170","Text":"I\u0027d like to mention that there are a whole bunch of techniques you"},{"Start":"02:36.170 ","End":"02:40.160","Text":"learned for computing the limit of a function at infinity."},{"Start":"02:40.160 ","End":"02:41.600","Text":"In the case of functions,"},{"Start":"02:41.600 ","End":"02:44.560","Text":"you would have x goes to infinity."},{"Start":"02:44.560 ","End":"02:48.170","Text":"Pretty much with only minor modifications,"},{"Start":"02:48.170 ","End":"02:51.965","Text":"all that theory applies for limit of sequences,"},{"Start":"02:51.965 ","End":"02:54.040","Text":"so we\u0027re not going to repeat it."},{"Start":"02:54.040 ","End":"02:59.929","Text":"But basically, limits of sequences are like limits of functions when x goes to infinity."},{"Start":"02:59.929 ","End":"03:05.520","Text":"That\u0027s all I want to say about limits of sequences of this stage, so we\u0027re done."}],"ID":27205},{"Watched":false,"Name":"Increasing and Decreasing Sequences","Duration":"7m 40s","ChapterTopicVideoID":26299,"CourseChapterTopicPlaylistID":254619,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26299.jpeg","UploadDate":"2021-09-12T14:07:20.5700000","DurationForVideoObject":"PT7M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.035","Text":"Our next topic as far as sequences go,"},{"Start":"00:04.035 ","End":"00:07.455","Text":"is increasing and decreasing sequences."},{"Start":"00:07.455 ","End":"00:11.100","Text":"These occur quite often and they\u0027re particularly"},{"Start":"00:11.100 ","End":"00:14.490","Text":"useful in theorems on limits of sequences."},{"Start":"00:14.490 ","End":"00:18.210","Text":"Many theorems require a sequence to be increasing or decreasing,"},{"Start":"00:18.210 ","End":"00:20.265","Text":"then we can deduce a lot from it."},{"Start":"00:20.265 ","End":"00:22.560","Text":"Let\u0027s give some examples."},{"Start":"00:22.560 ","End":"00:28.845","Text":"Here are 3 different sequences to consider."},{"Start":"00:28.845 ","End":"00:32.010","Text":"In the first sequence, 2, 4, 6,"},{"Start":"00:32.010 ","End":"00:34.925","Text":"8 and assuming this pattern continues,"},{"Start":"00:34.925 ","End":"00:43.780","Text":"each term in the sequence is bigger than the previous 1 and so in this case,"},{"Start":"00:43.780 ","End":"00:50.320","Text":"we say that the sequence is increasing and in the second 1,"},{"Start":"00:50.320 ","End":"00:51.350","Text":"it\u0027s going to be the opposite."},{"Start":"00:51.350 ","End":"00:59.345","Text":"If you look at it, then each term is smaller than the previous 1 and so it\u0027s decreasing."},{"Start":"00:59.345 ","End":"01:00.950","Text":"But in the third case,"},{"Start":"01:00.950 ","End":"01:03.170","Text":"it\u0027s going down then up."},{"Start":"01:03.170 ","End":"01:06.950","Text":"It\u0027s neither increasing nor decreasing."},{"Start":"01:06.950 ","End":"01:12.825","Text":"Now let\u0027s see if we can write something a little more formal than that."},{"Start":"01:12.825 ","End":"01:16.775","Text":"Suppose we have sequence a_n,"},{"Start":"01:16.775 ","End":"01:21.094","Text":"it\u0027s called increasing if each term"},{"Start":"01:21.094 ","End":"01:26.435","Text":"is less than or equal to the following term or if you look at the other way around,"},{"Start":"01:26.435 ","End":"01:30.860","Text":"the following term is bigger or equal to the current term."},{"Start":"01:30.860 ","End":"01:33.055","Text":"If this is true for all n,"},{"Start":"01:33.055 ","End":"01:36.145","Text":"then it\u0027s increasing and"},{"Start":"01:36.145 ","End":"01:41.060","Text":"decreasing means that each term is bigger than the following term."},{"Start":"01:41.060 ","End":"01:47.510","Text":"Or I\u0027d rather say the following term is always less than or equal to the current term."},{"Start":"01:47.510 ","End":"01:50.460","Text":"If it\u0027s increasing, if n is 1,"},{"Start":"01:50.460 ","End":"01:55.350","Text":"we have here a_1 and less than or equal to a_2."},{"Start":"01:55.350 ","End":"01:56.790","Text":"If put n equals 2,"},{"Start":"01:56.790 ","End":"02:04.425","Text":"we get that a_2 is less than or equal to a_3, and so on."},{"Start":"02:04.425 ","End":"02:07.700","Text":"Here the decreasing, if it at n equal 1,"},{"Start":"02:07.700 ","End":"02:10.625","Text":"we get that a_1 bigger or equal to a_2,"},{"Start":"02:10.625 ","End":"02:18.300","Text":"put an equals 2 you get a_2 bigger or equal to a_3, and so on."},{"Start":"02:18.650 ","End":"02:24.140","Text":"Note that in each of these inequalities are used less than or equal,"},{"Start":"02:24.140 ","End":"02:26.030","Text":"greater than or equal."},{"Start":"02:26.030 ","End":"02:30.080","Text":"We\u0027re going to say that a sequence is"},{"Start":"02:30.080 ","End":"02:35.340","Text":"strictly increasing if we have the strict inequality here."},{"Start":"02:35.340 ","End":"02:40.924","Text":"Similarly, if it\u0027s a strict inequality that each successive term gets smaller,"},{"Start":"02:40.924 ","End":"02:43.070","Text":"then we say strictly decreasing."},{"Start":"02:43.070 ","End":"02:48.499","Text":"That\u0027s the word strictly if the inequalities are strict."},{"Start":"02:48.499 ","End":"02:53.960","Text":"If a sequence is either increasing or decreasing,"},{"Start":"02:53.960 ","End":"02:55.865","Text":"there\u0027s a common name for that."},{"Start":"02:55.865 ","End":"02:59.170","Text":"It\u0027s called a monotonic sequence."},{"Start":"02:59.170 ","End":"03:02.000","Text":"I think you can even say strictly"},{"Start":"03:02.000 ","End":"03:05.675","Text":"monotonic if it\u0027s strictly increasing or strictly decreasing,"},{"Start":"03:05.675 ","End":"03:08.510","Text":"but this is getting too subtle."},{"Start":"03:08.510 ","End":"03:13.040","Text":"Now I want to know if you\u0027ve learned differentiation already,"},{"Start":"03:13.040 ","End":"03:17.870","Text":"then there are quite a few tools to decide if"},{"Start":"03:17.870 ","End":"03:20.855","Text":"a sequence is increasing or decreasing"},{"Start":"03:20.855 ","End":"03:24.395","Text":"using the derivative or what is called differentiation,"},{"Start":"03:24.395 ","End":"03:27.580","Text":"this will turn up in the following example."},{"Start":"03:27.580 ","End":"03:30.695","Text":"Here\u0027s our example exercise."},{"Start":"03:30.695 ","End":"03:32.240","Text":"We have to determine if"},{"Start":"03:32.240 ","End":"03:37.955","Text":"the following sequence is increasing or decreasing or it could be neither."},{"Start":"03:37.955 ","End":"03:40.885","Text":"Let\u0027s write a few terms and see what\u0027s going on."},{"Start":"03:40.885 ","End":"03:44.300","Text":"The numerator is always 1 more than the denominator,"},{"Start":"03:44.300 ","End":"03:48.025","Text":"2/1, 3/2, 4/3, 5/4,"},{"Start":"03:48.025 ","End":"03:50.300","Text":"looks like it\u0027s decreasing."},{"Start":"03:50.300 ","End":"03:53.465","Text":"We\u0027ll prove it in 2 ways."},{"Start":"03:53.465 ","End":"03:58.740","Text":"Let\u0027s first of all do the first way which is straight from the definition."},{"Start":"04:01.820 ","End":"04:07.445","Text":"Actually I didn\u0027t say strictly so we can have bigger or equal to."},{"Start":"04:07.445 ","End":"04:10.490","Text":"This is actually going to be strictly decreasing."},{"Start":"04:10.490 ","End":"04:14.965","Text":"Anyway, don\u0027t worry about the greater than or greater than or equal to."},{"Start":"04:14.965 ","End":"04:20.239","Text":"This is what we have to prove and then just rewrite it in terms of the definition."},{"Start":"04:20.239 ","End":"04:22.805","Text":"A_n is this, I just copied it,"},{"Start":"04:22.805 ","End":"04:27.155","Text":"A_n plus 1 means replace n by n plus 1."},{"Start":"04:27.155 ","End":"04:29.580","Text":"We have n plus 1 plus 1,"},{"Start":"04:29.580 ","End":"04:32.490","Text":"which is I\u0027m plus 2 and here m plus 1."},{"Start":"04:32.490 ","End":"04:34.215","Text":"If we can show this,"},{"Start":"04:34.215 ","End":"04:36.315","Text":"then this is true."},{"Start":"04:36.315 ","End":"04:41.160","Text":"All these quantities are positive, especially the denominators."},{"Start":"04:41.160 ","End":"04:45.935","Text":"We can say that this is bigger than this if and only if"},{"Start":"04:45.935 ","End":"04:50.960","Text":"the cross multiplication this times this is bigger than this times this,"},{"Start":"04:50.960 ","End":"04:57.190","Text":"it\u0027s equivalent and let\u0027s see if we can show that."},{"Start":"04:57.190 ","End":"05:02.450","Text":"Expanding this is equivalent to showing that this is bigger than this."},{"Start":"05:02.450 ","End":"05:07.475","Text":"Subtract m squared plus 2n from both sides and get 1 is bigger than 0."},{"Start":"05:07.475 ","End":"05:10.945","Text":"Anyway, it\u0027s obvious and that concludes the proof."},{"Start":"05:10.945 ","End":"05:12.230","Text":"Because 1 is bigger than 0,"},{"Start":"05:12.230 ","End":"05:14.600","Text":"this is bigger than this, so this is bigger than this."},{"Start":"05:14.600 ","End":"05:19.925","Text":"We trace it backwards we\u0027ve got a_n is actually strictly bigger than a_n plus 1."},{"Start":"05:19.925 ","End":"05:24.875","Text":"We\u0027ve even proved that the sequence is strictly decreasing."},{"Start":"05:24.875 ","End":"05:27.995","Text":"Now let\u0027s do the solution another way,"},{"Start":"05:27.995 ","End":"05:31.760","Text":"but only if you\u0027ve learned derivatives and differentiation."},{"Start":"05:31.760 ","End":"05:37.250","Text":"If you haven\u0027t learned derivatives yet then skip the rest of this clip."},{"Start":"05:37.250 ","End":"05:47.510","Text":"I just want to remind you which is scrolled off that a_n was n plus 1."},{"Start":"05:47.510 ","End":"05:52.205","Text":"Now remember we talked about sequences of functions."},{"Start":"05:52.205 ","End":"05:57.200","Text":"This n plus 1 is a function of n. What we\u0027re going to do"},{"Start":"05:57.200 ","End":"06:02.820","Text":"is extend it to a function of real numbers."},{"Start":"06:02.820 ","End":"06:07.335","Text":"All x\u0027s or at least x bigger or equal to 1,"},{"Start":"06:07.335 ","End":"06:11.270","Text":"we\u0027ll define this function and if"},{"Start":"06:11.270 ","End":"06:15.515","Text":"x is bigger or equal to 1 and we won\u0027t have any problem with the denominator either,"},{"Start":"06:15.515 ","End":"06:22.390","Text":"and then a_n is just f of n. Now we\u0027re going to use derivatives to"},{"Start":"06:22.390 ","End":"06:29.210","Text":"show that f is decreasing as a function of x for x bigger or equal to 1."},{"Start":"06:29.210 ","End":"06:31.045","Text":"To show it\u0027s decreasing,"},{"Start":"06:31.045 ","End":"06:35.705","Text":"we show that the derivative is negative."},{"Start":"06:35.705 ","End":"06:39.745","Text":"Here\u0027s a computation of the derivative."},{"Start":"06:39.745 ","End":"06:42.805","Text":"Hope you remember the quotient rule."},{"Start":"06:42.805 ","End":"06:44.859","Text":"This is a quotient,"},{"Start":"06:44.859 ","End":"06:49.675","Text":"so we take the derivative of the numerator times denominator"},{"Start":"06:49.675 ","End":"06:56.935","Text":"minus the derivative of the denominator times the numerator over the denominator squared."},{"Start":"06:56.935 ","End":"07:00.235","Text":"Anyway, it comes out to be minus 1/x squared."},{"Start":"07:00.235 ","End":"07:02.430","Text":"This thing is positive."},{"Start":"07:02.430 ","End":"07:04.850","Text":"Minus 1 is negative,"},{"Start":"07:04.850 ","End":"07:06.410","Text":"negative and positive is negative."},{"Start":"07:06.410 ","End":"07:08.330","Text":"It\u0027s always negative."},{"Start":"07:08.330 ","End":"07:13.125","Text":"So the derivative of the function is negative."},{"Start":"07:13.125 ","End":"07:15.800","Text":"The function of x is decreasing."},{"Start":"07:15.800 ","End":"07:20.600","Text":"If f is decreasing for all x bigger or equal to 1,"},{"Start":"07:20.600 ","End":"07:22.955","Text":"it\u0027s also decreasing if I just take"},{"Start":"07:22.955 ","End":"07:29.360","Text":"whole numbers and the sequence is decreasing for n equals 1,"},{"Start":"07:29.360 ","End":"07:31.710","Text":"2, 3, and so on."},{"Start":"07:33.040 ","End":"07:41.290","Text":"That\u0027s that for increasing and decreasing and that concludes this clip."}],"ID":27206},{"Watched":false,"Name":"Bounded Sequences","Duration":"15m 38s","ChapterTopicVideoID":26300,"CourseChapterTopicPlaylistID":254619,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26300.jpeg","UploadDate":"2021-09-12T14:11:33.9300000","DurationForVideoObject":"PT15M38S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.720","Text":"In this clip, we\u0027ll talk about bounded sequences."},{"Start":"00:03.720 ","End":"00:06.050","Text":"We\u0027ve already learned increasing sequences,"},{"Start":"00:06.050 ","End":"00:09.090","Text":"decreasing sequences, limits of the sequence."},{"Start":"00:09.090 ","End":"00:12.195","Text":"Now it\u0027s time for bounded sequences."},{"Start":"00:12.195 ","End":"00:16.440","Text":"First I need to talk about upper and lower bounds."},{"Start":"00:16.440 ","End":"00:18.480","Text":"To help me illustrate this concept,"},{"Start":"00:18.480 ","End":"00:20.670","Text":"let\u0027s consider the following 2 sequences."},{"Start":"00:20.670 ","End":"00:21.990","Text":"This one, 2, 4, 6,"},{"Start":"00:21.990 ","End":"00:23.940","Text":"8, the even numbers,"},{"Start":"00:23.940 ","End":"00:25.560","Text":"and here 1, 1/2,"},{"Start":"00:25.560 ","End":"00:28.600","Text":"1/3, 1/4, we\u0027ve seen this before."},{"Start":"00:29.870 ","End":"00:33.420","Text":"We\u0027ll consider the first sequence."},{"Start":"00:33.420 ","End":"00:37.535","Text":"First, I\u0027m going to use it to illustrate the concept of a lower bound."},{"Start":"00:37.535 ","End":"00:40.460","Text":"Notice that all the elements,"},{"Start":"00:40.460 ","End":"00:42.320","Text":"no matter how far I go,"},{"Start":"00:42.320 ","End":"00:44.840","Text":"they\u0027re all bigger or equal to 2,"},{"Start":"00:44.840 ","End":"00:47.010","Text":"there are at least 2."},{"Start":"00:47.260 ","End":"00:49.340","Text":"This being the case,"},{"Start":"00:49.340 ","End":"00:56.850","Text":"we say that the sequence is bounded below, that\u0027s general."},{"Start":"00:58.640 ","End":"01:03.980","Text":"Specifically, 2 is a lower bound of the sequence."},{"Start":"01:03.980 ","End":"01:10.355","Text":"Because 2 is less than or equal to all the elements of the sequence, all the terms."},{"Start":"01:10.355 ","End":"01:13.130","Text":"Now, lower bounds are not unique."},{"Start":"01:13.130 ","End":"01:15.775","Text":"If I have a lower bound in this case 2"},{"Start":"01:15.775 ","End":"01:18.800","Text":"anything smaller than 2 will also be a lower bound."},{"Start":"01:18.800 ","End":"01:21.740","Text":"For example, 1 is also a lower bound because it\u0027s"},{"Start":"01:21.740 ","End":"01:25.340","Text":"also less than or equal to all the elements in the sequence."},{"Start":"01:25.340 ","End":"01:27.595","Text":"That\u0027s just something to note."},{"Start":"01:27.595 ","End":"01:36.080","Text":"I\u0027m going to use the second sequence to illustrate the concept of an upper bound."},{"Start":"01:36.080 ","End":"01:40.895","Text":"Notice that in the second sequence,"},{"Start":"01:40.895 ","End":"01:44.270","Text":"all the elements are less than or equal to 1,"},{"Start":"01:44.270 ","End":"01:47.880","Text":"they\u0027re all at most 1."},{"Start":"01:47.990 ","End":"01:52.235","Text":"In this case, we say the sequence is bounded from above"},{"Start":"01:52.235 ","End":"01:56.590","Text":"and 1 is an upper bound of the sequence."},{"Start":"01:56.590 ","End":"02:00.380","Text":"Here too, we don\u0027t have uniqueness because anything"},{"Start":"02:00.380 ","End":"02:04.400","Text":"that\u0027s bigger than an upper bound is also going to be an upper bound."},{"Start":"02:04.400 ","End":"02:06.620","Text":"7 is bigger than 1."},{"Start":"02:06.620 ","End":"02:08.180","Text":"It\u0027s also going to be an upper bound."},{"Start":"02:08.180 ","End":"02:12.840","Text":"7 is bigger or equal to all the elements of the sequence."},{"Start":"02:13.120 ","End":"02:20.630","Text":"Now it just so happens that this second sequence also has a lower bound."},{"Start":"02:20.630 ","End":"02:22.190","Text":"It\u0027s bounded from below."},{"Start":"02:22.190 ","End":"02:24.605","Text":"Because if I take 0,"},{"Start":"02:24.605 ","End":"02:32.390","Text":"all these terms are positive and they\u0027re all bigger than or equal to 0."},{"Start":"02:32.390 ","End":"02:35.590","Text":"It also has a lower bound."},{"Start":"02:35.590 ","End":"02:40.450","Text":"This has an upper bound and a lower bound."},{"Start":"02:40.450 ","End":"02:44.405","Text":"Because this sequence has both an upper and a lower bound,"},{"Start":"02:44.405 ","End":"02:45.620","Text":"we say it\u0027s bounded."},{"Start":"02:45.620 ","End":"02:49.655","Text":"In general, the sequence is bounded above and below,"},{"Start":"02:49.655 ","End":"02:51.650","Text":"then we just say it\u0027s bounded."},{"Start":"02:51.650 ","End":"02:53.930","Text":"Here we had an example that was bounded,"},{"Start":"02:53.930 ","End":"02:56.420","Text":"but the previous one, what was it?"},{"Start":"02:56.420 ","End":"03:01.415","Text":"It was 2, 4, 6, 8, etc."},{"Start":"03:01.415 ","End":"03:04.370","Text":"This one isn\u0027t bounded."},{"Start":"03:04.370 ","End":"03:07.685","Text":"We showed that it\u0027s bounded from below,"},{"Start":"03:07.685 ","End":"03:10.160","Text":"but it has no bound from above,"},{"Start":"03:10.160 ","End":"03:11.944","Text":"and no upper bound."},{"Start":"03:11.944 ","End":"03:15.170","Text":"It can\u0027t be just bounded."},{"Start":"03:15.170 ","End":"03:19.195","Text":"It\u0027s half bounded, if you want it bounded from below."},{"Start":"03:19.195 ","End":"03:24.230","Text":"Now I\u0027m going to introduce 2 more concepts."},{"Start":"03:24.230 ","End":"03:29.600","Text":"One of them is the greatest lower bound and the other one is the least upper bound."},{"Start":"03:29.600 ","End":"03:31.610","Text":"They have alternative names,"},{"Start":"03:31.610 ","End":"03:35.705","Text":"Latin-sounding names, more international."},{"Start":"03:35.705 ","End":"03:41.270","Text":"The greatest lower bound is called the infimum."},{"Start":"03:41.270 ","End":"03:45.515","Text":"The least upper bound will also be called the supremum."},{"Start":"03:45.515 ","End":"03:47.525","Text":"Let\u0027s start with one of them,"},{"Start":"03:47.525 ","End":"03:49.675","Text":"the greatest lower bound."},{"Start":"03:49.675 ","End":"03:52.240","Text":"What is this?"},{"Start":"03:52.240 ","End":"03:54.530","Text":"Let\u0027s go back to that example."},{"Start":"03:54.530 ","End":"03:55.790","Text":"We had the sequence 2,"},{"Start":"03:55.790 ","End":"03:58.540","Text":"4, 6, 8 and so on."},{"Start":"03:58.540 ","End":"04:02.390","Text":"We found that 2 was a lower bound,"},{"Start":"04:02.390 ","End":"04:07.310","Text":"but that wasn\u0027t the only one that anything less than 2 is also a lower bound."},{"Start":"04:07.310 ","End":"04:09.680","Text":"It has lots of lower bounds."},{"Start":"04:09.680 ","End":"04:11.515","Text":"In fact, infinitely many."},{"Start":"04:11.515 ","End":"04:16.215","Text":"For example, 2, 1, minus 17,"},{"Start":"04:16.215 ","End":"04:22.770","Text":"minus a million, 0, lots of them."},{"Start":"04:22.770 ","End":"04:25.440","Text":"But there\u0027s 1 special 1."},{"Start":"04:25.440 ","End":"04:31.640","Text":"In some sense 2 is a special lower bound out of all the lower bounds."},{"Start":"04:31.640 ","End":"04:35.350","Text":"You might ask what\u0027s so special about it?"},{"Start":"04:35.350 ","End":"04:40.405","Text":"Well, it\u0027s the greatest from all the lower bounds of this sequence,"},{"Start":"04:40.405 ","End":"04:43.285","Text":"2 is actually the greatest."},{"Start":"04:43.285 ","End":"04:48.830","Text":"There\u0027s no other lower bound that\u0027s greater than 2."},{"Start":"04:49.140 ","End":"04:53.920","Text":"Because anything greater than 2"},{"Start":"04:53.920 ","End":"04:59.255","Text":"will no longer be greater than all the terms of the sequence."},{"Start":"04:59.255 ","End":"05:06.940","Text":"It\u0027s not surprising that we give it the name greatest lower bound and often abbreviated."},{"Start":"05:06.940 ","End":"05:12.655","Text":"In fact, I\u0027ll typically be abbreviating it as GLB in this context."},{"Start":"05:12.655 ","End":"05:21.030","Text":"As I mentioned, it\u0027s also called the infimum of the sequence."},{"Start":"05:21.030 ","End":"05:23.280","Text":"For our sequence, it was 2,"},{"Start":"05:23.280 ","End":"05:26.740","Text":"but this property exists in general."},{"Start":"05:26.770 ","End":"05:32.285","Text":"Very similar to the greatest lower bound will be the least upper bound."},{"Start":"05:32.285 ","End":"05:36.140","Text":"It\u0027s just going to work on the opposite side."},{"Start":"05:36.140 ","End":"05:40.650","Text":"Instead of taking lower bounds,"},{"Start":"05:40.650 ","End":"05:42.495","Text":"we\u0027re going to take upper bound."},{"Start":"05:42.495 ","End":"05:45.980","Text":"Let\u0027s return to the sequence we had before."},{"Start":"05:45.980 ","End":"05:47.510","Text":"This one, 1, 1/2, 1/3,"},{"Start":"05:47.510 ","End":"05:49.405","Text":"1/4, and so on."},{"Start":"05:49.405 ","End":"05:53.525","Text":"We said that 1 is an upper bound,"},{"Start":"05:53.525 ","End":"05:57.710","Text":"but anything greater than 1 is also an upper bound."},{"Start":"05:57.710 ","End":"06:00.415","Text":"There\u0027s infinitely many."},{"Start":"06:00.415 ","End":"06:06.225","Text":"For example, 1, 2, 13, 100,"},{"Start":"06:06.225 ","End":"06:09.660","Text":"245, they are all upper bounds,"},{"Start":"06:09.660 ","End":"06:14.035","Text":"they are all bigger or equal to everything in the sequence."},{"Start":"06:14.035 ","End":"06:20.285","Text":"Amongst all these infinitely many upper bounds is 1 special 1,"},{"Start":"06:20.285 ","End":"06:22.805","Text":"and that is the number 1."},{"Start":"06:22.805 ","End":"06:25.475","Text":"What\u0027s special about it?"},{"Start":"06:25.475 ","End":"06:33.365","Text":"Well, earlier we talked about the greatest lower bound and it was the greatest."},{"Start":"06:33.365 ","End":"06:40.530","Text":"This 1 has the opposite property that it\u0027s the smallest or the least."},{"Start":"06:40.530 ","End":"06:43.845","Text":"There\u0027s no other upper bound less than it."},{"Start":"06:43.845 ","End":"06:52.909","Text":"Anything that\u0027s less than 1 is not going to be an upper bound."},{"Start":"06:52.909 ","End":"06:57.650","Text":"Because an upper bound has to be bigger or equal to all the elements of the sequence,"},{"Start":"06:57.650 ","End":"07:00.390","Text":"so has to be at least 1."},{"Start":"07:01.710 ","End":"07:08.770","Text":"Such an upper bound that\u0027s the least 1 is just called the least upper bound,"},{"Start":"07:08.770 ","End":"07:12.230","Text":"and the abbreviated LUB,"},{"Start":"07:12.570 ","End":"07:16.705","Text":"also called the supremum of the sequence."},{"Start":"07:16.705 ","End":"07:26.590","Text":"Now we also noted earlier that this sequence also has a greatest lower bound."},{"Start":"07:26.590 ","End":"07:29.380","Text":"Well, we noted that it has a lower bound,"},{"Start":"07:29.380 ","End":"07:34.705","Text":"that 0 is a lower bound because all of these are positive and are big or equal to 0."},{"Start":"07:34.705 ","End":"07:41.980","Text":"In fact, 0 is the greatest lower bound."},{"Start":"07:41.980 ","End":"07:46.315","Text":"Nothing greater than 0 will be less than or equal to all of these."},{"Start":"07:46.315 ","End":"07:48.865","Text":"Can you think why?"},{"Start":"07:48.865 ","End":"07:51.625","Text":"Well, I\u0027ll tell you."},{"Start":"07:51.625 ","End":"07:54.475","Text":"Let\u0027s see. I want to keep it inside."},{"Start":"07:54.475 ","End":"07:58.690","Text":"Well, if something is greater than 0,"},{"Start":"07:58.690 ","End":"08:02.350","Text":"then at some point any number that\u0027s larger than 0,"},{"Start":"08:02.350 ","End":"08:08.125","Text":"there\u0027s going to be some n such that 1 over n is less than any positive number."},{"Start":"08:08.125 ","End":"08:11.770","Text":"There\u0027ll be some members of the sequence that will be smaller than it,"},{"Start":"08:11.770 ","End":"08:17.660","Text":"so it can\u0027t possibly be a lower bound,"},{"Start":"08:17.660 ","End":"08:20.080","Text":"so that\u0027s a proof by contradiction."},{"Start":"08:20.080 ","End":"08:24.025","Text":"Anyway, I don\u0027t want to dwell on that too much."},{"Start":"08:24.025 ","End":"08:30.685","Text":"What I want to do is make things a bit more formal."},{"Start":"08:30.685 ","End":"08:33.969","Text":"We define things pretty vaguely."},{"Start":"08:33.969 ","End":"08:38.365","Text":"Let\u0027s just go over these concepts again."},{"Start":"08:38.365 ","End":"08:43.430","Text":"We start off in general with the sequence a_n."},{"Start":"08:43.680 ","End":"08:54.400","Text":"If we have a number capital M such that a_n is less than or equal to M for all n,"},{"Start":"08:54.400 ","End":"09:00.610","Text":"in other words, M is bigger or equal to all the terms in the sequence,"},{"Start":"09:00.610 ","End":"09:04.460","Text":"then it\u0027s called an upper bound of the sequence."},{"Start":"09:04.500 ","End":"09:09.655","Text":"If the sequence has an upper bound and not all sequences do,"},{"Start":"09:09.655 ","End":"09:14.590","Text":"then it said to be bounded above or bounded from above,"},{"Start":"09:14.590 ","End":"09:17.500","Text":"optional the word from."},{"Start":"09:17.500 ","End":"09:20.365","Text":"Some upper bound are special."},{"Start":"09:20.365 ","End":"09:26.035","Text":"An upper bound of a sequence is called the least upper bound, LUB,"},{"Start":"09:26.035 ","End":"09:33.400","Text":"or supremum of the sequence if there is no smaller upper bound of the sequence."},{"Start":"09:33.400 ","End":"09:34.600","Text":"In other words, it\u0027s the smallest."},{"Start":"09:34.600 ","End":"09:36.740","Text":"There\u0027s nothing smaller than it."},{"Start":"09:37.440 ","End":"09:42.699","Text":"All this is pretty much repeated for the lower bounds."},{"Start":"09:42.699 ","End":"09:48.460","Text":"We take a little m such that a_n is bigger or equal to m for all n,"},{"Start":"09:48.460 ","End":"09:53.589","Text":"meaning that little m is less than or equal to all the members of the sequence,"},{"Start":"09:53.589 ","End":"09:55.810","Text":"so it\u0027s called the lower bound."},{"Start":"09:55.810 ","End":"09:58.030","Text":"If the sequence has a lower bound,"},{"Start":"09:58.030 ","End":"10:00.280","Text":"it\u0027s bounded from below."},{"Start":"10:00.280 ","End":"10:02.530","Text":"A lower bound might be special."},{"Start":"10:02.530 ","End":"10:10.224","Text":"It could be the greatest lower bound or infimum if it\u0027s the greatest,"},{"Start":"10:10.224 ","End":"10:14.275","Text":"if there\u0027s no other lower bound that\u0027s greater than it."},{"Start":"10:14.275 ","End":"10:17.020","Text":"Those are the formal definitions."},{"Start":"10:17.020 ","End":"10:20.210","Text":"Next, let\u0027s move on to an example."},{"Start":"10:20.280 ","End":"10:23.890","Text":"In this example exercise,"},{"Start":"10:23.890 ","End":"10:25.780","Text":"we\u0027re given the sequence a_n,"},{"Start":"10:25.780 ","End":"10:29.500","Text":"by this formula, minus 1^n plus 1 over n squared,"},{"Start":"10:29.500 ","End":"10:31.255","Text":"and these 3 parts."},{"Start":"10:31.255 ","End":"10:38.875","Text":"First of all, write a few upper bounds of the sequence and write its least upper bound."},{"Start":"10:38.875 ","End":"10:43.060","Text":"Then part b is to write a few lower bounds of"},{"Start":"10:43.060 ","End":"10:48.190","Text":"the sequence and what is its greatest lower bound."},{"Start":"10:48.190 ","End":"10:49.990","Text":"Finally, the question is,"},{"Start":"10:49.990 ","End":"10:52.580","Text":"is the sequence bounded?"},{"Start":"10:53.010 ","End":"10:55.675","Text":"For the solution."},{"Start":"10:55.675 ","End":"10:57.610","Text":"Before we get properly started,"},{"Start":"10:57.610 ","End":"10:59.379","Text":"let\u0027s just write a few terms,"},{"Start":"10:59.379 ","End":"11:00.880","Text":"see what\u0027s going on."},{"Start":"11:00.880 ","End":"11:03.850","Text":"Let n equal 1,"},{"Start":"11:03.850 ","End":"11:05.485","Text":"then 2, then 3,"},{"Start":"11:05.485 ","End":"11:11.170","Text":"and 4, 5, 6."},{"Start":"11:11.170 ","End":"11:14.560","Text":"You\u0027ll see that these are what we get."},{"Start":"11:14.560 ","End":"11:17.920","Text":"There\u0027s something that happens alternately."},{"Start":"11:17.920 ","End":"11:20.890","Text":"We have a minus 1, a minus 1, a minus 1,"},{"Start":"11:20.890 ","End":"11:25.340","Text":"and here we have a 1, a 1, and a 1."},{"Start":"11:25.340 ","End":"11:28.290","Text":"Now, if you just stare at it a while,"},{"Start":"11:28.290 ","End":"11:33.975","Text":"you\u0027ll see that this term is the least upper bound."},{"Start":"11:33.975 ","End":"11:36.435","Text":"First of all, it\u0027s an upper bound"},{"Start":"11:36.435 ","End":"11:43.540","Text":"because all the terms in the odd places are negative or 0."},{"Start":"11:43.540 ","End":"11:45.910","Text":"Well, this 1 is 0, but this 1 is negative,"},{"Start":"11:45.910 ","End":"11:48.370","Text":"negative, and so on."},{"Start":"11:48.370 ","End":"11:51.100","Text":"Then in the 2nd, 4th,"},{"Start":"11:51.100 ","End":"11:53.890","Text":"and 6th terms, it keeps getting smaller."},{"Start":"11:53.890 ","End":"11:58.810","Text":"We have 1 1/4, 1 1/16, 1 and 1/36."},{"Start":"11:58.810 ","End":"12:01.870","Text":"This is going to be bigger than all of them."},{"Start":"12:01.870 ","End":"12:04.900","Text":"Since it\u0027s 1 of the terms in the sequence,"},{"Start":"12:04.900 ","End":"12:08.365","Text":"it has to be the least upper bound."},{"Start":"12:08.365 ","End":"12:12.910","Text":"Because any upper bound has to be at least 1 1/4,"},{"Start":"12:12.910 ","End":"12:17.080","Text":"it\u0027s going to be bigger than all the terms in the sequence."},{"Start":"12:17.080 ","End":"12:19.180","Text":"Once we have the least upper bound,"},{"Start":"12:19.180 ","End":"12:25.225","Text":"we can write a few more by just taking any few numbers larger than this."},{"Start":"12:25.225 ","End":"12:28.150","Text":"For example, 2 is an upper bound,"},{"Start":"12:28.150 ","End":"12:34.435","Text":"then 100 and 13 1/2 and Pi and whatever."},{"Start":"12:34.435 ","End":"12:38.620","Text":"Just as long as it\u0027s bigger than 1 1/4."},{"Start":"12:38.620 ","End":"12:41.755","Text":"Now, in part b,"},{"Start":"12:41.755 ","End":"12:44.965","Text":"we want some lower bounds."},{"Start":"12:44.965 ","End":"12:49.764","Text":"Once again, it\u0027s easiest to start with the greatest lower bound."},{"Start":"12:49.764 ","End":"12:57.055","Text":"Notice that the terms in the even places are all positive."},{"Start":"12:57.055 ","End":"13:00.385","Text":"They\u0027re all 1 and something: 1 plus this 1, plus this 1, plus this."},{"Start":"13:00.385 ","End":"13:02.155","Text":"So let\u0027s leave those out."},{"Start":"13:02.155 ","End":"13:04.690","Text":"If we take the terms in odd places,"},{"Start":"13:04.690 ","End":"13:06.985","Text":"we have here minus 1 plus 1 is 0,"},{"Start":"13:06.985 ","End":"13:08.485","Text":"minus 1 plus 1/9,"},{"Start":"13:08.485 ","End":"13:11.125","Text":"minus 1 plus 1/25."},{"Start":"13:11.125 ","End":"13:16.240","Text":"It\u0027s always minus 1 plus a bit and that bit keeps getting smaller."},{"Start":"13:16.240 ","End":"13:21.580","Text":"It looks like that minus 1 is going to be the greatest lower bound."},{"Start":"13:21.580 ","End":"13:26.125","Text":"First of all, it\u0027s a lower bound because as I said,"},{"Start":"13:26.125 ","End":"13:28.060","Text":"these terms, this 1,"},{"Start":"13:28.060 ","End":"13:30.340","Text":"and this 1, and this 1 are positive."},{"Start":"13:30.340 ","End":"13:32.170","Text":"In the odd places,"},{"Start":"13:32.170 ","End":"13:35.470","Text":"it\u0027s minus 1 plus something positive,"},{"Start":"13:35.470 ","End":"13:41.300","Text":"so they\u0027re all bigger than minus 1."},{"Start":"13:42.480 ","End":"13:49.810","Text":"Now, why is it the greatest lower bound?"},{"Start":"13:49.810 ","End":"13:53.590","Text":"Because if I take something bigger than minus 1,"},{"Start":"13:53.590 ","End":"13:56.995","Text":"it\u0027s going to be minus 1 plus something positive."},{"Start":"13:56.995 ","End":"14:01.630","Text":"Sooner or later, 1/4,"},{"Start":"14:01.630 ","End":"14:04.465","Text":"1/16, 1/36, or get to 1 over something,"},{"Start":"14:04.465 ","End":"14:07.525","Text":"that\u0027s less than that something positive,"},{"Start":"14:07.525 ","End":"14:12.860","Text":"so it\u0027ll be less than anything that\u0027s bigger than minus 1."},{"Start":"14:12.860 ","End":"14:20.760","Text":"I think it\u0027s intuitively clear that minus 1 is the greatest lower bound."},{"Start":"14:21.310 ","End":"14:26.745","Text":"We were asked to write a few more lower bounds."},{"Start":"14:26.745 ","End":"14:32.950","Text":"Is there anything smaller than minus 1?"},{"Start":"14:32.950 ","End":"14:36.070","Text":"Minus 1 1/4, minus 2,"},{"Start":"14:36.070 ","End":"14:40.700","Text":"minus 17, minus 100, minus a zillion."},{"Start":"14:43.140 ","End":"14:46.915","Text":"Part c, we were asked if the sequence is bounded."},{"Start":"14:46.915 ","End":"14:49.780","Text":"The answer of course is yes,"},{"Start":"14:49.780 ","End":"14:55.570","Text":"because it has some upper bound,"},{"Start":"14:55.570 ","End":"14:58.970","Text":"it\u0027s bounded above, and it has lower bound,"},{"Start":"14:58.970 ","End":"15:00.545","Text":"so it\u0027s bounded below,"},{"Start":"15:00.545 ","End":"15:03.125","Text":"and if it\u0027s bounded above and below,"},{"Start":"15:03.125 ","End":"15:05.565","Text":"then it\u0027s just bounded."},{"Start":"15:05.565 ","End":"15:10.175","Text":"That answers that example exercise."},{"Start":"15:10.175 ","End":"15:13.775","Text":"I want to end this clip with a useful theorem,"},{"Start":"15:13.775 ","End":"15:16.865","Text":"turns out to be very useful, in fact."},{"Start":"15:16.865 ","End":"15:23.410","Text":"If you have a sequence that\u0027s monotonic and it\u0027s bounded above and below,"},{"Start":"15:23.410 ","End":"15:26.135","Text":"that means of course, then it converges,"},{"Start":"15:26.135 ","End":"15:28.530","Text":"meaning it has a limit."},{"Start":"15:28.710 ","End":"15:33.620","Text":"If a sequence is increasing or decreasing and is also bounded,"},{"Start":"15:33.620 ","End":"15:35.870","Text":"then it has a limit."},{"Start":"15:35.870 ","End":"15:39.360","Text":"I\u0027m ending this clip here."}],"ID":27207}],"Thumbnail":null,"ID":254619},{"Name":"Convergence of a Sequence, Monotone Sequences","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"2m 33s","ChapterTopicVideoID":26301,"CourseChapterTopicPlaylistID":254620,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26301.jpeg","UploadDate":"2021-09-12T14:13:21.1870000","DurationForVideoObject":"PT2M33S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.450","Text":"In this exercise, we have A,"},{"Start":"00:03.450 ","End":"00:06.210","Text":"a non-empty subset of the reels."},{"Start":"00:06.210 ","End":"00:09.540","Text":"It doesn\u0027t say, but it\u0027s bounded below."},{"Start":"00:09.540 ","End":"00:12.720","Text":"We know this because we\u0027re told that it has an infimum,"},{"Start":"00:12.720 ","End":"00:15.840","Text":"a greatest lower bound, which is Alpha."},{"Start":"00:15.840 ","End":"00:21.960","Text":"Our task is to show that there is a sequence a_n where all the members come"},{"Start":"00:21.960 ","End":"00:29.775","Text":"from A and the sequence converges to Alpha."},{"Start":"00:29.775 ","End":"00:37.360","Text":"For each n, Alpha is the greatest lower bound of the sequence."},{"Start":"00:37.360 ","End":"00:42.020","Text":"Alpha plus 1 over n can\u0027t be a lower bound because if it was,"},{"Start":"00:42.020 ","End":"00:44.335","Text":"it would be greater than the greatest."},{"Start":"00:44.335 ","End":"00:47.675","Text":"Now, what does it mean that it\u0027s not a lower bound?"},{"Start":"00:47.675 ","End":"00:51.050","Text":"There is some element of a,"},{"Start":"00:51.050 ","End":"00:53.484","Text":"I call it a_n,"},{"Start":"00:53.484 ","End":"01:01.505","Text":"such that a_n is less than Alpha plus 1 over n, strictly less than."},{"Start":"01:01.505 ","End":"01:07.880","Text":"By the way, a_n is still bigger or equal to Alpha because Alpha is a lower bound."},{"Start":"01:07.880 ","End":"01:12.710","Text":"We repeat this procedure for each n and choose some a_n less"},{"Start":"01:12.710 ","End":"01:17.630","Text":"than Alpha plus 1 over n and we get the sequence that it\u0027s all members of a,"},{"Start":"01:17.630 ","End":"01:19.340","Text":"so it\u0027s a subset."},{"Start":"01:19.340 ","End":"01:26.510","Text":"All we have to do now is show that the limit as n goes to infinity of a_n is Alpha."},{"Start":"01:26.510 ","End":"01:34.005","Text":"We\u0027re going to use the Epsilon N definition of the limit."},{"Start":"01:34.005 ","End":"01:39.020","Text":"Let\u0027s take Epsilon bigger than 0 arbitrary."},{"Start":"01:39.020 ","End":"01:44.905","Text":"We can choose an integer capital N bigger than 1 over Epsilon."},{"Start":"01:44.905 ","End":"01:49.175","Text":"What we have is that for all n bigger than or equal to N,"},{"Start":"01:49.175 ","End":"01:51.800","Text":"the absolute value of a_n minus Alpha,"},{"Start":"01:51.800 ","End":"01:57.350","Text":"we can drop the absolute value because a_n is bigger or equal to Alpha."},{"Start":"01:57.350 ","End":"01:59.465","Text":"So this is a_n minus Alpha."},{"Start":"01:59.465 ","End":"02:02.060","Text":"That\u0027s less than 1 over n,"},{"Start":"02:02.060 ","End":"02:09.085","Text":"because a_n is less than Alpha plus 1 over n. Since n is bigger or equal to N,"},{"Start":"02:09.085 ","End":"02:12.350","Text":"then this reciprocal inverts the direction of"},{"Start":"02:12.350 ","End":"02:15.935","Text":"the inequality and this is less than Epsilon."},{"Start":"02:15.935 ","End":"02:19.830","Text":"In short, this absolute value of a_n minus Alpha less than"},{"Start":"02:19.830 ","End":"02:24.725","Text":"Epsilon for all N bigger or equal to n and this proves the convergence."},{"Start":"02:24.725 ","End":"02:28.850","Text":"We\u0027ve shown that for each Epsilon there is an N depending on Epsilon"},{"Start":"02:28.850 ","End":"02:34.060","Text":"such that this is less than this and so we\u0027re done."}],"ID":27208},{"Watched":false,"Name":"Exercise 2","Duration":"1m 12s","ChapterTopicVideoID":26302,"CourseChapterTopicPlaylistID":254620,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26302.jpeg","UploadDate":"2021-09-12T14:13:37.3130000","DurationForVideoObject":"PT1M12S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.004","Text":"I\u0027m going to rephrase this exercise in simple words."},{"Start":"00:04.004 ","End":"00:05.700","Text":"What we have to show,"},{"Start":"00:05.700 ","End":"00:08.040","Text":"is that if we have a rational number,"},{"Start":"00:08.040 ","End":"00:10.515","Text":"we can find a sequence of irrationals,"},{"Start":"00:10.515 ","End":"00:12.720","Text":"that converge to it."},{"Start":"00:12.720 ","End":"00:15.120","Text":"There are many ways of doing this."},{"Start":"00:15.120 ","End":"00:18.390","Text":"It\u0027s constructive proof."},{"Start":"00:18.390 ","End":"00:26.310","Text":"For example, you could take xn as x naught plus pi over n. For each n,"},{"Start":"00:26.310 ","End":"00:28.800","Text":"x naught is rational."},{"Start":"00:28.800 ","End":"00:33.060","Text":"Pi over N is irrational because Pi is irrational."},{"Start":"00:33.060 ","End":"00:38.070","Text":"The sum of a rational and an irrational gives us an irrational."},{"Start":"00:38.070 ","End":"00:41.325","Text":"Now we\u0027ll have to show is that,"},{"Start":"00:41.325 ","End":"00:45.160","Text":"xn converges to x naught."},{"Start":"00:46.390 ","End":"00:51.535","Text":"Limit as n goes to infinity of xn,"},{"Start":"00:51.535 ","End":"00:54.800","Text":"is, now this x naught is a constant,"},{"Start":"00:54.800 ","End":"00:57.170","Text":"so I can take a constant in front of the limit,"},{"Start":"00:57.170 ","End":"01:01.100","Text":"so it\u0027s x-naught plus the limit of Pi over N,"},{"Start":"01:01.100 ","End":"01:05.175","Text":"and Pi over n goes to 0."},{"Start":"01:05.175 ","End":"01:07.350","Text":"We end up with just x naught,"},{"Start":"01:07.350 ","End":"01:12.430","Text":"and that\u0027s this part. We\u0027re done."}],"ID":27209},{"Watched":false,"Name":"Exercise 3","Duration":"3m 10s","ChapterTopicVideoID":26303,"CourseChapterTopicPlaylistID":254620,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26303.jpeg","UploadDate":"2021-09-12T14:14:29.6030000","DurationForVideoObject":"PT3M10S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.570","Text":"In this exercise, we have a sequence x_n of"},{"Start":"00:03.570 ","End":"00:08.955","Text":"real numbers and we have to prove or disprove the following 2 statements."},{"Start":"00:08.955 ","End":"00:18.795","Text":"A says that if x_n goes to 0 and y_n is bounded then the product x_n y_n goes to 0."},{"Start":"00:18.795 ","End":"00:26.100","Text":"In part B similar this time x_n goes to infinity and also y_n is a bounded sequence."},{"Start":"00:26.100 ","End":"00:30.525","Text":"Then the question is, does x_n y_n go to infinity?"},{"Start":"00:30.525 ","End":"00:32.505","Text":"Well, we\u0027ll do them 1 at a time."},{"Start":"00:32.505 ","End":"00:34.050","Text":"First of all A,"},{"Start":"00:34.050 ","End":"00:36.060","Text":"this turns out to be true."},{"Start":"00:36.060 ","End":"00:40.250","Text":"It\u0027s actually a useful result and you should remember this,"},{"Start":"00:40.250 ","End":"00:41.869","Text":"it will be useful in future."},{"Start":"00:41.869 ","End":"00:48.620","Text":"In words, it says that a null sequence times a bounded sequence is a null sequence."},{"Start":"00:48.620 ","End":"00:50.975","Text":"Null sequence is 1 that goes to 0."},{"Start":"00:50.975 ","End":"00:52.940","Text":"Anyway, let\u0027s prove this."},{"Start":"00:52.940 ","End":"00:56.600","Text":"Let M be a bound for y_n."},{"Start":"00:56.600 ","End":"00:59.015","Text":"If it\u0027s bounded, then it has a bound."},{"Start":"00:59.015 ","End":"01:04.790","Text":"What that means is that all the y_n are bounded in absolute value by this"},{"Start":"01:04.790 ","End":"01:11.840","Text":"M. Note that 0 is less than or equal to the absolute value of x_n y_n."},{"Start":"01:11.840 ","End":"01:14.500","Text":"The absolute value of anything is bigger or equal to 0."},{"Start":"01:14.500 ","End":"01:18.060","Text":"We can break this up because absolute value of x_n is"},{"Start":"01:18.060 ","End":"01:21.990","Text":"non-negative and absolute value of y_n is less than or equal to M,"},{"Start":"01:21.990 ","End":"01:24.170","Text":"we get this inequality."},{"Start":"01:24.170 ","End":"01:29.960","Text":"That means that the limit of this part is 0,"},{"Start":"01:29.960 ","End":"01:33.365","Text":"the absolute value of x_n also goes to 0."},{"Start":"01:33.365 ","End":"01:35.420","Text":"I\u0027ll say more on this in a moment."},{"Start":"01:35.420 ","End":"01:42.710","Text":"The limit of absolute value of x_n y_n goes to 0 by the sandwich theorem,"},{"Start":"01:42.710 ","End":"01:45.170","Text":"it\u0027s sandwiched between this and 0,"},{"Start":"01:45.170 ","End":"01:48.240","Text":"which you can think of as a 0 sequence."},{"Start":"01:48.980 ","End":"01:55.555","Text":"That means that the limit of x_n y_n without the absolute value is 0."},{"Start":"01:55.555 ","End":"02:00.110","Text":"Now I said I\u0027d comment on both this and this."},{"Start":"02:00.110 ","End":"02:07.280","Text":"In general, a_n goes to 0 if and only if the absolute value of a_n goes to 0."},{"Start":"02:07.280 ","End":"02:11.700","Text":"That explains why absolute value of x_n goes to 0."},{"Start":"02:11.700 ","End":"02:15.365","Text":"It also explains the other way how we got from this to this."},{"Start":"02:15.365 ","End":"02:16.610","Text":"For null sequence is,"},{"Start":"02:16.610 ","End":"02:19.835","Text":"it doesn\u0027t matter if you put an absolute value or not,"},{"Start":"02:19.835 ","End":"02:23.395","Text":"they\u0027re both null sequences or neither 1 is."},{"Start":"02:23.395 ","End":"02:26.775","Text":"Now part B turns out to be false."},{"Start":"02:26.775 ","End":"02:33.545","Text":"I mean, it might go to infinity and it might not but in general, not."},{"Start":"02:33.545 ","End":"02:36.395","Text":"All you have to do is give 1 counterexample."},{"Start":"02:36.395 ","End":"02:43.890","Text":"Let x_n be n and y_n be 1 over n. Now,"},{"Start":"02:43.890 ","End":"02:52.355","Text":"certainly x_n goes to infinity and also the sequence y_n is bounded."},{"Start":"02:52.355 ","End":"02:56.420","Text":"We can see that the absolute value of y_n is always less than or equal to 1."},{"Start":"02:56.420 ","End":"02:59.390","Text":"But the limit of x_n,"},{"Start":"02:59.390 ","End":"03:02.420","Text":"y_n is the limit of n times 1 over n, which is the limit of 1,"},{"Start":"03:02.420 ","End":"03:06.400","Text":"which is 1 and that\u0027s not infinity."},{"Start":"03:06.500 ","End":"03:10.390","Text":"That concludes B and we\u0027re done."}],"ID":27210},{"Watched":false,"Name":"Exercise 4","Duration":"2m 39s","ChapterTopicVideoID":26304,"CourseChapterTopicPlaylistID":254620,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26304.jpeg","UploadDate":"2021-09-12T14:15:10.7270000","DurationForVideoObject":"PT2M39S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.110","Text":"In this exercise, x_n is a sequence of real numbers and"},{"Start":"00:04.110 ","End":"00:08.715","Text":"we have to prove or disprove a and b."},{"Start":"00:08.715 ","End":"00:16.635","Text":"A says that if the sequence x_n plus 1 over n x_n converges, then x_n converges."},{"Start":"00:16.635 ","End":"00:18.990","Text":"Turns out this 1 is true."},{"Start":"00:18.990 ","End":"00:23.460","Text":"In b, it\u0027s very similar except there\u0027s a squared here."},{"Start":"00:23.460 ","End":"00:25.710","Text":"It turns out that b is false."},{"Start":"00:25.710 ","End":"00:28.020","Text":"Let\u0027s get started with a."},{"Start":"00:28.020 ","End":"00:32.685","Text":"Let y_n be this sequence here."},{"Start":"00:32.685 ","End":"00:37.590","Text":"Notice that it is just x_n times 1 plus 1 over"},{"Start":"00:37.590 ","End":"00:42.165","Text":"n. Let z_n be"},{"Start":"00:42.165 ","End":"00:48.660","Text":"just this part here so we can write x_n as y_n over z_n."},{"Start":"00:48.660 ","End":"00:57.450","Text":"It\u0027s this, divided by just the 1 plus 1 over n. Y_n converges,"},{"Start":"00:57.450 ","End":"01:02.430","Text":"that\u0027s given, and z_n converges to 1,"},{"Start":"01:02.430 ","End":"01:04.875","Text":"and is also non-0."},{"Start":"01:04.875 ","End":"01:13.300","Text":"We can look at the quotient of 2 converging sequences and conclude that x_n converges."},{"Start":"01:13.300 ","End":"01:18.140","Text":"Whenever the numerator converges and the denominator is not 0 and converges,"},{"Start":"01:18.140 ","End":"01:21.200","Text":"then the quotient converges to the quotient."},{"Start":"01:21.200 ","End":"01:23.815","Text":"Actually, we can say more."},{"Start":"01:23.815 ","End":"01:29.130","Text":"The limit of x_n is the limit of y_n over the limit of z_n."},{"Start":"01:29.130 ","End":"01:33.285","Text":"The limit of z_n is 1,"},{"Start":"01:33.285 ","End":"01:36.440","Text":"so we\u0027re just left with the limit of y_n."},{"Start":"01:36.440 ","End":"01:43.100","Text":"It turns out that x_n converges to the same limit as what this converges to."},{"Start":"01:43.100 ","End":"01:47.090","Text":"Now in part B, it\u0027s false so we need to provide a counterexample."},{"Start":"01:47.090 ","End":"01:48.425","Text":"There\u0027s many possible."},{"Start":"01:48.425 ","End":"01:50.195","Text":"I\u0027m just giving you an example."},{"Start":"01:50.195 ","End":"01:54.340","Text":"Take x_n, which is minus 1 to the n,"},{"Start":"01:54.340 ","End":"02:00.515","Text":"the oscillating sequence that goes back and forth from minus 1 to 1, it doesn\u0027t converge."},{"Start":"02:00.515 ","End":"02:07.964","Text":"But if we take x_n squared plus 1 over n x_n, x_n squared is 1."},{"Start":"02:07.964 ","End":"02:12.065","Text":"1 over n x_n is this over n,"},{"Start":"02:12.065 ","End":"02:15.840","Text":"and this certainly converges to 0."},{"Start":"02:15.880 ","End":"02:20.000","Text":"We could use the previous exercise,"},{"Start":"02:20.000 ","End":"02:25.100","Text":"something bounded minus 1 to the n times something converging to 0,"},{"Start":"02:25.100 ","End":"02:30.540","Text":"1 over n also converges to 0 so this thing goes to 1."},{"Start":"02:30.640 ","End":"02:35.450","Text":"That\u0027s our counterexample because this converges,"},{"Start":"02:35.450 ","End":"02:39.300","Text":"but this doesn\u0027t, and we\u0027re done."}],"ID":27211},{"Watched":false,"Name":"Exercise 5","Duration":"5m 5s","ChapterTopicVideoID":26305,"CourseChapterTopicPlaylistID":254620,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26305.jpeg","UploadDate":"2021-09-12T14:16:31.8430000","DurationForVideoObject":"PT5M5S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.875","Text":"In this exercise, we have 2 numbers,"},{"Start":"00:04.875 ","End":"00:09.120","Text":"b_1 and a_1, both positive b_1 less than a_1."},{"Start":"00:09.120 ","End":"00:13.095","Text":"We\u0027re going to define a sequence recursively,"},{"Start":"00:13.095 ","End":"00:16.275","Text":"a sequence a_n, and a sequence b_n,"},{"Start":"00:16.275 ","End":"00:22.560","Text":"a_n plus 1 will be the arithmetic mean of a_n and b_n,"},{"Start":"00:22.560 ","End":"00:28.140","Text":"and b_n plus 1 will be the geometric mean of a_n and b_n."},{"Start":"00:28.140 ","End":"00:31.670","Text":"For all n belonging to the set of natural numbers,"},{"Start":"00:31.670 ","End":"00:34.354","Text":"which means that this is recursive definition."},{"Start":"00:34.354 ","End":"00:40.100","Text":"Now, our task is to show that both sequences a_n and b_n converge,"},{"Start":"00:40.100 ","End":"00:44.090","Text":"and more than that, that they converge to the same limit."},{"Start":"00:44.090 ","End":"00:49.240","Text":"As a hint, we\u0027re going to use the AM-GM inequality,"},{"Start":"00:49.240 ","End":"00:51.815","Text":"this mean, the arithmetic mean, geometric mean."},{"Start":"00:51.815 ","End":"00:57.410","Text":"It\u0027s known that the geometric mean of"},{"Start":"00:57.410 ","End":"01:03.070","Text":"2 positive numbers is less than the arithmetic mean,"},{"Start":"01:03.070 ","End":"01:05.955","Text":"provided that the 2 numbers are not equal,"},{"Start":"01:05.955 ","End":"01:09.465","Text":"otherwise, we get equality here."},{"Start":"01:09.465 ","End":"01:16.545","Text":"The first claim is that for all n 0s less than b_n and less than a_n,"},{"Start":"01:16.545 ","End":"01:20.575","Text":"and a proof by induction, abbreviated proof."},{"Start":"01:20.575 ","End":"01:24.940","Text":"N equals 1, we can see by ourselves that from here."},{"Start":"01:24.940 ","End":"01:27.175","Text":"This is just the inductive step,"},{"Start":"01:27.175 ","End":"01:30.805","Text":"assuming it\u0027s true for a_n that show it\u0027s true for n plus 1."},{"Start":"01:30.805 ","End":"01:36.205","Text":"So b_n plus 1 is the square root of a_n, b n. Now,"},{"Start":"01:36.205 ","End":"01:38.965","Text":"each of these is positive,"},{"Start":"01:38.965 ","End":"01:43.840","Text":"so the square root of this is also going to be positive."},{"Start":"01:43.840 ","End":"01:46.735","Text":"That\u0027s the bigger than 0 part."},{"Start":"01:46.735 ","End":"01:50.425","Text":"Now I need to show that b_n is less than a_n."},{"Start":"01:50.425 ","End":"01:53.020","Text":"Well, I mean, for n plus 1,"},{"Start":"01:53.020 ","End":"01:56.485","Text":"b_n plus 1 is the square root of a _n b_n."},{"Start":"01:56.485 ","End":"02:02.370","Text":"By the hint, this is less than a_n plus b_n/ 2,"},{"Start":"02:02.370 ","End":"02:04.365","Text":"and that\u0027s just a_n plus 1."},{"Start":"02:04.365 ","End":"02:07.005","Text":"So this is less than this."},{"Start":"02:07.005 ","End":"02:09.815","Text":"We\u0027ve proven this by induction."},{"Start":"02:09.815 ","End":"02:17.760","Text":"The next claim is that the sequence a_n is decreasing, again by induction."},{"Start":"02:18.100 ","End":"02:22.400","Text":"Well, a_1 is whatever it is."},{"Start":"02:22.400 ","End":"02:31.895","Text":"All I have to show is that each subsequent term is less than the previous term,"},{"Start":"02:31.895 ","End":"02:34.715","Text":"so a_n plus 1,"},{"Start":"02:34.715 ","End":"02:39.270","Text":"which is this, the arithmetic mean."},{"Start":"02:41.320 ","End":"02:46.190","Text":"We can also write less than or equal to a_n plus a_n/2."},{"Start":"02:46.190 ","End":"02:51.914","Text":"In other words, I\u0027ve replaced b_n with a_n because b_n is less than a_n,"},{"Start":"02:51.914 ","End":"02:55.460","Text":"and so We get that a_n plus 1 is less than a_n."},{"Start":"02:55.460 ","End":"02:58.700","Text":"Well, less than or equal to is all we need for decreasing."},{"Start":"02:58.700 ","End":"03:01.620","Text":"It\u0027s actually strictly decreasing."},{"Start":"03:01.850 ","End":"03:07.770","Text":"Similarly, b_n is increasing and we have to"},{"Start":"03:07.770 ","End":"03:13.290","Text":"show is that b_n plus 1 is greater than b_n."},{"Start":"03:13.290 ","End":"03:16.935","Text":"Well, b_n plus 1 is the square root of a_n, b_n."},{"Start":"03:16.935 ","End":"03:22.460","Text":"That\u0027s bigger or equal to actually bigger than square root of b_n times b_n,"},{"Start":"03:22.460 ","End":"03:26.150","Text":"since a_n is bigger than b_n and that\u0027s b_n."},{"Start":"03:26.150 ","End":"03:30.155","Text":"That\u0027s the b_n part."},{"Start":"03:30.155 ","End":"03:34.730","Text":"Now this is increasing and this is decreasing."},{"Start":"03:34.730 ","End":"03:39.960","Text":"Claim next is that both are bounded."},{"Start":"03:40.300 ","End":"03:45.930","Text":"That\u0027s easy to see because each b_n is bigger or equal"},{"Start":"03:45.930 ","End":"03:50.775","Text":"to b_ 1 and b_n is less than or equal to a_n,"},{"Start":"03:50.775 ","End":"03:53.160","Text":"and a_n is less than or equal to a_1,"},{"Start":"03:53.160 ","End":"04:03.300","Text":"so b_n and a_n are both bounded in the interval from b_1 to a_1."},{"Start":"04:03.300 ","End":"04:09.230","Text":"Now we apply the monotone convergence criterion for increasing,"},{"Start":"04:09.230 ","End":"04:14.435","Text":"unbounded or decreasing unbounded and conclude that they both converge."},{"Start":"04:14.435 ","End":"04:18.365","Text":"The last part we have to show is that the limits are equal."},{"Start":"04:18.365 ","End":"04:23.030","Text":"Let\u0027s call this limit a and this limit b,"},{"Start":"04:23.030 ","End":"04:29.680","Text":"so the limit of a_n plus 1 is the same as the limit of a_n."},{"Start":"04:29.680 ","End":"04:31.330","Text":"If you just shift by 1,"},{"Start":"04:31.330 ","End":"04:33.040","Text":"it doesn\u0027t change the limit."},{"Start":"04:33.040 ","End":"04:35.840","Text":"What we get is that a,"},{"Start":"04:35.840 ","End":"04:39.255","Text":"which is the limit of a_n plus 1,"},{"Start":"04:39.255 ","End":"04:42.835","Text":"is the limit of a_n plus 1,"},{"Start":"04:42.835 ","End":"04:46.720","Text":"which is a_n plus b_n/2,"},{"Start":"04:46.720 ","End":"04:51.025","Text":"and that\u0027s equal to a plus b/2."},{"Start":"04:51.025 ","End":"04:55.055","Text":"Now once we have that, a equals a plus b/2,"},{"Start":"04:55.055 ","End":"04:58.250","Text":"by simple algebra, we get 2 a equals a plus b,"},{"Start":"04:58.250 ","End":"05:00.425","Text":"so a equals b."},{"Start":"05:00.425 ","End":"05:03.290","Text":"That\u0027s this last part."},{"Start":"05:03.290 ","End":"05:05.670","Text":"We are done."}],"ID":27212},{"Watched":false,"Name":"Exercise 6","Duration":"3m 14s","ChapterTopicVideoID":26306,"CourseChapterTopicPlaylistID":254620,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26306.jpeg","UploadDate":"2021-09-12T14:17:05.8070000","DurationForVideoObject":"PT3M14S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.705","Text":"In this exercise, we\u0027re given a positive number a,"},{"Start":"00:03.705 ","End":"00:07.170","Text":"and we\u0027re also given some x_1 which is positive,"},{"Start":"00:07.170 ","End":"00:13.065","Text":"and from this we build a sequence x_n recursively by"},{"Start":"00:13.065 ","End":"00:19.485","Text":"x_n plus 1 is 1/2 of x_n plus a over x_n."},{"Start":"00:19.485 ","End":"00:24.840","Text":"We have to show that the sequence x_n converges to the square root of a."},{"Start":"00:24.840 ","End":"00:28.799","Text":"First, we\u0027ll show that all the x_n in the sequence are positive,"},{"Start":"00:28.799 ","End":"00:30.705","Text":"and this is by induction."},{"Start":"00:30.705 ","End":"00:36.795","Text":"X_1 is positive, so here\u0027s just the inductive step from k to k plus 1."},{"Start":"00:36.795 ","End":"00:38.580","Text":"If x_k is positive,"},{"Start":"00:38.580 ","End":"00:41.610","Text":"a over x_k is positive."},{"Start":"00:41.610 ","End":"00:45.275","Text":"Then if we take the average of x_k and a over x_k,"},{"Start":"00:45.275 ","End":"00:48.150","Text":"that will also be positive."},{"Start":"00:48.190 ","End":"00:52.430","Text":"Now we apply the AM-GM inequality,"},{"Start":"00:52.430 ","End":"00:57.500","Text":"which says that the arithmetic mean is bigger or equal to the geometric mean."},{"Start":"00:57.500 ","End":"00:59.390","Text":"So x_n plus 1,"},{"Start":"00:59.390 ","End":"01:00.995","Text":"which is equal to this,"},{"Start":"01:00.995 ","End":"01:03.860","Text":"which is the arithmetic mean of these 2,"},{"Start":"01:03.860 ","End":"01:07.715","Text":"is bigger or equal to the geometric mean of these 2,"},{"Start":"01:07.715 ","End":"01:11.430","Text":"and that\u0027s equal to square root of a."},{"Start":"01:11.900 ","End":"01:18.920","Text":"All the members of the sequence x_n are bigger or equal to the square root of a,"},{"Start":"01:18.920 ","End":"01:21.765","Text":"at least from 2 onwards."},{"Start":"01:21.765 ","End":"01:26.885","Text":"The next thing we\u0027ll show is that the sequence is monotonically decreasing,"},{"Start":"01:26.885 ","End":"01:32.555","Text":"and we\u0027ll evaluate the difference between 2 consecutive terms,"},{"Start":"01:32.555 ","End":"01:35.170","Text":"x_n plus 1 minus x_n,"},{"Start":"01:35.170 ","End":"01:37.725","Text":"but the definition of x_n plus 1,"},{"Start":"01:37.725 ","End":"01:40.395","Text":"it\u0027s equal to this minus x_n,"},{"Start":"01:40.395 ","End":"01:43.680","Text":"and a bit of simplification gives us,"},{"Start":"01:43.680 ","End":"01:48.455","Text":"this is equal to 1/2 of a minus x_n squared over x_n."},{"Start":"01:48.455 ","End":"01:53.270","Text":"Now, this is negative or at least less than or equal to"},{"Start":"01:53.270 ","End":"01:58.220","Text":"0 because x_n is bigger or equal to square root of a."},{"Start":"01:58.220 ","End":"02:01.540","Text":"So this numerator is non-positive."},{"Start":"02:01.540 ","End":"02:04.880","Text":"We\u0027ve just shown that x_n is decreasing and we"},{"Start":"02:04.880 ","End":"02:10.375","Text":"showed that it\u0027s bounded below by square root of a."},{"Start":"02:10.375 ","End":"02:13.625","Text":"By the monotone sequence theorem,"},{"Start":"02:13.625 ","End":"02:17.840","Text":"we have that x_n converges to some limit,"},{"Start":"02:17.840 ","End":"02:23.390","Text":"call it L. From this definition of x_n plus 1,"},{"Start":"02:23.390 ","End":"02:26.090","Text":"what we can get if we let n go to infinity,"},{"Start":"02:26.090 ","End":"02:29.000","Text":"we can put the limits in here and here."},{"Start":"02:29.000 ","End":"02:32.490","Text":"Notice that all the x_n is non-zero,"},{"Start":"02:32.490 ","End":"02:34.080","Text":"so we can do that."},{"Start":"02:34.080 ","End":"02:38.055","Text":"What this gives us is that L,"},{"Start":"02:38.055 ","End":"02:41.750","Text":"I mean the limit of x_n is the same as the limit of x_n plus 1,"},{"Start":"02:41.750 ","End":"02:47.270","Text":"is equal to 1/2 of L plus a over L. Now what we need is a bit of algebra."},{"Start":"02:47.270 ","End":"02:54.095","Text":"From this, we get the 2L squared is L squared plus a multiplying both sides by 2L."},{"Start":"02:54.095 ","End":"02:59.630","Text":"Then we can get that L squared equals a so that L is the square root of a."},{"Start":"02:59.630 ","End":"03:07.050","Text":"Obviously L is positive because all the x_n\u0027s are positive,"},{"Start":"03:07.050 ","End":"03:14.340","Text":"and that\u0027s what we have to show that the sequence converges to route a so we\u0027re done."}],"ID":27213},{"Watched":false,"Name":"Exercise 7","Duration":"2m 43s","ChapterTopicVideoID":26307,"CourseChapterTopicPlaylistID":254620,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26307.jpeg","UploadDate":"2021-09-12T14:17:30.2000000","DurationForVideoObject":"PT2M43S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:06.269","Text":"In this exercise, we have a sequence x_n in the open interval 0, 1."},{"Start":"00:06.269 ","End":"00:08.640","Text":"Suppose we have the inequality,"},{"Start":"00:08.640 ","End":"00:11.865","Text":"as written here for all n,"},{"Start":"00:11.865 ","End":"00:15.660","Text":"we have to show that the sequence is monotone,"},{"Start":"00:15.660 ","End":"00:18.670","Text":"and to find its limit."},{"Start":"00:19.270 ","End":"00:25.095","Text":"From this inequality, we can get that"},{"Start":"00:25.095 ","End":"00:33.570","Text":"the square root of x_n times 1 minus x_n plus 1 is bigger than,"},{"Start":"00:33.570 ","End":"00:35.774","Text":"we take the 4 over to the other side,"},{"Start":"00:35.774 ","End":"00:37.305","Text":"and then take the square root,"},{"Start":"00:37.305 ","End":"00:39.420","Text":"so that\u0027s a half,"},{"Start":"00:39.420 ","End":"00:42.380","Text":"and from the arithmetic mean,"},{"Start":"00:42.380 ","End":"00:47.120","Text":"geometric mean inequality, what we get is that"},{"Start":"00:47.120 ","End":"00:54.050","Text":"the average arithmetic of x_n and 1 minus x_n plus 1 is bigger,"},{"Start":"00:54.050 ","End":"00:56.960","Text":"or equal to the geometric mean of the same 2 quantities,"},{"Start":"00:56.960 ","End":"01:00.125","Text":"but this, as we just saw,"},{"Start":"01:00.125 ","End":"01:03.485","Text":"is bigger than 1.5."},{"Start":"01:03.485 ","End":"01:09.645","Text":"That x_n plus 1 minus x_n plus 1 is bigger than 1,"},{"Start":"01:09.645 ","End":"01:12.870","Text":"which means take away the 1 from"},{"Start":"01:12.870 ","End":"01:16.620","Text":"both sides and bring the x_n plus 1 to the other side, we get this."},{"Start":"01:16.620 ","End":"01:20.850","Text":"The sequence is monotone decreasing."},{"Start":"01:20.850 ","End":"01:25.845","Text":"It\u0027s also bounded, it\u0027s bounded in the interval from 0 to 1,"},{"Start":"01:25.845 ","End":"01:31.230","Text":"so x_n must have some limit, call it x-naught."},{"Start":"01:32.770 ","End":"01:38.610","Text":"Back to this inequality."},{"Start":"01:38.840 ","End":"01:42.345","Text":"We take the limit of both sides,"},{"Start":"01:42.345 ","End":"01:45.635","Text":"and when we take the limit of an inequality,"},{"Start":"01:45.635 ","End":"01:49.250","Text":"it could go from strict to non-strict."},{"Start":"01:49.250 ","End":"01:52.680","Text":"Otherwise, we could get a big or equal to here."},{"Start":"01:53.840 ","End":"01:56.655","Text":"This gives us that,"},{"Start":"01:56.655 ","End":"01:59.010","Text":"taking the limit, x_n goes to x-naught,"},{"Start":"01:59.010 ","End":"02:00.780","Text":"x_n plus 1 also goes to x-naught,"},{"Start":"02:00.780 ","End":"02:03.470","Text":"that we get this inequality,"},{"Start":"02:03.470 ","End":"02:05.650","Text":"which you can rewrite like this,"},{"Start":"02:05.650 ","End":"02:07.970","Text":"and we\u0027ve got a quadratic inequality,"},{"Start":"02:07.970 ","End":"02:09.800","Text":"but this is a perfect square."},{"Start":"02:09.800 ","End":"02:13.205","Text":"It\u0027s 2 x-naught minus 1 squared."},{"Start":"02:13.205 ","End":"02:17.835","Text":"When you have something squared less than or equal to 0,"},{"Start":"02:17.835 ","End":"02:20.780","Text":"the square has to be bigger or equal to 0 also,"},{"Start":"02:20.780 ","End":"02:22.970","Text":"so it has to be equal to 0,"},{"Start":"02:22.970 ","End":"02:27.165","Text":"and that\u0027s only possible when this is 0."},{"Start":"02:27.165 ","End":"02:31.475","Text":"Extracting x-naught, it gives us that x-naught is a half."},{"Start":"02:31.475 ","End":"02:36.420","Text":"X-naught is the limit of the sequence,"},{"Start":"02:36.420 ","End":"02:38.760","Text":"so that\u0027s the answer."},{"Start":"02:38.760 ","End":"02:40.995","Text":"The limit, as n goes to infinity of x_n,"},{"Start":"02:40.995 ","End":"02:44.260","Text":"is 1.5, and we\u0027re done."}],"ID":27214},{"Watched":false,"Name":"Exercise 8","Duration":"2m 50s","ChapterTopicVideoID":26308,"CourseChapterTopicPlaylistID":254620,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26308.jpeg","UploadDate":"2021-09-12T14:18:02.5900000","DurationForVideoObject":"PT2M50S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.340","Text":"In this exercise, we have a non-empty set of real numbers A,"},{"Start":"00:05.340 ","End":"00:10.230","Text":"and we have another real number, x naught."},{"Start":"00:10.230 ","End":"00:19.050","Text":"Our task is to show that there\u0027s a sequence a_ n of elements in A such"},{"Start":"00:19.050 ","End":"00:23.300","Text":"that the distance from x naught to a_ n"},{"Start":"00:23.300 ","End":"00:28.770","Text":"converges to the distance from x naught to the set A."},{"Start":"00:28.770 ","End":"00:30.040","Text":"In case you\u0027ve forgotten,"},{"Start":"00:30.040 ","End":"00:36.320","Text":"the definition of a distance from a point to a set is the infimum or"},{"Start":"00:36.320 ","End":"00:43.890","Text":"greatest lower bound of all the individual distances of x to members of A."},{"Start":"00:44.870 ","End":"00:48.090","Text":"The solution. First of all,"},{"Start":"00:48.090 ","End":"00:56.370","Text":"let\u0027s denote S as this set that\u0027s in the curly brackets,"},{"Start":"00:56.370 ","End":"01:02.000","Text":"and the set of all distances of x to a point in A so"},{"Start":"01:02.000 ","End":"01:08.910","Text":"that the infimum is what we call d of x naught,"},{"Start":"01:08.910 ","End":"01:14.885","Text":"A, and we\u0027ll call that capital D. Now we\u0027re going to build a sequence."},{"Start":"01:14.885 ","End":"01:18.110","Text":"Choose any arbitrary n,"},{"Start":"01:18.110 ","End":"01:20.255","Text":"which is a natural number."},{"Start":"01:20.255 ","End":"01:29.510","Text":"Then D plus 1 over n is not a lower bound of S because D is the greatest lower bound,"},{"Start":"01:29.510 ","End":"01:31.820","Text":"so there can\u0027t be another lower bound bigger than it."},{"Start":"01:31.820 ","End":"01:35.135","Text":"Now, what does it mean that it\u0027s not a lower bound?"},{"Start":"01:35.135 ","End":"01:45.165","Text":"It means that some element of the set is less than D plus 1 over n. Now,"},{"Start":"01:45.165 ","End":"01:50.340","Text":"s is the absolute value of x naught minus a."},{"Start":"01:51.680 ","End":"01:55.640","Text":"This particular a that we chose for this n,"},{"Start":"01:55.640 ","End":"01:57.515","Text":"we\u0027ll call it a_ n,"},{"Start":"01:57.515 ","End":"02:03.645","Text":"and so we get a sequence a_ n. What it satisfies,"},{"Start":"02:03.645 ","End":"02:07.185","Text":"just plug a_n instead of a here,"},{"Start":"02:07.185 ","End":"02:14.930","Text":"is that x naught minus a_ n is less than D plus 1 over n. But it\u0027s"},{"Start":"02:14.930 ","End":"02:23.790","Text":"also bigger or equal to D because D is the infimum of all such expressions."},{"Start":"02:24.260 ","End":"02:28.040","Text":"Now, if we let n go to infinity,"},{"Start":"02:28.040 ","End":"02:30.275","Text":"we get a sandwich here."},{"Start":"02:30.275 ","End":"02:31.850","Text":"The limit, on the 1 hand,"},{"Start":"02:31.850 ","End":"02:35.220","Text":"is bigger or equal to D,"},{"Start":"02:35.930 ","End":"02:40.010","Text":"and it\u0027s also less than or equal to the limit of this,"},{"Start":"02:40.010 ","End":"02:43.415","Text":"which is D, so the limit is exactly D,"},{"Start":"02:43.415 ","End":"02:46.220","Text":"which is d of x naught,"},{"Start":"02:46.220 ","End":"02:50.460","Text":"A as required, and we are done."}],"ID":27215},{"Watched":false,"Name":"Exercise 9","Duration":"2m 49s","ChapterTopicVideoID":26309,"CourseChapterTopicPlaylistID":254620,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26309.jpeg","UploadDate":"2021-09-12T14:18:26.7470000","DurationForVideoObject":"PT2M49S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:04.770","Text":"In this exercise, we have a bounded sequence, a_k,"},{"Start":"00:04.770 ","End":"00:07.860","Text":"k goes from 1 to infinity, say,"},{"Start":"00:07.860 ","End":"00:09.855","Text":"and for every n,"},{"Start":"00:09.855 ","End":"00:12.990","Text":"we define another sequence, x_n."},{"Start":"00:12.990 ","End":"00:20.100","Text":"By x_n is the supremum of all the a_k in this sequence,"},{"Start":"00:20.100 ","End":"00:23.250","Text":"where the index k is less than n."},{"Start":"00:23.250 ","End":"00:28.215","Text":"This is the same as the maximum because it\u0027s a finite set."},{"Start":"00:28.215 ","End":"00:34.275","Text":"We have to show that the new sequence, x_n converges."},{"Start":"00:34.275 ","End":"00:38.955","Text":"Now, I\u0027m going to state something that may seem obvious."},{"Start":"00:38.955 ","End":"00:47.255","Text":"If I have 2 subsets of the real numbers and they\u0027re both bounded above,"},{"Start":"00:47.255 ","End":"00:54.500","Text":"then the supremum of A has to be less than or equal to the supremum of B."},{"Start":"00:54.500 ","End":"00:57.589","Text":"In other words, if you increase the set,"},{"Start":"00:57.589 ","End":"01:01.580","Text":"you can only increase the supremum."},{"Start":"01:01.580 ","End":"01:05.460","Text":"It seems clear because there\u0027s more terms here,"},{"Start":"01:05.460 ","End":"01:07.400","Text":"so the maximum could grow,"},{"Start":"01:07.400 ","End":"01:08.840","Text":"it could stay the same,"},{"Start":"01:08.840 ","End":"01:11.630","Text":"but it certainly can\u0027t be less."},{"Start":"01:11.630 ","End":"01:15.440","Text":"I\u0027ll give you a formal proof of that."},{"Start":"01:15.440 ","End":"01:20.035","Text":"If you have any element a in A,"},{"Start":"01:20.035 ","End":"01:21.740","Text":"then it also belongs to B."},{"Start":"01:21.740 ","End":"01:24.375","Text":"That\u0027s the definition of set containment."},{"Start":"01:24.375 ","End":"01:28.490","Text":"As a member of B, it\u0027s less than or equal to the supremum of B."},{"Start":"01:28.490 ","End":"01:33.985","Text":"Supremum of B is an upper bound for the set A."},{"Start":"01:33.985 ","End":"01:35.660","Text":"If it\u0027s an upper bound,"},{"Start":"01:35.660 ","End":"01:39.200","Text":"it has to be bigger or equal to the least upper bound."},{"Start":"01:39.200 ","End":"01:42.485","Text":"If m is less than n,"},{"Start":"01:42.485 ","End":"01:50.390","Text":"then the set of all a_k where k is less than m is contained in the set of a_k,"},{"Start":"01:50.390 ","End":"01:51.830","Text":"where k is less than n,"},{"Start":"01:51.830 ","End":"01:56.620","Text":"that means there can only be more elements here."},{"Start":"01:56.620 ","End":"01:58.564","Text":"But what we said above,"},{"Start":"01:58.564 ","End":"02:02.090","Text":"the supremum of this is less than or equal to supremum of this."},{"Start":"02:02.090 ","End":"02:05.940","Text":"In other words, x_m is less than x_n."},{"Start":"02:05.940 ","End":"02:08.460","Text":"So x_n is increasing."},{"Start":"02:08.460 ","End":"02:12.770","Text":"Actually it should be less than or equal to here, but never mind."},{"Start":"02:12.770 ","End":"02:18.790","Text":"Now clearly, if M is an upper bound for the infinite sequence a_k,"},{"Start":"02:18.790 ","End":"02:25.820","Text":"it\u0027s also an upper bound for the partial finite sequence from 1 to n."},{"Start":"02:25.820 ","End":"02:29.555","Text":"This is just x_n,"},{"Start":"02:29.555 ","End":"02:31.480","Text":"the least upper bound."},{"Start":"02:31.480 ","End":"02:36.810","Text":"So x_n is less than or equal to M."},{"Start":"02:36.810 ","End":"02:42.410","Text":"Now what we have is that the sequence x_n is increasing and bounded."},{"Start":"02:42.410 ","End":"02:45.460","Text":"Also by the theorem it must converge."},{"Start":"02:45.460 ","End":"02:50.140","Text":"That\u0027s what we have to show and so we are done."}],"ID":27216}],"Thumbnail":null,"ID":254620},{"Name":"Cauchy Criterion, Bolzano - Weierstrass Theorem","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"5m 21s","ChapterTopicVideoID":26317,"CourseChapterTopicPlaylistID":254621,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26317.jpeg","UploadDate":"2021-09-12T14:28:35.3270000","DurationForVideoObject":"PT5M21S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.120","Text":"This exercise is 3 in 1."},{"Start":"00:03.120 ","End":"00:09.495","Text":"In each part, we have to show that the sequence x_n satisfies the Cauchy criterion."},{"Start":"00:09.495 ","End":"00:12.790","Text":"Let\u0027s start with part a."},{"Start":"00:13.190 ","End":"00:18.360","Text":"It\u0027s easy to see that all the x_n bigger or equal to 1,"},{"Start":"00:18.360 ","End":"00:20.310","Text":"and in particular they\u0027re positive."},{"Start":"00:20.310 ","End":"00:23.070","Text":"It\u0027s just a simple induction for n equals 1,"},{"Start":"00:23.070 ","End":"00:27.630","Text":"that\u0027s clear, and if x_n is bigger or equal to 1,"},{"Start":"00:27.630 ","End":"00:30.620","Text":"and in particular positive, the next n plus 1,"},{"Start":"00:30.620 ","End":"00:32.135","Text":"which is this expression,"},{"Start":"00:32.135 ","End":"00:33.530","Text":"is clearly bigger than 1,"},{"Start":"00:33.530 ","End":"00:36.020","Text":"so bigger or equal to 1."},{"Start":"00:36.020 ","End":"00:41.765","Text":"The plan is to show that the sequence x_n satisfies the contractive condition,"},{"Start":"00:41.765 ","End":"00:45.020","Text":"which implies the Cauchy criterion."},{"Start":"00:45.020 ","End":"00:52.040","Text":"Let\u0027s compute x_n plus 2 minus x_n plus 1, estimate anyway."},{"Start":"00:52.040 ","End":"00:57.720","Text":"Well, this is equal to this by definition and this is"},{"Start":"00:57.720 ","End":"01:03.705","Text":"equal to the 1 \u0027s cancels so it\u0027s just 1/x_n plus 1 minus 1/x_n,"},{"Start":"01:03.705 ","End":"01:08.884","Text":"and that\u0027s equal to just cross multiply and subtract."},{"Start":"01:08.884 ","End":"01:11.105","Text":"We got this expression."},{"Start":"01:11.105 ","End":"01:16.400","Text":"Now I\u0027m going to show that this denominator is bigger or equal to 2."},{"Start":"01:16.400 ","End":"01:20.300","Text":"Here, just replace x_n plus 1 by what it\u0027s equal"},{"Start":"01:20.300 ","End":"01:25.550","Text":"and then this comes out if you multiply out to x_n plus 1."},{"Start":"01:25.550 ","End":"01:29.360","Text":"But everything in the absolute value is positive,"},{"Start":"01:29.360 ","End":"01:32.740","Text":"so this thing is bigger or equal to 2,"},{"Start":"01:32.740 ","End":"01:37.685","Text":"and now we can return to this point now that we\u0027ve estimated the denominator."},{"Start":"01:37.685 ","End":"01:42.830","Text":"What we have is, I\u0027m just copying this is equal to this,"},{"Start":"01:42.830 ","End":"01:45.920","Text":"and this is less than or equal to."},{"Start":"01:45.920 ","End":"01:49.900","Text":"Notice the inversion of the sign because we\u0027re dividing now,"},{"Start":"01:49.900 ","End":"01:52.805","Text":"so it comes less than or equal to"},{"Start":"01:52.805 ","End":"01:57.395","Text":"1/2 and then the absolute value of x_n plus 1 minus x_n."},{"Start":"01:57.395 ","End":"02:03.690","Text":"Now if you look at the half as Alpha in the contractive condition,"},{"Start":"02:03.690 ","End":"02:05.940","Text":"Alpha\u0027s between 0 and 1,"},{"Start":"02:05.940 ","End":"02:12.215","Text":"so x_n satisfies the condition and so is a Cauchy sequence."},{"Start":"02:12.215 ","End":"02:15.360","Text":"Now on to part b,"},{"Start":"02:15.360 ","End":"02:18.650","Text":"and this is a reminder of what it was and we\u0027re also"},{"Start":"02:18.650 ","End":"02:22.215","Text":"going to do this with the contractive condition."},{"Start":"02:22.215 ","End":"02:28.649","Text":"First of all, note that every x_n is between 0 and 1,"},{"Start":"02:28.649 ","End":"02:33.600","Text":"it\u0027s true for n equals 1 and by induction you can see that if"},{"Start":"02:33.600 ","End":"02:38.885","Text":"x_n is bigger or equal to 0, this is positive,"},{"Start":"02:38.885 ","End":"02:44.540","Text":"and also the denominator is bigger or equal to 2,"},{"Start":"02:44.540 ","End":"02:49.220","Text":"so certainly bigger or equal to 1 so the reciprocal\u0027s less than or equal to 1."},{"Start":"02:49.220 ","End":"02:52.940","Text":"Now let\u0027s compute this difference to substitute what"},{"Start":"02:52.940 ","End":"02:57.109","Text":"it equals from the definition of the sequence."},{"Start":"02:57.109 ","End":"03:00.920","Text":"We get this and then we subtract the fractions."},{"Start":"03:00.920 ","End":"03:03.590","Text":"We get the product of this minus the product of this,"},{"Start":"03:03.590 ","End":"03:08.315","Text":"the 2 \u0027s cancel and we\u0027re left with this and here it\u0027s just this times this."},{"Start":"03:08.315 ","End":"03:10.955","Text":"Now from here to here, we need a couple of things."},{"Start":"03:10.955 ","End":"03:13.550","Text":"This is a difference of squares,"},{"Start":"03:13.550 ","End":"03:17.300","Text":"so we can write it like this and then in the denominator,"},{"Start":"03:17.300 ","End":"03:20.580","Text":"this is bigger or equal to 2 and this is"},{"Start":"03:20.580 ","End":"03:24.720","Text":"bigger or equal to 2 because it\u0027s 2 plus something in each case."},{"Start":"03:24.720 ","End":"03:28.975","Text":"In the denominator that makes it less than or equal to,"},{"Start":"03:28.975 ","End":"03:32.870","Text":"so we get this expression is 2 times 2 is 4."},{"Start":"03:32.870 ","End":"03:36.995","Text":"This plus this is less than or equal to 2"},{"Start":"03:36.995 ","End":"03:41.985","Text":"because each of the x_n is less than or equal to 1,"},{"Start":"03:41.985 ","End":"03:44.265","Text":"and so 2/4 is 2."},{"Start":"03:44.265 ","End":"03:47.625","Text":"What we get is 1/2x_n plus 1 minus x_n."},{"Start":"03:47.625 ","End":"03:55.545","Text":"This 1/2 is precisely the Alpha that we need for the contractive condition."},{"Start":"03:55.545 ","End":"03:58.250","Text":"So basically we\u0027re done with part b,"},{"Start":"03:58.250 ","End":"04:03.130","Text":"and now on to part c. This is the exercise."},{"Start":"04:03.130 ","End":"04:10.280","Text":"Note that x _n is less than or equal to 2 and is also non-negative for all n,"},{"Start":"04:10.280 ","End":"04:12.970","Text":"and you can do that by induction."},{"Start":"04:12.970 ","End":"04:15.710","Text":"It\u0027s certainly true for x_1,"},{"Start":"04:15.710 ","End":"04:18.485","Text":"so we just need the inductive step."},{"Start":"04:18.485 ","End":"04:22.835","Text":"Assuming this, let\u0027s see what x_n plus 1 is."},{"Start":"04:22.835 ","End":"04:27.320","Text":"It\u0027s equal to this by the definition and"},{"Start":"04:27.320 ","End":"04:32.410","Text":"x_n squared is less than or equal to 2 squared so we get this,"},{"Start":"04:32.410 ","End":"04:36.985","Text":"now 2 squared is 4 plus 8 is 12 over 6 is 2,"},{"Start":"04:36.985 ","End":"04:41.045","Text":"so we have this double inequality."},{"Start":"04:41.045 ","End":"04:47.510","Text":"Now let\u0027s evaluate this difference because we\u0027re going to use the contractive condition."},{"Start":"04:47.510 ","End":"04:52.280","Text":"What we can get here is that this is equal to difference of squares,"},{"Start":"04:52.280 ","End":"04:58.760","Text":"so we split it this way and then we can say that this plus this,"},{"Start":"04:58.760 ","End":"04:59.990","Text":"but they\u0027re all non-negative."},{"Start":"04:59.990 ","End":"05:02.450","Text":"It\u0027s just this plus this without the absolute value,"},{"Start":"05:02.450 ","End":"05:05.120","Text":"and each of them is less than or equal to 2,"},{"Start":"05:05.120 ","End":"05:06.380","Text":"so 2 plus 2 is 4."},{"Start":"05:06.380 ","End":"05:07.700","Text":"That\u0027s this 4."},{"Start":"05:07.700 ","End":"05:12.860","Text":"4 over 6 is 2/3 so we\u0027re left with this and 2/3 is our Alpha,"},{"Start":"05:12.860 ","End":"05:14.860","Text":"which is between 0 and 1."},{"Start":"05:14.860 ","End":"05:20.075","Text":"So again, we have the contractive condition and so the Cauchy criterion,"},{"Start":"05:20.075 ","End":"05:22.770","Text":"and we are done."}],"ID":27224},{"Watched":false,"Name":"Exercise 2","Duration":"4m 8s","ChapterTopicVideoID":26318,"CourseChapterTopicPlaylistID":254621,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26318.jpeg","UploadDate":"2021-09-12T14:29:19.6530000","DurationForVideoObject":"PT4M8S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.850","Text":"In this exercise, x_n is a sequence of positive real numbers."},{"Start":"00:05.850 ","End":"00:10.575","Text":"Then there are 3 statements then we have to prove or disprove each of them."},{"Start":"00:10.575 ","End":"00:17.475","Text":"Part a states that if x_n plus 1 minus x_n tends to 0, then x_n converges."},{"Start":"00:17.475 ","End":"00:19.965","Text":"It turns out that this is false."},{"Start":"00:19.965 ","End":"00:21.585","Text":"The reverse is true."},{"Start":"00:21.585 ","End":"00:24.900","Text":"If x_n converges, then this difference tends to 0,"},{"Start":"00:24.900 ","End":"00:27.720","Text":"but not the other way round."},{"Start":"00:27.720 ","End":"00:34.455","Text":"As an example, we can choose x_n to be square root of n. Obviously,"},{"Start":"00:34.455 ","End":"00:36.260","Text":"x_n doesn\u0027t converge, in fact,"},{"Start":"00:36.260 ","End":"00:37.790","Text":"it goes to infinity."},{"Start":"00:37.790 ","End":"00:44.875","Text":"However, if we take the difference of 2 successive numbers,"},{"Start":"00:44.875 ","End":"00:49.130","Text":"then this is equal to square root of n plus 1 minus the square root of n,"},{"Start":"00:49.130 ","End":"00:53.030","Text":"multiply top and bottom by the conjugate,"},{"Start":"00:53.030 ","End":"00:58.695","Text":"and then we get the denominator is this,"},{"Start":"00:58.695 ","End":"01:02.450","Text":"and the numerator by difference of squares is n plus 1 minus n,"},{"Start":"01:02.450 ","End":"01:03.775","Text":"which is just 1."},{"Start":"01:03.775 ","End":"01:06.260","Text":"Obviously, the denominator goes to infinity,"},{"Start":"01:06.260 ","End":"01:08.945","Text":"so this goes to 0."},{"Start":"01:08.945 ","End":"01:13.805","Text":"This goes to 0, but this doesn\u0027t converge. That was part a."},{"Start":"01:13.805 ","End":"01:18.200","Text":"Now part b, if you look at what it says,"},{"Start":"01:18.200 ","End":"01:24.890","Text":"what it is is that successive differences get smaller in absolute value."},{"Start":"01:24.890 ","End":"01:28.295","Text":"The claim is that if this is true for all n,"},{"Start":"01:28.295 ","End":"01:30.560","Text":"then the sequence converges,"},{"Start":"01:30.560 ","End":"01:34.790","Text":"and it turns out that this is also false."},{"Start":"01:34.790 ","End":"01:38.185","Text":"In fact, we can use the same example,"},{"Start":"01:38.185 ","End":"01:40.860","Text":"x_n equals square root n as above,"},{"Start":"01:40.860 ","End":"01:44.620","Text":"that we can even reuse some of the calculations."},{"Start":"01:44.620 ","End":"01:48.705","Text":"This difference, x_n plus 1 minus x_n."},{"Start":"01:48.705 ","End":"01:51.765","Text":"Well, we can drop the absolute value,"},{"Start":"01:51.765 ","End":"01:55.940","Text":"it\u0027s positive and we just get what is written here."},{"Start":"01:55.940 ","End":"01:59.020","Text":"If we put n plus 1 in place of n,"},{"Start":"01:59.020 ","End":"02:01.320","Text":"then we get this."},{"Start":"02:01.320 ","End":"02:04.035","Text":"This is less than this,"},{"Start":"02:04.035 ","End":"02:08.780","Text":"and that\u0027s because this denominator is bigger than this denominator."},{"Start":"02:08.780 ","End":"02:12.110","Text":"This part, the n plus 1 is common,"},{"Start":"02:12.110 ","End":"02:15.680","Text":"and certainly root n plus 2 is bigger than root n,"},{"Start":"02:15.680 ","End":"02:18.065","Text":"so this is less than this."},{"Start":"02:18.065 ","End":"02:21.580","Text":"This disproves part b."},{"Start":"02:22.100 ","End":"02:29.360","Text":"Part c is basically the reverse of the contractive condition."},{"Start":"02:29.360 ","End":"02:32.525","Text":"If the sequence has the contractive condition,"},{"Start":"02:32.525 ","End":"02:36.690","Text":"then the sequence is a Cauchy sequence."},{"Start":"02:36.690 ","End":"02:40.475","Text":"But the other way around turns out to be false."},{"Start":"02:40.475 ","End":"02:47.015","Text":"Here\u0027s an example, let x_n equal 1 over n. Now,"},{"Start":"02:47.015 ","End":"02:49.640","Text":"this is a convergent sequence,"},{"Start":"02:49.640 ","End":"02:54.380","Text":"it goes to 0 and therefore it satisfies the Cauchy criterion."},{"Start":"02:54.380 ","End":"02:56.390","Text":"We\u0027ll show that it doesn\u0027t satisfy"},{"Start":"02:56.390 ","End":"03:01.330","Text":"the contractive condition for any Alpha between 0 and 1."},{"Start":"03:01.330 ","End":"03:08.105","Text":"Let\u0027s suppose we do it by contradiction that there is an Alpha between 0 and 1,"},{"Start":"03:08.105 ","End":"03:12.410","Text":"such that this is less than or equal to this."},{"Start":"03:12.410 ","End":"03:16.860","Text":"This expression is just x_n plus 2 minus x_n plus 1,"},{"Start":"03:16.860 ","End":"03:19.080","Text":"x_n plus 1 minus x_n."},{"Start":"03:19.080 ","End":"03:22.820","Text":"This is less than or equal to Alpha times this for"},{"Start":"03:22.820 ","End":"03:30.080","Text":"all n. This means if we multiply both sides by n plus 1 times n,"},{"Start":"03:30.080 ","End":"03:31.894","Text":"that we have this."},{"Start":"03:31.894 ","End":"03:40.160","Text":"n plus 1 cancels and we get n over n plus 2 is less than or equal to Alpha."},{"Start":"03:40.160 ","End":"03:43.670","Text":"Now we can take the limit of both sides."},{"Start":"03:43.670 ","End":"03:45.140","Text":"The right side is a constant,"},{"Start":"03:45.140 ","End":"03:49.115","Text":"so it just stays so we get the limit of this less than or equal to Alpha."},{"Start":"03:49.115 ","End":"03:52.500","Text":"But the limit of this is just 1,"},{"Start":"03:52.500 ","End":"03:55.309","Text":"so we get 1 less than or equal to Alpha,"},{"Start":"03:55.309 ","End":"04:02.130","Text":"which contradicts the condition that we\u0027re given that Alpha is less than 1,"},{"Start":"04:02.130 ","End":"04:04.170","Text":"it\u0027s between 0 and 1."},{"Start":"04:04.170 ","End":"04:08.920","Text":"That takes care of part c and we are done."}],"ID":27225},{"Watched":false,"Name":"Exercise 3","Duration":"2m 46s","ChapterTopicVideoID":26319,"CourseChapterTopicPlaylistID":254621,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26319.jpeg","UploadDate":"2021-09-12T14:29:47.2330000","DurationForVideoObject":"PT2M46S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.800","Text":"In this exercise, x_n is a sequence of integers and such"},{"Start":"00:05.800 ","End":"00:12.025","Text":"that the difference between 2 consecutive terms is bigger or equal to 1."},{"Start":"00:12.025 ","End":"00:17.470","Text":"We have to prove or disprove the following 2 statements."},{"Start":"00:17.470 ","End":"00:23.205","Text":"A, the sequence x_n doesn\u0027t satisfy the Cauchy criterion,"},{"Start":"00:23.205 ","End":"00:27.910","Text":"and b, the sequence can\u0027t have a convergent subsequence."},{"Start":"00:27.910 ","End":"00:31.060","Text":"Starting with a, a turns out to be"},{"Start":"00:31.060 ","End":"00:36.830","Text":"true that the sequence doesn\u0027t satisfy the Cauchy criterion."},{"Start":"00:36.830 ","End":"00:40.450","Text":"Let\u0027s do this by contradiction."},{"Start":"00:40.450 ","End":"00:46.700","Text":"Suppose that x_n does satisfy the Cauchy criterion,"},{"Start":"00:46.700 ","End":"00:55.190","Text":"let Epsilon equal 1 in the definition with the Epsilon n. Then for this Epsilon,"},{"Start":"00:55.190 ","End":"01:01.065","Text":"there is a N such that the difference between x_m and"},{"Start":"01:01.065 ","End":"01:07.950","Text":"x_n is less than 1 for all m and n bigger or equal to this N. Now,"},{"Start":"01:07.950 ","End":"01:14.795","Text":"in particular, if we let n and m be big N and big N plus 1,"},{"Start":"01:14.795 ","End":"01:18.160","Text":"then we get that this is less than 1."},{"Start":"01:18.160 ","End":"01:25.325","Text":"But this is a contradiction because it has to be bigger or equal to 1."},{"Start":"01:25.325 ","End":"01:28.445","Text":"Can\u0027t be bigger or equal to 1 and less than 1."},{"Start":"01:28.445 ","End":"01:35.340","Text":"The statement that it doesn\u0027t satisfy the Cauchy criterion is true."},{"Start":"01:35.710 ","End":"01:39.835","Text":"Part b turns out to be false,"},{"Start":"01:39.835 ","End":"01:43.605","Text":"i.e., it can have a convergent subsequence."},{"Start":"01:43.605 ","End":"01:46.860","Text":"What we have to do is bring a counterexample,"},{"Start":"01:46.860 ","End":"01:49.500","Text":"and this is 1 possible example."},{"Start":"01:49.500 ","End":"01:53.480","Text":"Let x_n be minus 1 to the power of n,"},{"Start":"01:53.480 ","End":"01:56.580","Text":"which is the alternating sequence minus 1,"},{"Start":"01:56.580 ","End":"01:57.810","Text":"1, minus 1, 1,"},{"Start":"01:57.810 ","End":"02:00.165","Text":"minus 1, 1, etc."},{"Start":"02:00.165 ","End":"02:05.670","Text":"Certainly, the difference between 2 consecutive terms is 2,"},{"Start":"02:05.670 ","End":"02:07.740","Text":"whether it\u0027s 1 and minus 1,"},{"Start":"02:07.740 ","End":"02:09.210","Text":"or whether it\u0027s minus 1 and 1."},{"Start":"02:09.210 ","End":"02:10.950","Text":"In either case, the difference is 2,"},{"Start":"02:10.950 ","End":"02:12.820","Text":"which is bigger or equal to 1."},{"Start":"02:12.820 ","End":"02:17.065","Text":"It does satisfy this condition."},{"Start":"02:17.065 ","End":"02:20.645","Text":"I claim it does have a convergent subsequence."},{"Start":"02:20.645 ","End":"02:29.190","Text":"Take the subsequence where nk is 2k, the even subscripts."},{"Start":"02:29.330 ","End":"02:36.900","Text":"That\u0027s equal to 1, because x_2k is minus 1 to the 2k and minus 1 to an even is 1."},{"Start":"02:36.900 ","End":"02:41.070","Text":"Just take the alternative terms and we get a convergent sub-sequence."},{"Start":"02:41.070 ","End":"02:46.860","Text":"The constant sequence is certainly convergent. We\u0027re done."}],"ID":27226},{"Watched":false,"Name":"Exercise 4","Duration":"2m 47s","ChapterTopicVideoID":26320,"CourseChapterTopicPlaylistID":254621,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26320.jpeg","UploadDate":"2021-09-12T14:30:12.8970000","DurationForVideoObject":"PT2M47S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:08.190","Text":"In this exercise, we have some number Alpha between 0 and 1 and we have a sequence x_n,"},{"Start":"00:08.190 ","End":"00:10.755","Text":"which satisfies the following condition."},{"Start":"00:10.755 ","End":"00:15.690","Text":"The difference between 2 consecutive terms x_n plus 1 minus x_n,"},{"Start":"00:15.690 ","End":"00:22.995","Text":"is less than or equal to Alpha to the power of n for all n equals 1, 2, 3, etc."},{"Start":"00:22.995 ","End":"00:31.020","Text":"We have to show that x_n satisfies the Cauchy criterion. Let\u0027s start."},{"Start":"00:31.020 ","End":"00:35.940","Text":"Let\u0027s assume that m is bigger than n. What we have to do is"},{"Start":"00:35.940 ","End":"00:42.025","Text":"estimate the absolute value of the difference between x_m and x_n."},{"Start":"00:42.025 ","End":"00:47.675","Text":"Basically, we have to show that this thing goes to 0 when n goes to infinity,"},{"Start":"00:47.675 ","End":"00:54.380","Text":"and m also goes to infinity because it\u0027s bigger than n. Using the triangle inequality,"},{"Start":"00:54.380 ","End":"00:57.185","Text":"x_m minus x_n is the sum."},{"Start":"00:57.185 ","End":"01:03.620","Text":"We can take it as the sum of differences of neighboring terms."},{"Start":"01:03.620 ","End":"01:09.215","Text":"We can get from m to n in a certain number of jumps,"},{"Start":"01:09.215 ","End":"01:10.850","Text":"so we have m, m minus 1,"},{"Start":"01:10.850 ","End":"01:12.455","Text":"m minus 1, m minus 2,"},{"Start":"01:12.455 ","End":"01:19.325","Text":"until we get to n plus 1 and n and then we take the absolute value."},{"Start":"01:19.325 ","End":"01:26.870","Text":"Now, given, this is less than Alpha to the m minus 1,"},{"Start":"01:26.870 ","End":"01:28.240","Text":"it\u0027s the index here,"},{"Start":"01:28.240 ","End":"01:32.660","Text":"this is less than or equal to Alpha to the m minus 2,"},{"Start":"01:32.660 ","End":"01:41.179","Text":"up to Alpha to the n. Let\u0027s just write it the other way around."},{"Start":"01:41.179 ","End":"01:47.195","Text":"Alpha to the n, and so on up to Alpha to the m minus 1,"},{"Start":"01:47.195 ","End":"01:53.845","Text":"on to have it in increasing order of exponents."},{"Start":"01:53.845 ","End":"01:57.800","Text":"Now, we can take Alpha to the n outside the brackets,"},{"Start":"01:57.800 ","End":"02:04.325","Text":"and we\u0027re left with 1 plus Alpha up to Alpha to the m minus 1 minus n. At any rate,"},{"Start":"02:04.325 ","End":"02:08.480","Text":"we can increase this by taking the infinite series."},{"Start":"02:08.480 ","End":"02:11.134","Text":"If we just take 1 plus Alpha plus etc,"},{"Start":"02:11.134 ","End":"02:15.260","Text":"to infinity, and we get a less than or equal to here."},{"Start":"02:15.260 ","End":"02:19.220","Text":"The reason we going to do this is we have a simple formula for the infinite sum."},{"Start":"02:19.220 ","End":"02:23.625","Text":"This sum is 1 over 1 minus Alpha."},{"Start":"02:23.625 ","End":"02:26.240","Text":"Together with the Alpha to the n, we get this."},{"Start":"02:26.240 ","End":"02:28.430","Text":"Now, Alpha is less than 1,"},{"Start":"02:28.430 ","End":"02:31.970","Text":"so Alpha to the n goes to 0 as n goes to"},{"Start":"02:31.970 ","End":"02:37.880","Text":"infinity and that means that x_n satisfies the Cauchy criterion."},{"Start":"02:37.880 ","End":"02:42.230","Text":"This difference goes to 0 or is smaller than"},{"Start":"02:42.230 ","End":"02:47.730","Text":"Epsilon whenever n is big enough, and we\u0027re done."}],"ID":27227},{"Watched":false,"Name":"Exercise 5","Duration":"5m 2s","ChapterTopicVideoID":26310,"CourseChapterTopicPlaylistID":254621,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26310.jpeg","UploadDate":"2021-09-12T14:22:15.0870000","DurationForVideoObject":"PT5M2S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.535","Text":"In this exercise, we have 2 real numbers, x_1 and x_2,"},{"Start":"00:05.535 ","End":"00:08.475","Text":"between 1 and 2 as follows,"},{"Start":"00:08.475 ","End":"00:13.230","Text":"and we define a sequence recursively."},{"Start":"00:13.230 ","End":"00:18.675","Text":"We can get _3 onwards from this recursive rule,"},{"Start":"00:18.675 ","End":"00:23.685","Text":"x_n plus 2 is the square root of x_n plus 1 times x_n."},{"Start":"00:23.685 ","End":"00:30.070","Text":"In simple words, each term is the geometric mean of the previous 2 terms."},{"Start":"00:30.110 ","End":"00:33.870","Text":"The main idea is to show that the sequence converges,"},{"Start":"00:33.870 ","End":"00:35.310","Text":"but we\u0027ll do it in 3 steps."},{"Start":"00:35.310 ","End":"00:42.545","Text":"We first show that the ratio of consecutive terms is being equal to a 1/2 for all n,"},{"Start":"00:42.545 ","End":"00:47.390","Text":"and then we show that this inequality is true,"},{"Start":"00:47.390 ","End":"00:52.640","Text":"which is the contractive condition with Alpha equals 2/3,"},{"Start":"00:52.640 ","End":"00:56.480","Text":"and finally, part C, that x_n converges."},{"Start":"00:56.480 ","End":"00:59.180","Text":"Let\u0027s start with part a."},{"Start":"00:59.180 ","End":"01:07.085","Text":"Now, I claim that all the members of the sequence are between 1 and 2."},{"Start":"01:07.085 ","End":"01:09.755","Text":"That\u0027s an easy induction."},{"Start":"01:09.755 ","End":"01:14.885","Text":"Well, it\u0027s true for n equals 1 and 2 by what\u0027s given."},{"Start":"01:14.885 ","End":"01:19.700","Text":"We just have to show that it\u0027s true onwards,"},{"Start":"01:19.700 ","End":"01:26.330","Text":"and suppose it\u0027s true up to x n for all the indices, 1, 2,"},{"Start":"01:26.330 ","End":"01:33.250","Text":"3 up to n. Then look at the following."},{"Start":"01:33.250 ","End":"01:35.775","Text":"X_n minus 1 is between 1 and 2,"},{"Start":"01:35.775 ","End":"01:37.770","Text":"and x_n is between 1 and 2."},{"Start":"01:37.770 ","End":"01:41.240","Text":"We get this inequality that the square root of this less than or equal to"},{"Start":"01:41.240 ","End":"01:45.860","Text":"square root of this is less than or equal to square root of this,"},{"Start":"01:45.860 ","End":"01:53.179","Text":"which means that x_n plus 1 is also between 1 and 2 and the rest by induction."},{"Start":"01:53.179 ","End":"02:00.575","Text":"Now, from this, we can conclude that this ratio is bigger or equal to a 1/2."},{"Start":"02:00.575 ","End":"02:01.985","Text":"Why is that?"},{"Start":"02:01.985 ","End":"02:06.175","Text":"X_n plus 1 is bigger or equal to 1,"},{"Start":"02:06.175 ","End":"02:09.055","Text":"and x_n is less than or equal to 2."},{"Start":"02:09.055 ","End":"02:14.450","Text":"If I increase the numerator and decrease the denominator,"},{"Start":"02:14.450 ","End":"02:17.045","Text":"I get something bigger."},{"Start":"02:17.045 ","End":"02:19.725","Text":"That proves part a."},{"Start":"02:19.725 ","End":"02:22.510","Text":"Now on to part B."},{"Start":"02:22.510 ","End":"02:24.950","Text":"Notice the following equality."},{"Start":"02:24.950 ","End":"02:28.545","Text":"It comes from the difference of squares in algebra."},{"Start":"02:28.545 ","End":"02:30.850","Text":"I got to this by trial and error."},{"Start":"02:30.850 ","End":"02:32.660","Text":"This is what works."},{"Start":"02:32.660 ","End":"02:35.290","Text":"X_n plus 2 squared."},{"Start":"02:35.290 ","End":"02:39.475","Text":"By the definition of x_n plus 2,"},{"Start":"02:39.475 ","End":"02:43.270","Text":"this is equal to the product of x_n plus 1 times x_n."},{"Start":"02:43.270 ","End":"02:46.170","Text":"Well, x_n plus 2 is the square root of this,"},{"Start":"02:46.170 ","End":"02:48.340","Text":"so we put the square here,"},{"Start":"02:48.340 ","End":"02:50.020","Text":"we can drop the square root."},{"Start":"02:50.020 ","End":"02:53.770","Text":"This is equal to take x_n plus 1 outside the brackets,"},{"Start":"02:53.770 ","End":"02:55.370","Text":"and we have this."},{"Start":"02:55.370 ","End":"03:03.475","Text":"Now, pull this expression here down to the denominator here under the x n plus 1,"},{"Start":"03:03.475 ","End":"03:05.020","Text":"and we get this."},{"Start":"03:05.020 ","End":"03:09.160","Text":"Next, we can apply the absolute value to both sides,"},{"Start":"03:09.160 ","End":"03:11.100","Text":"and this is what we get."},{"Start":"03:11.100 ","End":"03:14.400","Text":"Next, I\u0027m going to estimate this expression,"},{"Start":"03:14.400 ","End":"03:18.485","Text":"actually going to show that this is less than or equal to 2/3."},{"Start":"03:18.485 ","End":"03:20.960","Text":"Now, at this point we\u0027ll use part a."},{"Start":"03:20.960 ","End":"03:22.325","Text":"We\u0027ll use this."},{"Start":"03:22.325 ","End":"03:25.495","Text":"Replacing n by n plus 1,"},{"Start":"03:25.495 ","End":"03:31.825","Text":"we get that x_n plus 2 is bigger or equal to a 1/2 x_n plus 1."},{"Start":"03:31.825 ","End":"03:36.305","Text":"You bring the x_n to the other side and then raise the index by 1,"},{"Start":"03:36.305 ","End":"03:38.150","Text":"we get the following,"},{"Start":"03:38.150 ","End":"03:39.860","Text":"that x_n plus 1."},{"Start":"03:39.860 ","End":"03:42.125","Text":"Well, just what it is here,"},{"Start":"03:42.125 ","End":"03:48.350","Text":"and replace this by something that it\u0027s bigger or equal to."},{"Start":"03:48.350 ","End":"03:50.675","Text":"Now, it\u0027s in the denominator."},{"Start":"03:50.675 ","End":"03:53.270","Text":"This is bigger or equal to this."},{"Start":"03:53.270 ","End":"03:55.220","Text":"This is what\u0027s here,"},{"Start":"03:55.220 ","End":"03:56.900","Text":"and since it\u0027s in the denominator,"},{"Start":"03:56.900 ","End":"04:01.605","Text":"we get a less than or equal to here. New page."},{"Start":"04:01.605 ","End":"04:06.255","Text":"Now, we can cancel the x_n plus 1 in all 3 places,"},{"Start":"04:06.255 ","End":"04:09.270","Text":"and what we get here is just 1 here is 1,"},{"Start":"04:09.270 ","End":"04:12.510","Text":"here\u0027s the 1/2, is 1 over a 1/2 plus 1,"},{"Start":"04:12.510 ","End":"04:17.900","Text":"and in your head you can figure out this is equal to 2/3."},{"Start":"04:17.900 ","End":"04:22.340","Text":"Putting this back in the equation that we had,"},{"Start":"04:22.340 ","End":"04:25.050","Text":"we replace this with 2/3,"},{"Start":"04:25.050 ","End":"04:29.120","Text":"we get an inequality that this is less than or equal to 2/3 of this,"},{"Start":"04:29.120 ","End":"04:32.320","Text":"and this is what we have to show for part B."},{"Start":"04:32.320 ","End":"04:35.015","Text":"Now, onto part C,"},{"Start":"04:35.015 ","End":"04:41.510","Text":"this inequality is what we called the contractive condition."},{"Start":"04:41.510 ","End":"04:46.985","Text":"That means that x_n satisfies this condition."},{"Start":"04:46.985 ","End":"04:49.955","Text":"We know that when it satisfies this condition,"},{"Start":"04:49.955 ","End":"04:53.765","Text":"then it\u0027s also satisfies the Cauchy criterion,"},{"Start":"04:53.765 ","End":"04:57.545","Text":"and if it satisfies the Cauchy criterion, then it converges."},{"Start":"04:57.545 ","End":"05:02.280","Text":"That\u0027s what we are asked to do in part C. We\u0027re done."}],"ID":27217},{"Watched":false,"Name":"Exercise 6","Duration":"5m 9s","ChapterTopicVideoID":26311,"CourseChapterTopicPlaylistID":254621,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26311.jpeg","UploadDate":"2021-09-12T14:23:31.7070000","DurationForVideoObject":"PT5M9S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:05.685","Text":"In this exercise, we have to show that a sequence x_n of real numbers has"},{"Start":"00:05.685 ","End":"00:14.355","Text":"no convergent subsequence if and only if x_n in absolute value goes to infinity."},{"Start":"00:14.355 ","End":"00:18.540","Text":"Note that it\u0027s if and only if."},{"Start":"00:18.540 ","End":"00:22.380","Text":"That means there\u0027s 2 parts to the question."},{"Start":"00:22.380 ","End":"00:26.445","Text":"In the first part, we\u0027ll assume that this is true,"},{"Start":"00:26.445 ","End":"00:31.425","Text":"and we\u0027ll show that x_n has no convergent subsequence."},{"Start":"00:31.425 ","End":"00:35.835","Text":"Suppose x_n goes to infinity in absolute value,"},{"Start":"00:35.835 ","End":"00:41.800","Text":"and let\u0027s take a subsequence x_n_k of x_n."},{"Start":"00:41.800 ","End":"00:48.605","Text":"The claim is that the absolute value of the subsequence also goes to infinity."},{"Start":"00:48.605 ","End":"00:51.250","Text":"Now we need to prove that."},{"Start":"00:51.250 ","End":"00:55.415","Text":"It\u0027s intuitively clear, but we still need to prove it."},{"Start":"00:55.415 ","End":"01:01.650","Text":"The definition of x_n goes to infinity is that,"},{"Start":"01:01.850 ","End":"01:05.475","Text":"for any M bigger than 0,"},{"Start":"01:05.475 ","End":"01:12.140","Text":"there\u0027s sum N natural number such that the absolute value of x_n is"},{"Start":"01:12.140 ","End":"01:15.440","Text":"bigger than M whenever n is bigger or"},{"Start":"01:15.440 ","End":"01:19.340","Text":"equal to big N. Now by the definition of subsequence,"},{"Start":"01:19.340 ","End":"01:26.405","Text":"this n_k is a strictly increasing sequence of numbers."},{"Start":"01:26.405 ","End":"01:32.820","Text":"There is some k such that n_k is bigger than"},{"Start":"01:32.820 ","End":"01:35.690","Text":"N for all k bigger than K over"},{"Start":"01:35.690 ","End":"01:39.410","Text":"little k bigger than or equal to capital K. From a certain point onwards,"},{"Start":"01:39.410 ","End":"01:45.180","Text":"this is also bigger or equal to N. From that point onwards,"},{"Start":"01:45.180 ","End":"01:47.060","Text":"in other words for all k bigger or equal to K,"},{"Start":"01:47.060 ","End":"01:52.500","Text":"the absolute value of x and k is bigger or equal to M. For each M,"},{"Start":"01:52.500 ","End":"01:55.880","Text":"we found k such that whenever k is bigger or equal to k,"},{"Start":"01:55.880 ","End":"01:58.640","Text":"this is bigger or equal to M. That\u0027s the definition of"},{"Start":"01:58.640 ","End":"02:02.245","Text":"the absolute value of x_n_k goes to infinity,"},{"Start":"02:02.245 ","End":"02:06.930","Text":"and so x_n_k doesn\u0027t converge."},{"Start":"02:06.930 ","End":"02:10.170","Text":"For the second half of the if and only if,"},{"Start":"02:10.170 ","End":"02:16.960","Text":"we can prove logical equivalent that if this doesn\u0027t go to infinity,"},{"Start":"02:16.960 ","End":"02:21.640","Text":"then x_n has a convergent subsequence."},{"Start":"02:21.640 ","End":"02:25.265","Text":"Thus just logically equivalent. Let\u0027s see."},{"Start":"02:25.265 ","End":"02:27.430","Text":"If we say that x_n goes to infinity,"},{"Start":"02:27.430 ","End":"02:28.555","Text":"like we said above,"},{"Start":"02:28.555 ","End":"02:33.370","Text":"what it means, put it in more precise logic terminology."},{"Start":"02:33.370 ","End":"02:35.830","Text":"For all M bigger than 0,"},{"Start":"02:35.830 ","End":"02:38.200","Text":"there exists an N bigger than 0,"},{"Start":"02:38.200 ","End":"02:41.215","Text":"such that for all n bigger or equal to N,"},{"Start":"02:41.215 ","End":"02:46.025","Text":"the absolute value of x_n is bigger than M. Now,"},{"Start":"02:46.025 ","End":"02:49.235","Text":"the logical inversion of this,"},{"Start":"02:49.235 ","End":"02:52.745","Text":"what we do is we replace the for all by,"},{"Start":"02:52.745 ","End":"02:57.289","Text":"there exists, this exists by for all."},{"Start":"02:57.289 ","End":"02:59.870","Text":"Again, for all by there exists."},{"Start":"02:59.870 ","End":"03:04.160","Text":"Then at the end we negate this part,"},{"Start":"03:04.160 ","End":"03:09.690","Text":"which says that absolute value of x_n less than or equal to M. Again,"},{"Start":"03:09.690 ","End":"03:15.740","Text":"there exists an M such that for all N bigger than 0,"},{"Start":"03:15.740 ","End":"03:18.155","Text":"there exists a little n,"},{"Start":"03:18.155 ","End":"03:19.820","Text":"which depends on big N,"},{"Start":"03:19.820 ","End":"03:21.395","Text":"I want to emphasize,"},{"Start":"03:21.395 ","End":"03:27.500","Text":"such that the absolute value of x_n is less than or equal to M. I want to use"},{"Start":"03:27.500 ","End":"03:33.470","Text":"a different notation instead of big N. I want to use k and n,"},{"Start":"03:33.470 ","End":"03:36.890","Text":"which depends on big N, not depends on k,"},{"Start":"03:36.890 ","End":"03:41.315","Text":"and I\u0027m going to use the subscript notation rather than the function notation."},{"Start":"03:41.315 ","End":"03:44.340","Text":"I mean the sequence is a function."},{"Start":"03:45.110 ","End":"03:50.745","Text":"The indices n_k go to infinity"},{"Start":"03:50.745 ","End":"03:56.895","Text":"because n_k is bigger or equal to k. Here exists n bigger or equal to big N,"},{"Start":"03:56.895 ","End":"04:01.065","Text":"and big N is k. Now,"},{"Start":"04:01.065 ","End":"04:05.805","Text":"to be precise, a subsequence has to have these n_k increasing."},{"Start":"04:05.805 ","End":"04:08.015","Text":"But if something goes to infinity,"},{"Start":"04:08.015 ","End":"04:10.580","Text":"we can adjust it so it\u0027s increasing."},{"Start":"04:10.580 ","End":"04:13.430","Text":"For example, we can just thin it out,"},{"Start":"04:13.430 ","End":"04:16.025","Text":"throw out some terms whenever it doesn\u0027t increase,"},{"Start":"04:16.025 ","End":"04:17.915","Text":"throw that term out."},{"Start":"04:17.915 ","End":"04:24.355","Text":"How we can get an increasing sequence of indices and they still go to infinity."},{"Start":"04:24.355 ","End":"04:27.295","Text":"We got a subsequence x_n_k,"},{"Start":"04:27.295 ","End":"04:32.465","Text":"such that the absolute value of x_n_k is less than or equal to M,"},{"Start":"04:32.465 ","End":"04:35.015","Text":"which means that it\u0027s bounded."},{"Start":"04:35.015 ","End":"04:39.830","Text":"Now, bounded and increasing gives us a hint we should apply"},{"Start":"04:39.830 ","End":"04:47.945","Text":"the Bolzano Weierstrass theorem that x_n_k has a convergent subsequence."},{"Start":"04:47.945 ","End":"04:56.745","Text":"A subsequence of a subsequence is also a subsequence so x_n has a convergent subsequence,"},{"Start":"04:56.745 ","End":"05:00.200","Text":"and we\u0027ve proved the logical equivalent of the other direction,"},{"Start":"05:00.200 ","End":"05:04.160","Text":"that if x_n absolute value doesn\u0027t go to infinity,"},{"Start":"05:04.160 ","End":"05:09.860","Text":"then our sequence has a convergent subsequence, and we\u0027re done."}],"ID":27218},{"Watched":false,"Name":"Exercise 7","Duration":"2m 59s","ChapterTopicVideoID":26312,"CourseChapterTopicPlaylistID":254621,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26312.jpeg","UploadDate":"2021-09-12T14:24:18.8530000","DurationForVideoObject":"PT2M59S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:07.335","Text":"In this exercise, we have a sequence x_n and as a point x_naught,"},{"Start":"00:07.335 ","End":"00:15.555","Text":"such that every subsequence of x_n has itself a subsequence converging to x_naught."},{"Start":"00:15.555 ","End":"00:18.295","Text":"So that would be a subsequence."},{"Start":"00:18.295 ","End":"00:22.810","Text":"We have to show that the original sequence tends to x_naught."},{"Start":"00:22.810 ","End":"00:26.085","Text":"We\u0027ll prove this by contradiction."},{"Start":"00:26.085 ","End":"00:30.075","Text":"We\u0027ll suppose the x_n does not converge to x_naught."},{"Start":"00:30.075 ","End":"00:33.875","Text":"Now let\u0027s see what this means precisely."},{"Start":"00:33.875 ","End":"00:38.540","Text":"First of all, let\u0027s interpret what it means that x_n does converge to x_naught."},{"Start":"00:38.540 ","End":"00:40.265","Text":"In terms of Epsilon,"},{"Start":"00:40.265 ","End":"00:42.440","Text":"means for all Epsilon bigger than 0,"},{"Start":"00:42.440 ","End":"00:44.495","Text":"there exists n bigger than 0."},{"Start":"00:44.495 ","End":"00:49.565","Text":"Such that for all n bigger or equal to N,"},{"Start":"00:49.565 ","End":"00:53.660","Text":"the absolute value of x_n minus x_naught is less than Epsilon."},{"Start":"00:53.660 ","End":"00:56.750","Text":"Now if we take the logical inverse of this,"},{"Start":"00:56.750 ","End":"01:03.740","Text":"then x_n not converge to x_naught means that there exists Epsilon bigger than 0,"},{"Start":"01:03.740 ","End":"01:07.220","Text":"such that for all N bigger than 0,"},{"Start":"01:07.220 ","End":"01:14.235","Text":"there exists n bigger or equal to N. Note that this n depends on"},{"Start":"01:14.235 ","End":"01:20.150","Text":"N. Such that the absolute value of x_n minus x_naught"},{"Start":"01:20.150 ","End":"01:22.010","Text":"is the opposite of less than Epsilon,"},{"Start":"01:22.010 ","End":"01:24.320","Text":"means bigger or equal to Epsilon."},{"Start":"01:24.320 ","End":"01:26.130","Text":"I\u0027m just going to relabel this."},{"Start":"01:26.130 ","End":"01:28.470","Text":"Instead of N, I\u0027ll use k,"},{"Start":"01:28.470 ","End":"01:30.140","Text":"and instead of function notation,"},{"Start":"01:30.140 ","End":"01:32.255","Text":"I\u0027ll use sequence notation."},{"Start":"01:32.255 ","End":"01:35.655","Text":"Then this gives us a sequence n_k because,"},{"Start":"01:35.655 ","End":"01:39.435","Text":"for all k, there exists some n_k."},{"Start":"01:39.435 ","End":"01:43.085","Text":"Since n_k is bigger or equal to k,"},{"Start":"01:43.085 ","End":"01:45.095","Text":"then it goes to infinity."},{"Start":"01:45.095 ","End":"01:46.610","Text":"To make it a subsequence,"},{"Start":"01:46.610 ","End":"01:52.370","Text":"we have to make sure that n_k is strictly increasing,"},{"Start":"01:52.370 ","End":"01:54.195","Text":"but it\u0027s easy to adjust."},{"Start":"01:54.195 ","End":"01:59.420","Text":"You just throw out all the terms where it\u0027s not increasing."},{"Start":"01:59.420 ","End":"02:00.874","Text":"If it goes down,"},{"Start":"02:00.874 ","End":"02:02.810","Text":"then just throw that term out,"},{"Start":"02:02.810 ","End":"02:06.325","Text":"so it\u0027s a standard adjustment."},{"Start":"02:06.325 ","End":"02:09.180","Text":"Now we found Epsilon bigger than 0,"},{"Start":"02:09.180 ","End":"02:11.100","Text":"that\u0027s this one from the exists,"},{"Start":"02:11.100 ","End":"02:13.675","Text":"and a subsequence x_n_k,"},{"Start":"02:13.675 ","End":"02:23.640","Text":"such that x_n_k minus x_naught is bigger or equal to Epsilon for all k. Now,"},{"Start":"02:23.640 ","End":"02:26.394","Text":"if this is true for all k,"},{"Start":"02:26.394 ","End":"02:30.950","Text":"then any subsequence of x_n_k,"},{"Start":"02:30.950 ","End":"02:33.830","Text":"call it x_n_k_i, will also"},{"Start":"02:33.830 ","End":"02:37.325","Text":"have this property that the difference is bigger or equal to Epsilon."},{"Start":"02:37.325 ","End":"02:39.110","Text":"I\u0027m just thinning this out,"},{"Start":"02:39.110 ","End":"02:41.690","Text":"but still, everyone satisfies this."},{"Start":"02:41.690 ","End":"02:48.455","Text":"What it means is that every subsequence of our subsequence doesn\u0027t converge to x_naught."},{"Start":"02:48.455 ","End":"02:53.560","Text":"Meaning, the x_n_k we chose has no subsequence converging to x_naught."},{"Start":"02:53.560 ","End":"02:59.580","Text":"That is a contradiction to what was given, and we\u0027re done."}],"ID":27219},{"Watched":false,"Name":"Exercise 8 - Bolzano - Weierstrass Theorem proof","Duration":"4m 40s","ChapterTopicVideoID":26313,"CourseChapterTopicPlaylistID":254621,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26313.jpeg","UploadDate":"2021-09-12T14:25:49.6770000","DurationForVideoObject":"PT4M40S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.330","Text":"This exercise is actually, if you look at the end,"},{"Start":"00:03.330 ","End":"00:08.265","Text":"an alternative proof of the Bolzano-Weierstrass theorem,"},{"Start":"00:08.265 ","End":"00:10.485","Text":"which states that, well,"},{"Start":"00:10.485 ","End":"00:12.870","Text":"that\u0027s part c actually."},{"Start":"00:12.870 ","End":"00:15.254","Text":"Let\u0027s start at the beginning,"},{"Start":"00:15.254 ","End":"00:22.695","Text":"x_n is a sequence of real numbers and we define a concept of a peak positive integer n,"},{"Start":"00:22.695 ","End":"00:25.530","Text":"the index, is called the peak of the sequence."},{"Start":"00:25.530 ","End":"00:28.785","Text":"If for all m bigger than n,"},{"Start":"00:28.785 ","End":"00:32.385","Text":"x_n is bigger than x_m."},{"Start":"00:32.385 ","End":"00:37.740","Text":"Part a says, if x_n has infinitely many peaks,"},{"Start":"00:37.740 ","End":"00:42.825","Text":"to show that it has a decreasing subsequence. I\u0027ll read it all though."},{"Start":"00:42.825 ","End":"00:46.550","Text":"Part b, which is the other case,"},{"Start":"00:46.550 ","End":"00:49.490","Text":"is if it only has finitely many peaks,"},{"Start":"00:49.490 ","End":"00:52.760","Text":"then it has an increasing subsequence."},{"Start":"00:52.760 ","End":"00:59.220","Text":"Then in c we combine a and b to show the Bolzano-Weierstrass theorem."},{"Start":"00:59.220 ","End":"01:04.294","Text":"Now, in part a, the peaks are an infinite subset of the natural numbers,"},{"Start":"01:04.294 ","End":"01:07.224","Text":"and as such, they can be ordered."},{"Start":"01:07.224 ","End":"01:11.690","Text":"The way you do it is you take the smallest element because there\u0027s"},{"Start":"01:11.690 ","End":"01:17.000","Text":"a theorem that every non-empty set of natural numbers has a least element."},{"Start":"01:17.000 ","End":"01:19.835","Text":"You take the smallest and remove it,"},{"Start":"01:19.835 ","End":"01:21.260","Text":"then the next smallest,"},{"Start":"01:21.260 ","End":"01:23.950","Text":"and then each time you keep taking the smallest one,"},{"Start":"01:23.950 ","End":"01:28.580","Text":"and so on, and then you get all the elements in an increasing order."},{"Start":"01:28.580 ","End":"01:32.035","Text":"This gives us a subsequence."},{"Start":"01:32.035 ","End":"01:34.590","Text":"The indices give us elements."},{"Start":"01:34.590 ","End":"01:36.575","Text":"Let me just take x sub,"},{"Start":"01:36.575 ","End":"01:38.675","Text":"whatever it is, we get x_n_k,"},{"Start":"01:38.675 ","End":"01:46.695","Text":"and this subsequence is decreasing because n_k plus 1 is bigger than n_k,"},{"Start":"01:46.695 ","End":"01:50.410","Text":"so the corresponding x has the reverse inequality,"},{"Start":"01:50.410 ","End":"01:51.950","Text":"later on in the sequence,"},{"Start":"01:51.950 ","End":"01:55.065","Text":"smaller elements, and that\u0027s part a."},{"Start":"01:55.065 ","End":"01:56.880","Text":"Now onto b."},{"Start":"01:56.880 ","End":"01:59.990","Text":"If there\u0027s only finitely many peaks,"},{"Start":"01:59.990 ","End":"02:05.130","Text":"then there is the last one and call it N."},{"Start":"02:05.130 ","End":"02:09.140","Text":"This is not quite accurate."},{"Start":"02:09.140 ","End":"02:14.760","Text":"Finitely many includes the possibility of an empty set,"},{"Start":"02:14.760 ","End":"02:16.700","Text":"no peaks at all."},{"Start":"02:16.700 ","End":"02:20.495","Text":"In that case, we\u0027ll just let N equals 1."},{"Start":"02:20.495 ","End":"02:24.680","Text":"We\u0027ve lumped together the finitely many and the non together."},{"Start":"02:24.680 ","End":"02:29.790","Text":"Now we\u0027ll define an increasing set of indices n_k,"},{"Start":"02:29.790 ","End":"02:37.145","Text":"such that the sequence x_n_k is increasing as opposed to decreasing in part a,"},{"Start":"02:37.145 ","End":"02:41.100","Text":"and we\u0027ll define it recursively as follows."},{"Start":"02:41.110 ","End":"02:46.085","Text":"We\u0027ll let n_1 be N plus 1."},{"Start":"02:46.085 ","End":"02:50.020","Text":"Now, suppose that we\u0027ve defined n_k,"},{"Start":"02:50.020 ","End":"02:53.040","Text":"how do we define n_k plus 1?"},{"Start":"02:53.040 ","End":"02:57.470","Text":"Well, note that n_k is not a peak since we\u0027re"},{"Start":"02:57.470 ","End":"03:03.060","Text":"past all the peaks by taking this, where was it?"},{"Start":"03:03.060 ","End":"03:05.610","Text":"The last one and going beyond the last."},{"Start":"03:05.610 ","End":"03:07.815","Text":"There are no more peaks at this point,"},{"Start":"03:07.815 ","End":"03:10.170","Text":"and here it is, precisely."},{"Start":"03:10.170 ","End":"03:15.090","Text":"We can choose n_k plus 1 bigger than n_k,"},{"Start":"03:15.090 ","End":"03:19.470","Text":"such that x_n_k plus 1 is bigger than x_n_k."},{"Start":"03:19.470 ","End":"03:22.005","Text":"If we couldn\u0027t choose anything bigger,"},{"Start":"03:22.005 ","End":"03:23.270","Text":"then it would be a peak,"},{"Start":"03:23.270 ","End":"03:25.520","Text":"it would be bigger or equal to all the following ones."},{"Start":"03:25.520 ","End":"03:27.875","Text":"There has to be at least one that\u0027s bigger than."},{"Start":"03:27.875 ","End":"03:30.170","Text":"Proceeding this way recursively,"},{"Start":"03:30.170 ","End":"03:33.160","Text":"we keep getting another big one and a big one and a big one,"},{"Start":"03:33.160 ","End":"03:36.690","Text":"and that gives us our increasing sequence,"},{"Start":"03:36.690 ","End":"03:39.345","Text":"and that does part b."},{"Start":"03:39.345 ","End":"03:42.845","Text":"Now, part c combines both,"},{"Start":"03:42.845 ","End":"03:45.470","Text":"we take both cases if we have a sequence."},{"Start":"03:45.470 ","End":"03:49.970","Text":"We either have infinitely many peaks,"},{"Start":"03:49.970 ","End":"03:52.010","Text":"in which case by part a,"},{"Start":"03:52.010 ","End":"03:54.260","Text":"it has a decreasing subsequence,"},{"Start":"03:54.260 ","End":"03:58.785","Text":"or it only has a finite number of peaks or no peaks,"},{"Start":"03:58.785 ","End":"04:02.970","Text":"and then by part b it has an increasing subsequence."},{"Start":"04:02.970 ","End":"04:07.235","Text":"Either way, it has a monotone subsequence,"},{"Start":"04:07.235 ","End":"04:14.900","Text":"and if this monotone sequence is also bounded, then it converges."},{"Start":"04:14.900 ","End":"04:18.065","Text":"There is some monotone sequence theorem."},{"Start":"04:18.065 ","End":"04:20.420","Text":"Either way, if we\u0027re bounded,"},{"Start":"04:20.420 ","End":"04:26.120","Text":"then we either have a monotone increasing or monotone decreasing sequence,"},{"Start":"04:26.120 ","End":"04:33.185","Text":"but we just need the bounded in order to conclude that it has a convergent subsequence,"},{"Start":"04:33.185 ","End":"04:37.470","Text":"and that does part c."},{"Start":"04:37.470 ","End":"04:40.809","Text":"We\u0027re finished with the exercise."}],"ID":27220},{"Watched":false,"Name":"Exercise 9","Duration":"3m 30s","ChapterTopicVideoID":26314,"CourseChapterTopicPlaylistID":254621,"HasSubtitles":true,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26314.jpeg","UploadDate":"2021-09-12T14:26:42.0800000","DurationForVideoObject":"PT3M30S","Description":null,"VideoComments":[],"Subtitles":[{"Start":"00:00.000 ","End":"00:03.780","Text":"In this exercise, we have a sequence x_n,"},{"Start":"00:03.780 ","End":"00:05.850","Text":"which is defined recursively."},{"Start":"00:05.850 ","End":"00:07.815","Text":"The first 2 elements,"},{"Start":"00:07.815 ","End":"00:11.890","Text":"x_1 and x_2 are 1 and 2."},{"Start":"00:11.960 ","End":"00:19.155","Text":"We get the recursive element x_n plus 2 from the 2 previous elements as follows;"},{"Start":"00:19.155 ","End":"00:23.460","Text":"It\u0027s 3/4 x_n plus 1/4 x_n plus 1,"},{"Start":"00:23.460 ","End":"00:25.815","Text":"that gives us x_n plus 2."},{"Start":"00:25.815 ","End":"00:28.770","Text":"That\u0027s for all n bigger or equal to 1,"},{"Start":"00:28.770 ","End":"00:31.590","Text":"so that gives us from x_3 onwards."},{"Start":"00:31.590 ","End":"00:35.490","Text":"We have to show that x_n converges,"},{"Start":"00:35.490 ","End":"00:39.400","Text":"and furthermore to find its limit."},{"Start":"00:40.070 ","End":"00:43.970","Text":"We\u0027ll show that it converges by showing that it satisfies"},{"Start":"00:43.970 ","End":"00:49.205","Text":"the contractive condition with Alpha equals 3/4 specifically."},{"Start":"00:49.205 ","End":"00:51.405","Text":"I need to do a computation,"},{"Start":"00:51.405 ","End":"00:54.660","Text":"x_n plus 2 minus x_n plus 1, the difference."},{"Start":"00:54.660 ","End":"00:59.445","Text":"Replace x_n plus 2 by its definition here,"},{"Start":"00:59.445 ","End":"01:01.264","Text":"and we get this."},{"Start":"01:01.264 ","End":"01:09.300","Text":"Then combining we have 3/4 x_n plus 1/4 x_n plus 1 minus x_n plus 1,"},{"Start":"01:09.300 ","End":"01:12.300","Text":"so that\u0027s minus 3/4 x_n plus 1."},{"Start":"01:12.300 ","End":"01:16.325","Text":"Take the 3/4 outside the brackets, and we have this."},{"Start":"01:16.325 ","End":"01:22.085","Text":"This is exactly the contractive condition with Alpha equals 3/4."},{"Start":"01:22.085 ","End":"01:26.000","Text":"It\u0027s a Cauchy sequence and it converges."},{"Start":"01:26.000 ","End":"01:28.519","Text":"The question is, what\u0027s the limit?"},{"Start":"01:28.519 ","End":"01:36.990","Text":"Now, note that if we take the expression x_n plus 2 plus 3/4 x_n plus 1,"},{"Start":"01:36.990 ","End":"01:42.105","Text":"we can get by replacing x_n plus 2 with what it equals,"},{"Start":"01:42.105 ","End":"01:44.625","Text":"and then we rearrange a bit,"},{"Start":"01:44.625 ","End":"01:51.100","Text":"we get x_n plus 1 from here and here, plus 3/4 x_n."},{"Start":"01:51.110 ","End":"01:54.860","Text":"Basically, this is equal to this,"},{"Start":"01:54.860 ","End":"01:59.060","Text":"which is the same thing but with the index lowered by 1."},{"Start":"01:59.060 ","End":"02:02.920","Text":"If we rename this to be y_n,"},{"Start":"02:02.920 ","End":"02:07.830","Text":"then what we have here is y_n plus 1 and what we\u0027ve just proved is"},{"Start":"02:07.830 ","End":"02:12.915","Text":"that y_n plus 1 equals y_n for all n bigger or equal to 1."},{"Start":"02:12.915 ","End":"02:18.000","Text":"Y_n is a constant sequence, and in fact,"},{"Start":"02:18.000 ","End":"02:22.485","Text":"what it equals is y_1 and y_1,"},{"Start":"02:22.485 ","End":"02:30.885","Text":"if you look at the definition of y_n is x_2 plus 3/4 x_1."},{"Start":"02:30.885 ","End":"02:33.095","Text":"X_2 is 2, x_1 is 1,"},{"Start":"02:33.095 ","End":"02:35.255","Text":"that gives us 11 over 4."},{"Start":"02:35.255 ","End":"02:38.565","Text":"Now let\u0027s compute the limit of x_n."},{"Start":"02:38.565 ","End":"02:46.430","Text":"What we have is that 11 over 4 is the limit as n goes to infinity of y_n."},{"Start":"02:46.430 ","End":"02:50.330","Text":"In fact, we show that all the y_n are 11 over 4, the constant sequence."},{"Start":"02:50.330 ","End":"02:55.985","Text":"But anyway, we phrase it in terms of x and we get this limit."},{"Start":"02:55.985 ","End":"02:59.330","Text":"Then by linearity of the limit,"},{"Start":"02:59.330 ","End":"03:02.030","Text":"we can put the limit of x_n plus 1,"},{"Start":"03:02.030 ","End":"03:03.320","Text":"which is the limit of x_n,"},{"Start":"03:03.320 ","End":"03:06.470","Text":"and that\u0027s L. Here we get 3/4 L,"},{"Start":"03:06.470 ","End":"03:10.285","Text":"so we have 7/4 L altogether."},{"Start":"03:10.285 ","End":"03:15.530","Text":"If you look at this 11 over 4 equals 7 over 4 L,"},{"Start":"03:15.530 ","End":"03:17.935","Text":"and then isolate L from that,"},{"Start":"03:17.935 ","End":"03:20.390","Text":"it\u0027s 11 over 4 divided by 7 over 4,"},{"Start":"03:20.390 ","End":"03:23.015","Text":"which is just 11 over 7."},{"Start":"03:23.015 ","End":"03:30.540","Text":"The limit as n goes to infinity of x_n is 11 over 7, and we\u0027re done."}],"ID":27221}],"Thumbnail":null,"ID":254621},{"Name":"Disproving Limits of Sequences","TopicPlaylistFirstVideoID":0,"Duration":null,"Videos":[{"Watched":false,"Name":"Exercise 1","Duration":"3m 25s","ChapterTopicVideoID":26321,"CourseChapterTopicPlaylistID":254622,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26321.jpeg","UploadDate":"2021-09-12T14:32:37.7170000","DurationForVideoObject":"PT3M25S","Description":null,"VideoComments":[],"Subtitles":[],"ID":27228},{"Watched":false,"Name":"Exercise 2","Duration":"3m 48s","ChapterTopicVideoID":26322,"CourseChapterTopicPlaylistID":254622,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26322.jpeg","UploadDate":"2021-09-12T14:33:30.5630000","DurationForVideoObject":"PT3M48S","Description":null,"VideoComments":[],"Subtitles":[],"ID":27229},{"Watched":false,"Name":"Exercise 3","Duration":"2m 52s","ChapterTopicVideoID":26323,"CourseChapterTopicPlaylistID":254622,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26323.jpeg","UploadDate":"2021-09-12T14:34:18.4970000","DurationForVideoObject":"PT2M52S","Description":null,"VideoComments":[],"Subtitles":[],"ID":27230},{"Watched":false,"Name":"Exercise 4","Duration":"3m 31s","ChapterTopicVideoID":26324,"CourseChapterTopicPlaylistID":254622,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26324.jpeg","UploadDate":"2021-09-12T14:35:09.2030000","DurationForVideoObject":"PT3M31S","Description":null,"VideoComments":[],"Subtitles":[],"ID":27231},{"Watched":false,"Name":"Exercise 5","Duration":"3m 41s","ChapterTopicVideoID":26325,"CourseChapterTopicPlaylistID":254622,"HasSubtitles":false,"ThumbnailPath":"https://www.proprep.uk/Images/Videos_Thumbnails/26325.jpeg","UploadDate":"2021-09-12T14:36:02.7100000","DurationForVideoObject":"PT3M41S","Description":null,"VideoComments":[],"Subtitles":[],"ID":27232}],"Thumbnail":null,"ID":254622}]

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