WEBVTT 00:00:00.000 --> 00:00:02.425 align:middle line:90% [SQUEAKING] 00:00:02.425 --> 00:00:04.365 align:middle line:90% [RUSTLING] 00:00:04.365 --> 00:00:06.305 align:middle line:90% [CLICKING] 00:00:06.305 --> 00:00:10.300 align:middle line:90% 00:00:10.300 --> 00:00:15.320 align:middle line:84% TOBIAS COLDING: OK, so let me just recall first 00:00:15.320 --> 00:00:21.920 align:middle line:84% that we were talking about a very important concept 00:00:21.920 --> 00:00:25.440 align:middle line:84% of a sequence, concept of a sequence. 00:00:25.440 --> 00:00:30.520 align:middle line:84% So a sequence, the formal definition 00:00:30.520 --> 00:00:39.100 align:middle line:84% is that you have a map from the natural number into R. 00:00:39.100 --> 00:00:43.800 align:middle line:84% So then the image of a particular natural number n, 00:00:43.800 --> 00:00:47.660 align:middle line:84% it's usually denoted like an, like this. 00:00:47.660 --> 00:00:51.280 align:middle line:84% It could be some other instead of a, it could be bn. 00:00:51.280 --> 00:00:53.660 align:middle line:84% Potentially, it could be another subscript. 00:00:53.660 --> 00:00:58.800 align:middle line:84% But this is the most common thing to denote it like this. 00:00:58.800 --> 00:01:01.130 align:middle line:90% But it really is a function. 00:01:01.130 --> 00:01:02.430 align:middle line:90% So that's a sequence. 00:01:02.430 --> 00:01:07.370 align:middle line:90% 00:01:07.370 --> 00:01:11.490 align:middle line:84% So a subsequence is that you have a sequence 00:01:11.490 --> 00:01:14.010 align:middle line:90% and then you have another map. 00:01:14.010 --> 00:01:18.690 align:middle line:84% But this map is not just any map from the integers 00:01:18.690 --> 00:01:19.850 align:middle line:90% to the integers. 00:01:19.850 --> 00:01:22.030 align:middle line:84% It has to be strictly increasing. 00:01:22.030 --> 00:01:25.410 align:middle line:90% 00:01:25.410 --> 00:01:27.690 align:middle line:84% And why does it have to be strictly increasing? 00:01:27.690 --> 00:01:31.290 align:middle line:84% Because we want, for the subsequence, 00:01:31.290 --> 00:01:34.690 align:middle line:84% if you think about the numbers that come out 00:01:34.690 --> 00:01:39.170 align:middle line:84% when you plug in a, when you're looking at the an, 00:01:39.170 --> 00:01:42.850 align:middle line:84% then you want some of those numbers 00:01:42.850 --> 00:01:44.470 align:middle line:90% to be in the subsequence. 00:01:44.470 --> 00:01:48.070 align:middle line:84% But you want the ordering to be the same. 00:01:48.070 --> 00:01:52.890 align:middle line:84% So that's why this map has to be strictly increasing. 00:01:52.890 --> 00:01:56.650 align:middle line:84% You can't pick the several elements many times 00:01:56.650 --> 00:01:59.950 align:middle line:84% if it wasn't picked many times in the original sequence. 00:01:59.950 --> 00:02:03.620 align:middle line:90% 00:02:03.620 --> 00:02:07.440 align:middle line:84% So a subsequence is where you have 00:02:07.440 --> 00:02:09.500 align:middle line:84% another map from the natural number 00:02:09.500 --> 00:02:12.340 align:middle line:84% to the natural number that is strictly increasing. 00:02:12.340 --> 00:02:16.100 align:middle line:84% And then you're looking at the composition of these two maps. 00:02:16.100 --> 00:02:21.000 align:middle line:84% So you're looking at this in here. 00:02:21.000 --> 00:02:22.780 align:middle line:90% And you call this here. 00:02:22.780 --> 00:02:29.580 align:middle line:84% And then that's usually denoted something like this-- 00:02:29.580 --> 00:02:31.780 align:middle line:90% ank. 00:02:31.780 --> 00:02:34.980 align:middle line:84% An is the original sequence, k means 00:02:34.980 --> 00:02:37.940 align:middle line:84% that it's really that you take the image of k 00:02:37.940 --> 00:02:40.380 align:middle line:84% under this strictly increasing map. 00:02:40.380 --> 00:02:43.300 align:middle line:84% And then you're looking at the element in the sequence. 00:02:43.300 --> 00:02:51.580 align:middle line:84% So, nk, really, you should think about it as g of k. 00:02:51.580 --> 00:02:56.290 align:middle line:90% So that's what a subsequence is. 00:02:56.290 --> 00:02:58.790 align:middle line:84% And again, it's important that it's strictly 00:02:58.790 --> 00:03:02.990 align:middle line:84% increasing because you want the elements to be in the same order 00:03:02.990 --> 00:03:05.190 align:middle line:84% than it was in the original sequence. 00:03:05.190 --> 00:03:10.362 align:middle line:84% And you don't even want things to be picked more times than it 00:03:10.362 --> 00:03:11.570 align:middle line:90% was in the original sequence. 00:03:11.570 --> 00:03:15.550 align:middle line:90% 00:03:15.550 --> 00:03:19.470 align:middle line:84% So this whole concept of a sequence and subsequence 00:03:19.470 --> 00:03:21.110 align:middle line:90% is really important. 00:03:21.110 --> 00:03:31.390 align:middle line:84% Then, the next really important concept is of convergence. 00:03:31.390 --> 00:03:34.230 align:middle line:84% So if you have a sequence an, then 00:03:34.230 --> 00:03:36.490 align:middle line:84% you say that it's converging to a. 00:03:36.490 --> 00:03:39.190 align:middle line:90% So a is another real number. 00:03:39.190 --> 00:03:40.270 align:middle line:90% It's not infinity. 00:03:40.270 --> 00:03:42.310 align:middle line:90% It's another real number. 00:03:42.310 --> 00:03:56.360 align:middle line:84% And so an converges to a if for all epsilon greater than 0, 00:03:56.360 --> 00:04:06.000 align:middle line:84% there exist N, an integer N, denoted by capital N, 00:04:06.000 --> 00:04:11.920 align:middle line:84% such that if little n is bigger or equal to this capital N, then 00:04:11.920 --> 00:04:21.640 align:middle line:84% an minus a have to be less than epsilon. 00:04:21.640 --> 00:04:25.440 align:middle line:84% So if you are sufficiently far out in the sequence, 00:04:25.440 --> 00:04:31.640 align:middle line:84% then everything in that sequence is bunched epsilon close to n. 00:04:31.640 --> 00:04:34.000 align:middle line:90% That's what it's saying. 00:04:34.000 --> 00:04:50.840 align:middle line:84% And so if you have a sequence, a sequence that is not convergent, 00:04:50.840 --> 00:04:52.990 align:middle line:90% it is said to be divergent. 00:04:52.990 --> 00:05:03.214 align:middle line:90% 00:05:03.214 --> 00:05:09.330 align:middle line:84% And the next thing, I'm just briefly 00:05:09.330 --> 00:05:12.810 align:middle line:84% reviewing this from last time-- the next thing for sequences 00:05:12.810 --> 00:05:25.230 align:middle line:84% are the algebraic laws or whatever you want to call them. 00:05:25.230 --> 00:05:27.810 align:middle line:90% 00:05:27.810 --> 00:05:31.090 align:middle line:84% And so that is that if you take a sequence, 00:05:31.090 --> 00:05:40.370 align:middle line:84% if you have a sequence an and an converts to a, bn converts to b, 00:05:40.370 --> 00:05:46.010 align:middle line:84% then if you form the sum of these two, 00:05:46.010 --> 00:05:52.140 align:middle line:84% an plus bn, so this is a new sequence where 00:05:52.140 --> 00:05:55.300 align:middle line:84% the n-th element is the sum of the n-th element 00:05:55.300 --> 00:05:57.500 align:middle line:90% of the other two sequences. 00:05:57.500 --> 00:06:01.060 align:middle line:84% Then this sequence here cn also converges. 00:06:01.060 --> 00:06:06.486 align:middle line:84% And the limit here is the sum of the limits. 00:06:06.486 --> 00:06:16.740 align:middle line:84% And likewise, if you define a sequence-- so this was 1-- 00:06:16.740 --> 00:06:19.540 align:middle line:84% 2 is that if you take a constant. 00:06:19.540 --> 00:06:22.420 align:middle line:90% So this is a constant. 00:06:22.420 --> 00:06:29.460 align:middle line:84% And you define cn to be c times an, then this cn also converge. 00:06:29.460 --> 00:06:34.000 align:middle line:84% And the limit here is just c times a. 00:06:34.000 --> 00:06:37.160 align:middle line:90% That's the second one. 00:06:37.160 --> 00:06:38.880 align:middle line:84% I might have done it in the other order. 00:06:38.880 --> 00:06:48.990 align:middle line:84% But there's no particular order to have it in. 00:06:48.990 --> 00:06:53.810 align:middle line:84% The third one is that if you are looking at the products. 00:06:53.810 --> 00:06:55.590 align:middle line:84% So if you define a sequence that is 00:06:55.590 --> 00:07:01.430 align:middle line:84% given by the product of the two, then cn converge also. 00:07:01.430 --> 00:07:05.670 align:middle line:84% And the limit is the product of the limits. 00:07:05.670 --> 00:07:13.430 align:middle line:84% And then the fourth rule is that if an is not equal to 0 00:07:13.430 --> 00:07:20.910 align:middle line:84% for all n and the limit here is also not equal to 0, 00:07:20.910 --> 00:07:25.470 align:middle line:84% then if you're looking at 1 over an, then those converge to 1 00:07:25.470 --> 00:07:27.830 align:middle line:90% over a. 00:07:27.830 --> 00:07:32.030 align:middle line:90% So those are the four rules. 00:07:32.030 --> 00:07:34.510 align:middle line:90% Let me just write it as 4. 00:07:34.510 --> 00:07:37.270 align:middle line:84% And again, it's not like there's any particular convention 00:07:37.270 --> 00:07:38.410 align:middle line:90% with the ordering. 00:07:38.410 --> 00:07:42.390 align:middle line:84% And I think I might have exchanged the order of 1 and 2 00:07:42.390 --> 00:07:43.170 align:middle line:90% last time. 00:07:43.170 --> 00:07:48.760 align:middle line:90% 00:07:48.760 --> 00:07:52.690 align:middle line:84% Now, let's see an example of how that works. 00:07:52.690 --> 00:07:58.480 align:middle line:90% 00:07:58.480 --> 00:08:04.920 align:middle line:84% Let's define a sequence n squared plus 1, 00:08:04.920 --> 00:08:10.760 align:middle line:90% n squared plus n plus 1. 00:08:10.760 --> 00:08:14.320 align:middle line:84% Let's look at this sequence here for each n that 00:08:14.320 --> 00:08:19.840 align:middle line:84% gives us, actually, a rational number, but it's a real number. 00:08:19.840 --> 00:08:22.520 align:middle line:84% Now in order to understand this, I 00:08:22.520 --> 00:08:31.160 align:middle line:84% can rewrite this one here, as I can factor out n squared here. 00:08:31.160 --> 00:08:32.620 align:middle line:90% So I can write it like this. 00:08:32.620 --> 00:08:44.049 align:middle line:90% 00:08:44.049 --> 00:08:44.710 align:middle line:90% Like that. 00:08:44.710 --> 00:08:46.170 align:middle line:84% And then, of course, I can get rid 00:08:46.170 --> 00:08:51.230 align:middle line:84% of the n squared in both numerator and denominator. 00:08:51.230 --> 00:08:53.750 align:middle line:90% 00:08:53.750 --> 00:08:56.790 align:middle line:84% So you may as well think about the sequence as this thing here. 00:08:56.790 --> 00:09:00.470 align:middle line:90% 00:09:00.470 --> 00:09:10.890 align:middle line:84% So now, I can just first look at the sequence which is 1 over n. 00:09:10.890 --> 00:09:15.690 align:middle line:84% If I'm looking at this sequence here, 1 over n, this 00:09:15.690 --> 00:09:17.390 align:middle line:90% here converges to 0. 00:09:17.390 --> 00:09:20.150 align:middle line:84% This essentially just follows from the Archimedean property. 00:09:20.150 --> 00:09:26.530 align:middle line:84% But we've already seen this, that 1 over n converges to 0. 00:09:26.530 --> 00:09:30.690 align:middle line:84% And the same holds for 1 over n squared. 00:09:30.690 --> 00:09:33.810 align:middle line:90% This also converges to 0. 00:09:33.810 --> 00:09:36.690 align:middle line:84% And so if you're looking at the sequence, 00:09:36.690 --> 00:09:42.500 align:middle line:84% you could even look at the sequence, let's call this xn. 00:09:42.500 --> 00:09:44.020 align:middle line:90% This here is yn. 00:09:44.020 --> 00:09:48.700 align:middle line:84% I'm going to run out of character very quickly, zn. 00:09:48.700 --> 00:09:53.380 align:middle line:84% This is just going to be the sequence, which is constant 1. 00:09:53.380 --> 00:09:59.060 align:middle line:84% Well, if it's constant 1, then obviously it's converging to 1. 00:09:59.060 --> 00:10:06.060 align:middle line:84% So now I can think about the sequence being 00:10:06.060 --> 00:10:18.140 align:middle line:84% to be equal to z plus y, zn plus yn. 00:10:18.140 --> 00:10:20.040 align:middle line:90% This here is a sequence. 00:10:20.040 --> 00:10:21.860 align:middle line:90% Here is what yn is. 00:10:21.860 --> 00:10:23.900 align:middle line:90% Here is what is zn is. 00:10:23.900 --> 00:10:28.060 align:middle line:84% So if you add these two, this is converging to 1. 00:10:28.060 --> 00:10:30.220 align:middle line:90% This is converging to 0. 00:10:30.220 --> 00:10:32.940 align:middle line:84% By the algebraic property of limits, 00:10:32.940 --> 00:10:36.160 align:middle line:84% this converge also to the sum of the limit. 00:10:36.160 --> 00:10:38.540 align:middle line:90% So this converges to 1. 00:10:38.540 --> 00:10:42.310 align:middle line:84% Now, I can also define a sequence 00:10:42.310 --> 00:10:47.630 align:middle line:84% where I'm just adding zn and then I'm adding xn 00:10:47.630 --> 00:10:48.870 align:middle line:90% and then I'm adding yn. 00:10:48.870 --> 00:10:52.812 align:middle line:90% 00:10:52.812 --> 00:10:58.310 align:middle line:84% This here converges to 1, this here to 0, this here to 0. 00:10:58.310 --> 00:11:02.850 align:middle line:84% So the sum here also converge to the sum of the limit. 00:11:02.850 --> 00:11:04.690 align:middle line:90% But the sum of the limit is 1. 00:11:04.690 --> 00:11:16.430 align:middle line:90% 00:11:16.430 --> 00:11:17.250 align:middle line:90% Put it up here. 00:11:17.250 --> 00:11:21.410 align:middle line:90% And then I'll just go over here. 00:11:21.410 --> 00:11:24.070 align:middle line:90% 00:11:24.070 --> 00:11:33.230 align:middle line:84% So now you have that cn converge to 1. 00:11:33.230 --> 00:11:35.710 align:middle line:90% And let's call this c. 00:11:35.710 --> 00:11:40.920 align:middle line:84% And so, in particular, all of these numbers here, 00:11:40.920 --> 00:11:43.180 align:middle line:84% these things here are all strictly positive. 00:11:43.180 --> 00:11:45.500 align:middle line:84% So each of the cn's are strictly positive. 00:11:45.500 --> 00:11:47.800 align:middle line:90% So they're definitely not 0. 00:11:47.800 --> 00:11:51.720 align:middle line:84% So the cn, so not only do they converge to 1, 00:11:51.720 --> 00:11:54.440 align:middle line:90% but the cn's are not equal to 0. 00:11:54.440 --> 00:11:57.380 align:middle line:90% And the limit c, well, that's 1. 00:11:57.380 --> 00:12:00.330 align:middle line:84% So that's, of course, not equal to 0 either. 00:12:00.330 --> 00:12:03.760 align:middle line:84% So you can apply the fourth rule here. 00:12:03.760 --> 00:12:10.480 align:middle line:84% The fourth rule for the sequence, for the sequence cn, 00:12:10.480 --> 00:12:13.600 align:middle line:84% so applying the fourth rule to the sequence cn, 00:12:13.600 --> 00:12:18.360 align:middle line:84% and then you have that 1 over cn converge to 1 over c, 00:12:18.360 --> 00:12:21.200 align:middle line:90% but c was 1, so this is 1. 00:12:21.200 --> 00:12:24.160 align:middle line:90% 00:12:24.160 --> 00:12:33.720 align:middle line:84% Now you can look at bn times 1 over cn. 00:12:33.720 --> 00:12:37.130 align:middle line:90% But bn times 1 over cn-- 00:12:37.130 --> 00:12:38.950 align:middle line:90% remember what bn was. 00:12:38.950 --> 00:12:40.570 align:middle line:90% Here's bn. 00:12:40.570 --> 00:12:44.890 align:middle line:84% And bn is really the numerator up here. 00:12:44.890 --> 00:12:46.890 align:middle line:84% It's the numerator up here when you 00:12:46.890 --> 00:12:51.970 align:middle line:84% have divided by n squared on both 00:12:51.970 --> 00:12:54.450 align:middle line:90% in the top and the numerator. 00:12:54.450 --> 00:13:00.070 align:middle line:84% So bn is the numerator and cn is the denominator. 00:13:00.070 --> 00:13:05.690 align:middle line:90% So bn divided by cn, this is an. 00:13:05.690 --> 00:13:08.310 align:middle line:90% And now this here converge. 00:13:08.310 --> 00:13:10.050 align:middle line:90% That's what we saw over there. 00:13:10.050 --> 00:13:13.530 align:middle line:84% Using the algebraic rule, it converges to 1. 00:13:13.530 --> 00:13:18.570 align:middle line:84% This here also converge to 1 by the algebraic rules. 00:13:18.570 --> 00:13:22.410 align:middle line:84% Number 3 of the algebraic rules is that for the product 00:13:22.410 --> 00:13:24.330 align:middle line:90% it will also converge. 00:13:24.330 --> 00:13:27.970 align:middle line:84% And it converges to the product of the limits. 00:13:27.970 --> 00:13:28.890 align:middle line:90% So that's 1. 00:13:28.890 --> 00:13:31.410 align:middle line:90% 00:13:31.410 --> 00:13:36.560 align:middle line:84% Obviously, this is like when you're looking at a sequence, 00:13:36.560 --> 00:13:40.260 align:middle line:84% it's often expressed in some form 00:13:40.260 --> 00:13:43.940 align:middle line:90% with some functions like this. 00:13:43.940 --> 00:13:46.580 align:middle line:90% And you do this reduction. 00:13:46.580 --> 00:13:50.280 align:middle line:84% And very quickly, this would be almost automatic. 00:13:50.280 --> 00:13:53.260 align:middle line:84% This is too easy almost, this one. 00:13:53.260 --> 00:13:54.900 align:middle line:84% This is the first time you see it, 00:13:54.900 --> 00:13:58.940 align:middle line:84% but you know in a couple of weeks this will feel too easy. 00:13:58.940 --> 00:14:01.360 align:middle line:84% Then there will be something more complicated. 00:14:01.360 --> 00:14:05.524 align:middle line:90% 00:14:05.524 --> 00:14:11.780 align:middle line:84% So that's the algebraic rules for sequences. 00:14:11.780 --> 00:14:13.540 align:middle line:84% The next thing I want to talk about 00:14:13.540 --> 00:14:19.940 align:middle line:84% is, I want to try to make more precise this about square. 00:14:19.940 --> 00:14:22.470 align:middle line:84% We made it pretty precise, but I want to think about-- 00:14:22.470 --> 00:14:25.180 align:middle line:90% 00:14:25.180 --> 00:14:28.380 align:middle line:84% I don't have a very good eraser, unfortunately. 00:14:28.380 --> 00:14:29.740 align:middle line:90% Let me just see. 00:14:29.740 --> 00:14:33.097 align:middle line:90% This is a better one. 00:14:33.097 --> 00:14:34.680 align:middle line:84% This one here will make it very dirty. 00:14:34.680 --> 00:14:38.630 align:middle line:90% 00:14:38.630 --> 00:14:44.050 align:middle line:84% So the next thing, we did define square root of 2. 00:14:44.050 --> 00:14:46.270 align:middle line:84% And we talked about that quite extensively, 00:14:46.270 --> 00:14:50.770 align:middle line:84% how square root of 2, how it really was defined. 00:14:50.770 --> 00:14:52.410 align:middle line:90% And it was a real number. 00:14:52.410 --> 00:14:57.910 align:middle line:84% And the real numbers followed from that, the field, 00:14:57.910 --> 00:15:01.350 align:middle line:90% the reals is complete. 00:15:01.350 --> 00:15:08.670 align:middle line:84% But we also, in the back of our mind, 00:15:08.670 --> 00:15:13.190 align:middle line:84% we had that we already sort of knew what square root of 2 00:15:13.190 --> 00:15:17.030 align:middle line:84% were because when you think about square root of 2, at least 00:15:17.030 --> 00:15:20.930 align:middle line:84% I think about square root of 2, I think about it as 1.4. 00:15:20.930 --> 00:15:24.030 align:middle line:84% Just because you can't really express it as a fraction. 00:15:24.030 --> 00:15:27.810 align:middle line:84% So you think about it as 1.414, et cetera. 00:15:27.810 --> 00:15:31.140 align:middle line:84% And then I don't remember the second, third, and fourth, 00:15:31.140 --> 00:15:31.640 align:middle line:90% whatever. 00:15:31.640 --> 00:15:34.826 align:middle line:84% But you may remember just a few of the digits. 00:15:34.826 --> 00:15:39.680 align:middle line:90% 00:15:39.680 --> 00:15:44.560 align:middle line:84% So that's sort of our intuition with 1.4-- 00:15:44.560 --> 00:15:46.920 align:middle line:90% with square root 2. 00:15:46.920 --> 00:15:49.720 align:middle line:84% But how do we make this more precise? 00:15:49.720 --> 00:15:53.920 align:middle line:84% And so to make this more precise, 00:15:53.920 --> 00:15:58.800 align:middle line:84% we want to look at particular sequences, what's 00:15:58.800 --> 00:16:02.800 align:middle line:90% called monotone sequences. 00:16:02.800 --> 00:16:07.080 align:middle line:90% 00:16:07.080 --> 00:16:10.440 align:middle line:90% So what is a monotone sequence? 00:16:10.440 --> 00:16:13.300 align:middle line:84% So there are two types of monotone sequences. 00:16:13.300 --> 00:16:16.410 align:middle line:84% There's one which is monotone increasing sequence. 00:16:16.410 --> 00:16:26.320 align:middle line:90% 00:16:26.320 --> 00:16:31.210 align:middle line:84% And then there's, similarly, monotone decreasing sequence. 00:16:31.210 --> 00:16:33.710 align:middle line:84% So a monotone increasing sequence, 00:16:33.710 --> 00:16:41.210 align:middle line:84% this just means that for all n, you 00:16:41.210 --> 00:16:45.450 align:middle line:84% have that the next element is larger 00:16:45.450 --> 00:16:48.290 align:middle line:84% or equal to the previous element. 00:16:48.290 --> 00:16:51.530 align:middle line:84% That's a monotone increasing sequence. 00:16:51.530 --> 00:16:54.910 align:middle line:84% Some people talk about strictly monotone increasing-- 00:16:54.910 --> 00:16:57.467 align:middle line:84% and then this has to be a strict inequality. 00:16:57.467 --> 00:17:01.290 align:middle line:90% 00:17:01.290 --> 00:17:04.450 align:middle line:84% The second thing is kind of the obvious. 00:17:04.450 --> 00:17:11.470 align:middle line:84% Parallel to this is monotone decreasing sequences. 00:17:11.470 --> 00:17:17.730 align:middle line:90% 00:17:17.730 --> 00:17:19.609 align:middle line:90% So there's these two types. 00:17:19.609 --> 00:17:25.050 align:middle line:84% And this just means that the next element, so an plus 1 00:17:25.050 --> 00:17:28.319 align:middle line:84% is less than or equal to the previous one. 00:17:28.319 --> 00:17:33.788 align:middle line:90% 00:17:33.788 --> 00:17:37.060 align:middle line:90% And so now there's a theorem. 00:17:37.060 --> 00:17:39.530 align:middle line:84% And I'll prove this in just a minute. 00:17:39.530 --> 00:17:43.100 align:middle line:90% 00:17:43.100 --> 00:17:45.940 align:middle line:84% And I want to come back to square root of 2. 00:17:45.940 --> 00:17:51.060 align:middle line:84% And then I want to look at another classical example 00:17:51.060 --> 00:17:52.290 align:middle line:90% of a monotone sequence. 00:17:52.290 --> 00:17:59.400 align:middle line:90% 00:17:59.400 --> 00:18:01.040 align:middle line:90% So this is a monotone sequence. 00:18:01.040 --> 00:18:03.910 align:middle line:84% And now let me state the monotone convergence theorem. 00:18:03.910 --> 00:18:17.415 align:middle line:90% 00:18:17.415 --> 00:18:18.790 align:middle line:84% So this is the following theorem. 00:18:18.790 --> 00:18:24.820 align:middle line:90% 00:18:24.820 --> 00:18:26.910 align:middle line:84% And there's two versions of this. 00:18:26.910 --> 00:18:29.890 align:middle line:84% There's one version for monotone increasing sequence. 00:18:29.890 --> 00:18:32.982 align:middle line:84% And then there's one version for monotone decreasing sequence. 00:18:32.982 --> 00:18:35.190 align:middle line:84% I'll just state one of them, because the other one is 00:18:35.190 --> 00:18:37.190 align:middle line:90% kind of obviously a parallel. 00:18:37.190 --> 00:18:48.270 align:middle line:84% So suppose that an is a monotone increasing 00:18:48.270 --> 00:18:57.640 align:middle line:90% sequence that is bounded. 00:18:57.640 --> 00:19:01.350 align:middle line:90% 00:19:01.350 --> 00:19:08.150 align:middle line:84% Bounded, so this just means that there exists, i.e. 00:19:08.150 --> 00:19:21.610 align:middle line:84% there exists some a, real number, so that you have a1. 00:19:21.610 --> 00:19:24.080 align:middle line:90% This says that it's monotone. 00:19:24.080 --> 00:19:26.280 align:middle line:84% This is monotone that they increase. 00:19:26.280 --> 00:19:30.960 align:middle line:84% But the claim is that now all of the an 00:19:30.960 --> 00:19:35.200 align:middle line:90% is less than or equal to this a. 00:19:35.200 --> 00:19:38.920 align:middle line:84% Any monotone increasing sequence is clearly bounded from below 00:19:38.920 --> 00:19:44.360 align:middle line:84% because already the first element is a lower bound. 00:19:44.360 --> 00:19:51.840 align:middle line:84% So if you have a monotone increasing sequence that is 00:19:51.840 --> 00:20:14.880 align:middle line:84% bounded, then it's convergent and with limit a, where a is 00:20:14.880 --> 00:20:17.640 align:middle line:90% the sup of the an's. 00:20:17.640 --> 00:20:28.690 align:middle line:90% 00:20:28.690 --> 00:20:34.730 align:middle line:84% So that is a monotone convergence sequence. 00:20:34.730 --> 00:20:39.730 align:middle line:84% And the main theorem about monotone convergence sequence 00:20:39.730 --> 00:20:42.690 align:middle line:84% is that if you have a monotone increasing 00:20:42.690 --> 00:20:47.690 align:middle line:84% sequence that is bounded, then it's also convergent. 00:20:47.690 --> 00:20:51.290 align:middle line:84% And likewise, if you have a monotone decreasing 00:20:51.290 --> 00:20:56.490 align:middle line:84% sequence that is bounded from below, 00:20:56.490 --> 00:20:59.330 align:middle line:90% then it is also convergent. 00:20:59.330 --> 00:21:03.370 align:middle line:90% And so maybe I will state it. 00:21:03.370 --> 00:21:04.920 align:middle line:84% But it's just the obvious parallel. 00:21:04.920 --> 00:21:09.382 align:middle line:90% 00:21:09.382 --> 00:21:14.580 align:middle line:84% So this is part 1 of the monotone convergence sequence. 00:21:14.580 --> 00:21:18.050 align:middle line:90% 00:21:18.050 --> 00:21:33.860 align:middle line:84% Part B is that if an is a monotone decreasing 00:21:33.860 --> 00:21:51.140 align:middle line:84% sequence that is bounded from below, 00:21:51.140 --> 00:22:00.940 align:middle line:84% then an is convergent with limit a, where a is now 00:22:00.940 --> 00:22:04.370 align:middle line:90% the inf of these numbers. 00:22:04.370 --> 00:22:16.150 align:middle line:90% 00:22:16.150 --> 00:22:19.910 align:middle line:84% Now, the first thing you might say-- 00:22:19.910 --> 00:22:22.210 align:middle line:84% and we'll come to square root of 2 in just a minute-- 00:22:22.210 --> 00:22:24.970 align:middle line:84% but you may say that this seems rather special. 00:22:24.970 --> 00:22:30.150 align:middle line:84% If you take a random sequence, obviously, it's 00:22:30.150 --> 00:22:31.656 align:middle line:90% not going to be monotone. 00:22:31.656 --> 00:22:36.590 align:middle line:90% 00:22:36.590 --> 00:22:39.430 align:middle line:90% But here's the thing. 00:22:39.430 --> 00:22:47.100 align:middle line:84% So let an be any bounded sequence. 00:22:47.100 --> 00:22:56.390 align:middle line:90% 00:22:56.390 --> 00:22:59.110 align:middle line:84% So bounded, again, means that it's bounded from above 00:22:59.110 --> 00:23:00.910 align:middle line:90% and bounded from below. 00:23:00.910 --> 00:23:05.070 align:middle line:84% So if you take any bounded sequence, 00:23:05.070 --> 00:23:15.682 align:middle line:84% then we can form two other sequences. 00:23:15.682 --> 00:23:17.140 align:middle line:84% Obviously, we can form many others. 00:23:17.140 --> 00:23:21.380 align:middle line:84% But here's two other very natural sequences. 00:23:21.380 --> 00:23:24.480 align:middle line:90% 00:23:24.480 --> 00:23:27.840 align:middle line:90% The first one I call bn. 00:23:27.840 --> 00:23:41.920 align:middle line:84% And bn is the sup, where you're starting at an, 00:23:41.920 --> 00:23:45.200 align:middle line:84% then you take the sup of all of the ones 00:23:45.200 --> 00:23:49.612 align:middle line:84% from an and outwards in the sequence. 00:23:49.612 --> 00:23:54.220 align:middle line:90% So this is one sequence. 00:23:54.220 --> 00:23:58.900 align:middle line:84% We know that the set of an's is bounded. 00:23:58.900 --> 00:24:03.180 align:middle line:84% So you can certainly take the sup, and you get a real number. 00:24:03.180 --> 00:24:05.880 align:middle line:90% That gives you another sequence. 00:24:05.880 --> 00:24:08.720 align:middle line:84% You can also look at cn similarly, 00:24:08.720 --> 00:24:10.640 align:middle line:84% where you take instead of the sup, 00:24:10.640 --> 00:24:20.730 align:middle line:84% you take the inf of the same set, like that. 00:24:20.730 --> 00:24:24.170 align:middle line:84% And so in particular, just to make sure 00:24:24.170 --> 00:24:34.410 align:middle line:84% it's totally clear that if you were looking at, say, bn, 00:24:34.410 --> 00:24:35.930 align:middle line:90% so what is bn? 00:24:35.930 --> 00:24:39.090 align:middle line:84% If you're looking at, say, b3, then this 00:24:39.090 --> 00:24:47.770 align:middle line:84% is the sup where you're starting at a3, a4, et cetera. 00:24:47.770 --> 00:24:52.170 align:middle line:84% And so you see, of course, that b3 here, 00:24:52.170 --> 00:24:57.850 align:middle line:84% well, obviously, b3 is bigger or equal to b2-- 00:24:57.850 --> 00:25:03.050 align:middle line:90% 00:25:03.050 --> 00:25:05.090 align:middle line:84% sorry, no, it's not bigger or equal to, 00:25:05.090 --> 00:25:08.490 align:middle line:84% it's less than or equal to b2 because, you 00:25:08.490 --> 00:25:14.660 align:middle line:84% see, in b2, the definition of b2, so let's just write b2 down. 00:25:14.660 --> 00:25:23.992 align:middle line:84% So b2 is when you take the sup here of a2 and then a3, a4, et 00:25:23.992 --> 00:25:26.540 align:middle line:90% cetera. 00:25:26.540 --> 00:25:29.060 align:middle line:90% But you see that b2 here-- 00:25:29.060 --> 00:25:31.420 align:middle line:84% both of them are defined to be sup-- 00:25:31.420 --> 00:25:36.500 align:middle line:84% but b2 is defined as sup over a larger set 00:25:36.500 --> 00:25:42.380 align:middle line:84% because, in addition to these elements, it also includes a2. 00:25:42.380 --> 00:25:48.940 align:middle line:84% So clearly, b2 here is bigger or equal to b3. 00:25:48.940 --> 00:26:00.900 align:middle line:84% And in fact, the same argument tells you that if you take bn, 00:26:00.900 --> 00:26:06.540 align:middle line:84% bn is bigger or equal to bn plus 1 00:26:06.540 --> 00:26:11.870 align:middle line:84% because this here is the sup over a larger set. 00:26:11.870 --> 00:26:18.030 align:middle line:84% It's sup over all the ones you take sup in here, plus an. 00:26:18.030 --> 00:26:21.590 align:middle line:90% So you have this property here. 00:26:21.590 --> 00:26:22.350 align:middle line:90% Yeah? 00:26:22.350 --> 00:26:24.505 align:middle line:84% AUDIENCE: When you say that the [INAUDIBLE] sup over a larger 00:26:24.505 --> 00:26:24.990 align:middle line:90% set-- 00:26:24.990 --> 00:26:25.350 align:middle line:90% TOBIAS COLDING: Yeah. 00:26:25.350 --> 00:26:27.267 align:middle line:84% AUDIENCE: The way that I was thinking about it 00:26:27.267 --> 00:26:31.030 align:middle line:84% is, like, well, if it said monotone increasing-- 00:26:31.030 --> 00:26:33.990 align:middle line:84% TOBIAS COLDING: If it's monotone increasing, 00:26:33.990 --> 00:26:40.590 align:middle line:84% then this here, then it becomes very special. 00:26:40.590 --> 00:26:48.630 align:middle line:84% So if the an was monotone increasing, then all of these, 00:26:48.630 --> 00:26:52.850 align:middle line:84% any elements here, would just be the a we talked about before. 00:26:52.850 --> 00:26:56.070 align:middle line:90% It would always be the limit. 00:26:56.070 --> 00:27:01.010 align:middle line:84% So it somehow wouldn't be that interesting a sequence. 00:27:01.010 --> 00:27:02.750 align:middle line:90% It would be a constant sequence. 00:27:02.750 --> 00:27:04.390 align:middle line:90% And it would be just the limit. 00:27:04.390 --> 00:27:05.390 align:middle line:90% AUDIENCE: OK, thank you. 00:27:05.390 --> 00:27:08.030 align:middle line:84% TOBIAS COLDING: But nevertheless, it's 00:27:08.030 --> 00:27:09.445 align:middle line:90% a good example. 00:27:09.445 --> 00:27:11.840 align:middle line:84% AUDIENCE: But I guess the inequality still holds, then. 00:27:11.840 --> 00:27:12.560 align:middle line:84% TOBIAS COLDING: Yeah, definitely, 00:27:12.560 --> 00:27:14.400 align:middle line:84% because it's just constant, exactly. 00:27:14.400 --> 00:27:14.960 align:middle line:90% AUDIENCE: Yeah, of course. 00:27:14.960 --> 00:27:16.460 align:middle line:84% TOBIAS COLDING: It's a good example. 00:27:16.460 --> 00:27:19.120 align:middle line:84% But it's just kind of almost like too simple 00:27:19.120 --> 00:27:23.280 align:middle line:84% because you get a constant sequence, right? 00:27:23.280 --> 00:27:28.340 align:middle line:84% OK, so for this sequence that we defined, so we have, 00:27:28.340 --> 00:27:32.360 align:middle line:84% again, if you take any sequence, then 00:27:32.360 --> 00:27:36.040 align:middle line:84% you have two sequences bn and cn. 00:27:36.040 --> 00:27:41.080 align:middle line:84% And bn here is now a decreasing sequence. 00:27:41.080 --> 00:27:44.940 align:middle line:84% And instead of writing decreasing sequence, 00:27:44.940 --> 00:27:49.720 align:middle line:84% then it's much easier to write it like this. 00:27:49.720 --> 00:27:51.170 align:middle line:90% This is pretty standard. 00:27:51.170 --> 00:27:54.040 align:middle line:90% 00:27:54.040 --> 00:27:57.600 align:middle line:84% If you just look quickly at it, you know what this means. 00:27:57.600 --> 00:28:00.160 align:middle line:90% You don't have to write it out. 00:28:00.160 --> 00:28:03.600 align:middle line:90% But let's look at cn also. 00:28:03.600 --> 00:28:08.910 align:middle line:84% Remember that we assume that an could be any sequence, 00:28:08.910 --> 00:28:11.290 align:middle line:84% but it is supposed to be bounded. 00:28:11.290 --> 00:28:15.250 align:middle line:84% So this is now a monotone decreasing sequence 00:28:15.250 --> 00:28:16.390 align:middle line:90% that is bounded. 00:28:16.390 --> 00:28:17.630 align:middle line:90% So it actually has a limit. 00:28:17.630 --> 00:28:20.330 align:middle line:90% 00:28:20.330 --> 00:28:22.340 align:middle line:90% The cn's, what about the cn's? 00:28:22.340 --> 00:28:28.130 align:middle line:90% 00:28:28.130 --> 00:28:36.810 align:middle line:84% The cn's, well, cn was defined to be the inf starting 00:28:36.810 --> 00:28:39.796 align:middle line:90% at an like that. 00:28:39.796 --> 00:28:45.522 align:middle line:90% 00:28:45.522 --> 00:28:49.130 align:middle line:84% So if you're starting like that, this 00:28:49.130 --> 00:28:56.070 align:middle line:84% means that if you take cn plus 1, then you take the infimum, 00:28:56.070 --> 00:28:59.470 align:middle line:84% but you don't include this element. 00:28:59.470 --> 00:29:02.690 align:middle line:84% So it's an infimum over fewer elements. 00:29:02.690 --> 00:29:08.500 align:middle line:84% But then this n must be larger or equal to that. 00:29:08.500 --> 00:29:14.740 align:middle line:84% So you see that the sequence cn is monotone 00:29:14.740 --> 00:29:17.300 align:middle line:90% increasing again, is bounded. 00:29:17.300 --> 00:29:22.760 align:middle line:84% And just copying from over there, the bn was decreasing. 00:29:22.760 --> 00:29:27.280 align:middle line:90% 00:29:27.280 --> 00:29:29.300 align:middle line:90% This is an important thing. 00:29:29.300 --> 00:29:31.060 align:middle line:84% It's also something that you would 00:29:31.060 --> 00:29:33.700 align:middle line:84% see in a lot of different contexts 00:29:33.700 --> 00:29:37.600 align:middle line:84% where you define something, some sort of something, 00:29:37.600 --> 00:29:40.420 align:middle line:90% and then you have this property. 00:29:40.420 --> 00:29:46.240 align:middle line:84% And so now the thing is that if you're looking at, so a priori, 00:29:46.240 --> 00:29:48.820 align:middle line:84% it might not be so clear to compare 00:29:48.820 --> 00:29:50.580 align:middle line:84% a random element in this sequence 00:29:50.580 --> 00:29:52.540 align:middle line:90% to a random element in that. 00:29:52.540 --> 00:30:03.310 align:middle line:84% But what is clear is that if you try to compare bn with cn, 00:30:03.310 --> 00:30:05.510 align:middle line:90% it's the same set. 00:30:05.510 --> 00:30:06.950 align:middle line:90% It's the same set. 00:30:06.950 --> 00:30:09.990 align:middle line:84% In one, you take the sup, the other you take the inf. 00:30:09.990 --> 00:30:14.370 align:middle line:84% So of course bn is bigger or equal to cn. 00:30:14.370 --> 00:30:15.870 align:middle line:90% So you have this, right? 00:30:15.870 --> 00:30:19.310 align:middle line:84% And so now you see that if you take like a random element 00:30:19.310 --> 00:30:28.650 align:middle line:84% here, if it was like further out, if it was cn, further out, 00:30:28.650 --> 00:30:31.370 align:middle line:84% I just write it as something further out. 00:30:31.370 --> 00:30:33.750 align:middle line:84% So you're adding something else to it. 00:30:33.750 --> 00:30:35.910 align:middle line:84% So this would be like further out. 00:30:35.910 --> 00:30:39.590 align:middle line:84% We know that this sequence here-- 00:30:39.590 --> 00:30:41.230 align:middle line:90% anyway, so you get-- 00:30:41.230 --> 00:30:47.010 align:middle line:84% sorry, now you're getting this here like that. 00:30:47.010 --> 00:30:50.030 align:middle line:90% 00:30:50.030 --> 00:30:52.870 align:middle line:90% And so you have this. 00:30:52.870 --> 00:30:57.190 align:middle line:84% What I should have done is that this sequence here, so 00:30:57.190 --> 00:31:00.230 align:middle line:90% let me draw it like this. 00:31:00.230 --> 00:31:05.080 align:middle line:84% So I have that, here is maybe c1. 00:31:05.080 --> 00:31:09.240 align:middle line:90% Here is b1. 00:31:09.240 --> 00:31:16.360 align:middle line:84% And now the next element, the sequence cn is increasing 00:31:16.360 --> 00:31:21.440 align:middle line:84% and the sequence bn is decreasing. 00:31:21.440 --> 00:31:23.520 align:middle line:90% But you always have this here. 00:31:23.520 --> 00:31:28.040 align:middle line:84% So c2 have to lie to the right of this. 00:31:28.040 --> 00:31:32.000 align:middle line:84% But it has to lie to the left of b2. 00:31:32.000 --> 00:31:34.560 align:middle line:90% So you're getting this here. 00:31:34.560 --> 00:31:40.840 align:middle line:84% So you see that this gives you a way to compare that the cn will 00:31:40.840 --> 00:31:44.080 align:middle line:84% always lie to the left of the bn's. 00:31:44.080 --> 00:31:47.000 align:middle line:90% 00:31:47.000 --> 00:31:55.280 align:middle line:84% So now what we have done here is we took any bounded sequence 00:31:55.280 --> 00:31:59.558 align:middle line:84% and we formed two monotone sequences. 00:31:59.558 --> 00:32:01.630 align:middle line:90% They are both monotone. 00:32:01.630 --> 00:32:08.570 align:middle line:90% 00:32:08.570 --> 00:32:11.810 align:middle line:84% So this means that now from the monotone convergence that we 00:32:11.810 --> 00:32:17.210 align:middle line:84% haven't quite proven yet, but from the monotone convergence 00:32:17.210 --> 00:32:20.170 align:middle line:84% theorem, because these are bounded, 00:32:20.170 --> 00:32:26.690 align:middle line:90% we have that bn are converging. 00:32:26.690 --> 00:32:27.880 align:middle line:90% They're going down. 00:32:27.880 --> 00:32:29.190 align:middle line:90% So they're converging. 00:32:29.190 --> 00:32:32.410 align:middle line:84% So one often would write it like this more suggestively. 00:32:32.410 --> 00:32:37.290 align:middle line:84% That is converging downwards to a limit b. 00:32:37.290 --> 00:32:42.430 align:middle line:84% And the cn is converging upwards to a limit c. 00:32:42.430 --> 00:32:45.690 align:middle line:90% 00:32:45.690 --> 00:32:49.630 align:middle line:84% And so what you get is because of all of these elements, 00:32:49.630 --> 00:32:54.370 align:middle line:84% all the cn's lie to the left of the bn's. 00:32:54.370 --> 00:32:57.960 align:middle line:84% This means that, and we'll prove this later, 00:32:57.960 --> 00:33:02.500 align:middle line:84% but it seems probably pretty obvious that the limit here, c 00:33:02.500 --> 00:33:04.200 align:middle line:90% is less than or equal to b. 00:33:04.200 --> 00:33:07.160 align:middle line:90% 00:33:07.160 --> 00:33:09.650 align:middle line:84% So we'll come back to this also next time. 00:33:09.650 --> 00:33:13.580 align:middle line:90% 00:33:13.580 --> 00:33:15.220 align:middle line:84% If I have time, I'll talk a little bit 00:33:15.220 --> 00:33:16.900 align:middle line:90% about Cauchy sequence. 00:33:16.900 --> 00:33:19.430 align:middle line:84% But I won't really prove too much about it. 00:33:19.430 --> 00:33:23.260 align:middle line:84% But this here will play an important role 00:33:23.260 --> 00:33:29.700 align:middle line:84% when we prove the basic theorem about Cauchy sequences. 00:33:29.700 --> 00:33:35.060 align:middle line:84% The takeaway here at the moment is just, 00:33:35.060 --> 00:33:39.740 align:middle line:84% you have this concept of a monotone either increasing 00:33:39.740 --> 00:33:41.740 align:middle line:90% or decreasing sequence. 00:33:41.740 --> 00:33:45.480 align:middle line:84% If such a sequence is bounded, then it's convergent. 00:33:45.480 --> 00:33:47.900 align:middle line:84% That's a monotone convergence theorem. 00:33:47.900 --> 00:33:53.840 align:middle line:84% And again, for any sequence, any bounded sequence, 00:33:53.840 --> 00:33:57.390 align:middle line:84% there's a very natural monotone increasing sequence 00:33:57.390 --> 00:33:59.800 align:middle line:84% and a very natural monotone decreasing sequence. 00:33:59.800 --> 00:34:04.030 align:middle line:90% 00:34:04.030 --> 00:34:08.409 align:middle line:84% Now, let's try to prove the monotone convergence theorem. 00:34:08.409 --> 00:34:13.550 align:middle line:84% And then let's get back how to think about square root of 2 00:34:13.550 --> 00:34:16.250 align:middle line:84% in a more intuitive way using this. 00:34:16.250 --> 00:34:28.670 align:middle line:90% 00:34:28.670 --> 00:34:32.530 align:middle line:84% OK, so now the proof of the monotone convergence theorem. 00:34:32.530 --> 00:34:54.560 align:middle line:90% 00:34:54.560 --> 00:34:58.120 align:middle line:84% So now let me just remind you what we have. 00:34:58.120 --> 00:35:02.240 align:middle line:84% So I'm only going to prove one of the two versions of it 00:35:02.240 --> 00:35:04.520 align:middle line:84% where the sequence is monotone increasing 00:35:04.520 --> 00:35:09.640 align:middle line:84% because the other one is just the same 00:35:09.640 --> 00:35:13.320 align:middle line:90% with obvious modifications. 00:35:13.320 --> 00:35:17.080 align:middle line:90% So this is the assumption. 00:35:17.080 --> 00:35:18.460 align:middle line:90% So this is assumption. 00:35:18.460 --> 00:35:24.760 align:middle line:90% 00:35:24.760 --> 00:35:29.640 align:middle line:84% So this is just that the next element is larger 00:35:29.640 --> 00:35:31.400 align:middle line:90% or equal to the previous one. 00:35:31.400 --> 00:35:35.880 align:middle line:84% And they're all bounded by this a. 00:35:35.880 --> 00:35:46.980 align:middle line:84% Now, we set little a equal to the sup of the an. 00:35:46.980 --> 00:35:59.110 align:middle line:90% 00:35:59.110 --> 00:36:10.290 align:middle line:84% We want to show that an converts to a. 00:36:10.290 --> 00:36:15.670 align:middle line:84% Now to do that what we need is we need this stuff about that, 00:36:15.670 --> 00:36:22.010 align:middle line:84% for a given epsilon, there exists a capital N. 00:36:22.010 --> 00:36:37.050 align:middle line:84% So now let me first make the observation that a here, 00:36:37.050 --> 00:36:41.880 align:middle line:90% since a here is the sup-- 00:36:41.880 --> 00:36:50.980 align:middle line:90% 00:36:50.980 --> 00:36:55.460 align:middle line:84% remember that the sup of a set is the smallest upper bound. 00:36:55.460 --> 00:37:00.620 align:middle line:90% 00:37:00.620 --> 00:37:03.760 align:middle line:84% So a is, in particular, an upper bound. 00:37:03.760 --> 00:37:11.900 align:middle line:84% So you have that an is less than a for all n, 00:37:11.900 --> 00:37:15.820 align:middle line:90% since a is an upper bound. 00:37:15.820 --> 00:37:17.480 align:middle line:84% It's actually the smallest upper bound. 00:37:17.480 --> 00:37:19.590 align:middle line:84% But for this you just need it's an upper bound. 00:37:19.590 --> 00:37:32.940 align:middle line:90% 00:37:32.940 --> 00:37:35.190 align:middle line:90% I'm going to look at a minus an. 00:37:35.190 --> 00:37:37.740 align:middle line:90% 00:37:37.740 --> 00:37:41.660 align:middle line:84% And what I want to prove is that for a given epsilon, 00:37:41.660 --> 00:37:47.020 align:middle line:84% if I'm sufficiently far out, if little n is 00:37:47.020 --> 00:37:51.350 align:middle line:84% bigger than some capital N, then this here is less than epsilon. 00:37:51.350 --> 00:37:52.910 align:middle line:90% That's what I want to prove. 00:37:52.910 --> 00:37:55.570 align:middle line:84% But this thing here, you see this here, 00:37:55.570 --> 00:37:58.350 align:middle line:84% I don't need the absolute value because I already know 00:37:58.350 --> 00:37:59.730 align:middle line:90% that this is bigger than that. 00:37:59.730 --> 00:38:02.990 align:middle line:84% So I don't need the absolute value. 00:38:02.990 --> 00:38:05.070 align:middle line:90% So this is what I want to prove. 00:38:05.070 --> 00:38:10.110 align:middle line:84% I want to prove that given epsilon greater than 0, 00:38:10.110 --> 00:38:10.700 align:middle line:90% they exists-- 00:38:10.700 --> 00:38:13.830 align:middle line:90% 00:38:13.830 --> 00:38:16.270 align:middle line:84% let me just write a little larger here. 00:38:16.270 --> 00:38:21.590 align:middle line:84% So what I want to prove is that given epsilon greater than 0, 00:38:21.590 --> 00:38:32.430 align:middle line:84% there exists a capital N such that if little n is bigger 00:38:32.430 --> 00:38:36.990 align:middle line:84% or equal to this capital N, then a minus an, which I know 00:38:36.990 --> 00:38:39.750 align:middle line:84% is positive, I just need to prove 00:38:39.750 --> 00:38:41.080 align:middle line:90% that this is less than epsilon. 00:38:41.080 --> 00:38:47.320 align:middle line:90% 00:38:47.320 --> 00:38:49.120 align:middle line:90% So now how do we do that? 00:38:49.120 --> 00:38:58.260 align:middle line:84% Well, at the moment I've just used that a was an upper bound. 00:38:58.260 --> 00:39:01.840 align:middle line:84% Now I'm going to use that as the least upper bound. 00:39:01.840 --> 00:39:15.800 align:middle line:84% So since a is the least upper bound, 00:39:15.800 --> 00:39:24.200 align:middle line:84% then a minus epsilon is not an upper bound. 00:39:24.200 --> 00:39:32.385 align:middle line:90% 00:39:32.385 --> 00:39:35.300 align:middle line:84% a minus epsilon is not an upper bound. 00:39:35.300 --> 00:39:37.150 align:middle line:84% So this means that there must exist-- 00:39:37.150 --> 00:39:39.920 align:middle line:90% 00:39:39.920 --> 00:39:50.180 align:middle line:84% there exists a capital N so that an is bigger or equal-- 00:39:50.180 --> 00:39:53.580 align:middle line:84% so capital N, just this one here, 00:39:53.580 --> 00:39:57.340 align:middle line:84% is bigger than a minus epsilon because otherwise, it 00:39:57.340 --> 00:39:59.620 align:middle line:90% would have been an upper bound. 00:39:59.620 --> 00:40:01.460 align:middle line:90% But it's actually strictly. 00:40:01.460 --> 00:40:04.740 align:middle line:84% Maybe I should write strictly because else 00:40:04.740 --> 00:40:05.880 align:middle line:90% would have [INAUDIBLE]. 00:40:05.880 --> 00:40:09.700 align:middle line:90% 00:40:09.700 --> 00:40:15.100 align:middle line:84% But I already know that this thing here is less than a. 00:40:15.100 --> 00:40:17.846 align:middle line:90% 00:40:17.846 --> 00:40:21.500 align:middle line:84% And also, in fact, I know more than this. 00:40:21.500 --> 00:40:26.660 align:middle line:84% I know that this thing here is less than or equal to a little n 00:40:26.660 --> 00:40:30.580 align:middle line:84% as long as little n is bigger than capital N. That was just 00:40:30.580 --> 00:40:32.780 align:middle line:84% because the sequence was increasing. 00:40:32.780 --> 00:40:37.420 align:middle line:84% And all of them is smaller than this upper bound. 00:40:37.420 --> 00:40:42.560 align:middle line:84% So now you see, so here you have a, 00:40:42.560 --> 00:40:46.990 align:middle line:84% all the elements an lie to the left. 00:40:46.990 --> 00:40:51.630 align:middle line:84% And if you're looking at just the one from a capital 00:40:51.630 --> 00:40:55.670 align:middle line:84% N outwards, then here you have a minus epsilon. 00:40:55.670 --> 00:41:00.110 align:middle line:84% And then they lie between this one here and that. 00:41:00.110 --> 00:41:03.110 align:middle line:84% In particular, they lie to the right of this 00:41:03.110 --> 00:41:04.790 align:middle line:90% and to the left of that. 00:41:04.790 --> 00:41:06.590 align:middle line:84% So this means that they're squished 00:41:06.590 --> 00:41:09.630 align:middle line:90% in this little interval. 00:41:09.630 --> 00:41:13.550 align:middle line:84% So this difference here, this difference from here to here 00:41:13.550 --> 00:41:16.430 align:middle line:84% is certainly bounded by the distance 00:41:16.430 --> 00:41:17.740 align:middle line:90% to here, which was epsilon. 00:41:17.740 --> 00:41:21.400 align:middle line:90% 00:41:21.400 --> 00:41:25.060 align:middle line:84% So this proves that this thing here is less than epsilon. 00:41:25.060 --> 00:41:28.355 align:middle line:90% 00:41:28.355 --> 00:41:33.310 align:middle line:84% And, well, an obvious modification-- 00:41:33.310 --> 00:41:36.450 align:middle line:90% the proof that if you take-- 00:41:36.450 --> 00:41:39.270 align:middle line:84% so this was an increasing sequence-- 00:41:39.270 --> 00:41:43.760 align:middle line:84% the proof that if you have a decreasing sequence that 00:41:43.760 --> 00:41:49.120 align:middle line:84% is bounded, that the limit of that is the inf, 00:41:49.120 --> 00:41:54.620 align:middle line:84% the proof with obvious changes is the same. 00:41:54.620 --> 00:41:59.470 align:middle line:90% 00:41:59.470 --> 00:42:03.240 align:middle line:84% So now let's come back to square root of 2. 00:42:03.240 --> 00:42:07.280 align:middle line:84% And so we have this idea that square root 2, 00:42:07.280 --> 00:42:14.720 align:middle line:84% in a way, the way to just get an idea about how large it is, 00:42:14.720 --> 00:42:18.580 align:middle line:84% you think about it as 1 and then a number of digits afterwards. 00:42:18.580 --> 00:42:20.540 align:middle line:90% That's a kind of approximation. 00:42:20.540 --> 00:42:23.412 align:middle line:90% 00:42:23.412 --> 00:42:26.170 align:middle line:84% So let's try to make this more precise. 00:42:26.170 --> 00:42:47.410 align:middle line:90% 00:42:47.410 --> 00:42:56.530 align:middle line:84% I'm going to define two sequences an and bn. 00:42:56.530 --> 00:43:10.370 align:middle line:84% So we'll define two sequences an and bn. 00:43:10.370 --> 00:43:18.930 align:middle line:84% And an will really be an, where an 00:43:18.930 --> 00:43:25.480 align:middle line:84% is going to be bn divided by 10 to the power n minus 1. 00:43:25.480 --> 00:43:28.250 align:middle line:90% 00:43:28.250 --> 00:43:34.410 align:middle line:84% Remember that anything, if it's 10 to the power 0, 00:43:34.410 --> 00:43:37.122 align:middle line:90% by definition, that's 1. 00:43:37.122 --> 00:43:39.060 align:middle line:84% So this here, in particular, means 00:43:39.060 --> 00:43:43.130 align:middle line:90% that a1 should be equal to b1. 00:43:43.130 --> 00:43:48.020 align:middle line:90% 00:43:48.020 --> 00:43:49.800 align:middle line:84% And we're going to define it like this. 00:43:49.800 --> 00:43:53.060 align:middle line:90% 00:43:53.060 --> 00:44:00.300 align:middle line:84% And we are going to define the bn. 00:44:00.300 --> 00:44:08.540 align:middle line:84% And the sequences here, so there will be some properties here, 00:44:08.540 --> 00:44:25.580 align:middle line:84% that the sequence an is monotone increasing. 00:44:25.580 --> 00:44:28.740 align:middle line:90% 00:44:28.740 --> 00:44:32.900 align:middle line:84% an are just real numbers that is monotone increasing. 00:44:32.900 --> 00:44:37.320 align:middle line:90% 00:44:37.320 --> 00:44:44.130 align:middle line:84% And the an is going to be defined. 00:44:44.130 --> 00:44:46.880 align:middle line:90% And an is defined like that. 00:44:46.880 --> 00:44:53.750 align:middle line:90% 00:44:53.750 --> 00:44:57.910 align:middle line:84% And the bn are just natural numbers. 00:44:57.910 --> 00:45:00.940 align:middle line:90% 00:45:00.940 --> 00:45:03.430 align:middle line:90% So an is defined from bn. 00:45:03.430 --> 00:45:06.210 align:middle line:84% It's really an that we are interested in. 00:45:06.210 --> 00:45:08.810 align:middle line:84% But bn is going to be a useful tool here. 00:45:08.810 --> 00:45:12.950 align:middle line:90% 00:45:12.950 --> 00:45:14.678 align:middle line:84% And the bn to some extent is very simple, 00:45:14.678 --> 00:45:16.220 align:middle line:84% because they're just natural numbers. 00:45:16.220 --> 00:45:20.550 align:middle line:90% 00:45:20.550 --> 00:45:25.150 align:middle line:90% So now we define bn. 00:45:25.150 --> 00:45:55.040 align:middle line:84% So bn is the largest natural number such that bn squared is 00:45:55.040 --> 00:46:03.780 align:middle line:84% lesser or equal to 2 times 10 to the power 2n minus 2. 00:46:03.780 --> 00:46:08.980 align:middle line:90% 00:46:08.980 --> 00:46:13.960 align:middle line:84% So the sets of natural numbers that is bounded by that, that's 00:46:13.960 --> 00:46:15.060 align:middle line:90% like a finite set. 00:46:15.060 --> 00:46:17.200 align:middle line:84% So you just take the max or sup or whatever, 00:46:17.200 --> 00:46:18.340 align:middle line:90% the max of that set. 00:46:18.340 --> 00:46:21.000 align:middle line:90% It's just the max of that set. 00:46:21.000 --> 00:46:22.290 align:middle line:90% So that's a natural number. 00:46:22.290 --> 00:46:29.840 align:middle line:90% 00:46:29.840 --> 00:46:39.170 align:middle line:84% And now, again, we define, so now we have the bn's. 00:46:39.170 --> 00:46:40.520 align:middle line:90% And then again an's. 00:46:40.520 --> 00:46:46.850 align:middle line:90% 00:46:46.850 --> 00:46:53.670 align:middle line:84% The an are defined to be bn divided by 10 to the n minus 1. 00:46:53.670 --> 00:47:00.110 align:middle line:90% 00:47:00.110 --> 00:47:02.890 align:middle line:84% So now the first thing you observe 00:47:02.890 --> 00:47:10.870 align:middle line:84% is that if n is equal to 1, then this thing here is 0. 00:47:10.870 --> 00:47:12.890 align:middle line:90% So this here is 1. 00:47:12.890 --> 00:47:15.790 align:middle line:84% So it's the largest natural number 00:47:15.790 --> 00:47:18.330 align:middle line:84% so that the square is less than 2. 00:47:18.330 --> 00:47:25.650 align:middle line:84% So this means that bn, so n equal to 1, bn is equal to 1. 00:47:25.650 --> 00:47:30.340 align:middle line:84% And because this thing here is also 0, 00:47:30.340 --> 00:47:33.140 align:middle line:90% so this denominator is 1. 00:47:33.140 --> 00:47:37.980 align:middle line:90% So a1 is 1. 00:47:37.980 --> 00:47:39.200 align:middle line:90% b1 is 1. 00:47:39.200 --> 00:47:42.700 align:middle line:90% And a1 has to be the same as b1. 00:47:42.700 --> 00:47:44.000 align:middle line:90% And so that's also 1. 00:47:44.000 --> 00:47:46.606 align:middle line:90% 00:47:46.606 --> 00:47:49.860 align:middle line:90% So that's how it starts. 00:47:49.860 --> 00:47:57.420 align:middle line:84% And now we want to prove that the sequence an is 00:47:57.420 --> 00:47:58.900 align:middle line:90% monotone increasing. 00:47:58.900 --> 00:48:01.700 align:middle line:84% We want to prove that an is monotone increasing. 00:48:01.700 --> 00:48:04.700 align:middle line:84% So let's just try to understand that. 00:48:04.700 --> 00:48:07.460 align:middle line:84% And so again, for monotone increasing, 00:48:07.460 --> 00:48:10.600 align:middle line:84% you just need to compare two consecutive elements. 00:48:10.600 --> 00:48:15.180 align:middle line:90% 00:48:15.180 --> 00:48:20.260 align:middle line:84% So this here, remember that there's 00:48:20.260 --> 00:48:23.460 align:middle line:84% ways to write a formal proof and then there's 00:48:23.460 --> 00:48:26.680 align:middle line:84% things you have on the scrap paper to get the idea. 00:48:26.680 --> 00:48:29.230 align:middle line:84% So this would be what we're going on, 00:48:29.230 --> 00:48:32.550 align:middle line:90% maybe the first statement is. 00:48:32.550 --> 00:48:35.390 align:middle line:84% You could certainly put it as part of the proof. 00:48:35.390 --> 00:48:42.930 align:middle line:84% We want to show that an is less than or equal to an plus 1. 00:48:42.930 --> 00:48:47.270 align:middle line:90% 00:48:47.270 --> 00:49:03.110 align:middle line:84% And so this is equivalent to show that since an here 00:49:03.110 --> 00:49:08.190 align:middle line:84% is equal to bn divided by 10 to the power n minus 2, 00:49:08.190 --> 00:49:11.590 align:middle line:84% you want to prove that this thing here is less than an 00:49:11.590 --> 00:49:23.030 align:middle line:84% plus 1, which is bn plus 1 over 10 to the n plus 1 minus 1. 00:49:23.030 --> 00:49:31.600 align:middle line:84% So this here is bn plus 1 over 10 to the power n. 00:49:31.600 --> 00:49:32.940 align:middle line:90% That's what you want to prove. 00:49:32.940 --> 00:49:37.140 align:middle line:90% 00:49:37.140 --> 00:49:45.160 align:middle line:84% So you see that we want to show, this is what we want to prove, 00:49:45.160 --> 00:49:48.000 align:middle line:84% that this thing here is less than this. 00:49:48.000 --> 00:49:52.920 align:middle line:84% If you multiply by 10 to the power n on both sides, 00:49:52.920 --> 00:49:56.560 align:middle line:90% then you get over here. 00:49:56.560 --> 00:50:02.240 align:middle line:84% And after you reduce it, you get 10 bn, 00:50:02.240 --> 00:50:18.040 align:middle line:84% so you need to show that 10 bn is less than 00:50:18.040 --> 00:50:22.360 align:middle line:90% or equal to bn plus 1. 00:50:22.360 --> 00:50:23.970 align:middle line:90% That's what we need to prove. 00:50:23.970 --> 00:50:27.250 align:middle line:84% And again, this just comes from that you 00:50:27.250 --> 00:50:30.370 align:middle line:84% want to prove-- forget about the a now-- you can just think about 00:50:30.370 --> 00:50:34.210 align:middle line:84% that we need to prove that this thing here is less than this. 00:50:34.210 --> 00:50:37.970 align:middle line:84% Multiply over by 10 to the n on both sides, 00:50:37.970 --> 00:50:40.590 align:middle line:84% then there's still 10 times bn left. 00:50:40.590 --> 00:50:44.110 align:middle line:90% And here it's just bn plus 1. 00:50:44.110 --> 00:50:46.950 align:middle line:84% So we need to show this one here. 00:50:46.950 --> 00:50:49.990 align:middle line:84% And so if I was to write it down, 00:50:49.990 --> 00:50:53.370 align:middle line:84% I would probably have some of this on scrap paper. 00:50:53.370 --> 00:50:56.210 align:middle line:84% And then I would instead establish this here 00:50:56.210 --> 00:50:59.450 align:middle line:84% first, and then say, therefore, and then I'll conclude why. 00:50:59.450 --> 00:51:02.170 align:middle line:90% 00:51:02.170 --> 00:51:05.230 align:middle line:84% But of course, that's not the way you're thinking about it. 00:51:05.230 --> 00:51:08.730 align:middle line:84% You first try to analyze what you have to prove. 00:51:08.730 --> 00:51:11.130 align:middle line:84% And then once you know that, then you 00:51:11.130 --> 00:51:12.400 align:middle line:90% try to write it down nicely. 00:51:12.400 --> 00:51:14.910 align:middle line:90% 00:51:14.910 --> 00:51:16.830 align:middle line:90% So we need to prove this here. 00:51:16.830 --> 00:51:22.180 align:middle line:90% 00:51:22.180 --> 00:51:30.260 align:middle line:84% So now the thing is that we have that the definition 00:51:30.260 --> 00:51:34.120 align:middle line:84% of bn, which is right here, that's the definition of bn. 00:51:34.120 --> 00:51:36.500 align:middle line:90% But let me copy it over here. 00:51:36.500 --> 00:51:40.220 align:middle line:84% So we know that bn squared is less than or equal to, 00:51:40.220 --> 00:51:43.820 align:middle line:84% it's the largest integer so that you have this thing here. 00:51:43.820 --> 00:51:47.780 align:middle line:90% 00:51:47.780 --> 00:51:50.980 align:middle line:84% But this means that if you're looking at 10 bn, 00:51:50.980 --> 00:51:52.860 align:middle line:90% that's still an integer. 00:51:52.860 --> 00:51:56.060 align:middle line:84% And if you square this here, then this 00:51:56.060 --> 00:52:02.020 align:middle line:84% is the same as 100 times bn squared. 00:52:02.020 --> 00:52:06.970 align:middle line:84% And so that actually has to be less than 100 times this. 00:52:06.970 --> 00:52:10.140 align:middle line:90% 00:52:10.140 --> 00:52:15.480 align:middle line:84% So it just becomes less than or equal to 2 10 to the 2n, 00:52:15.480 --> 00:52:17.100 align:middle line:90% like that. 00:52:17.100 --> 00:52:24.550 align:middle line:84% And this here you can also write as 2 to the 10 times 10 00:52:24.550 --> 00:52:30.110 align:middle line:90% to the power 2n plus 1 minus 2. 00:52:30.110 --> 00:52:36.710 align:middle line:84% But now, you see that bn, so 10 bn is an integer 00:52:36.710 --> 00:52:38.550 align:middle line:90% and it has this property. 00:52:38.550 --> 00:52:43.870 align:middle line:84% But bn plus 1 was supposed to be the largest integer so that 00:52:43.870 --> 00:52:52.870 align:middle line:84% the square here is less than 2 times 10 to 2n plus 1 minus 2. 00:52:52.870 --> 00:52:59.390 align:middle line:84% So you see that bn plus 1 is the largest one with this property. 00:52:59.390 --> 00:53:11.190 align:middle line:84% So you conclude that since bn plus 1 is the largest 00:53:11.190 --> 00:53:18.120 align:middle line:84% and both are integers, and bn is an integer, 00:53:18.120 --> 00:53:26.280 align:middle line:84% then bn plus 1 must be bigger or equal to bn-- 00:53:26.280 --> 00:53:29.345 align:middle line:84% must be bigger than 10 times bn, right? 00:53:29.345 --> 00:53:32.960 align:middle line:90% 00:53:32.960 --> 00:53:35.732 align:middle line:84% And that's exactly what we wanted to prove, 00:53:35.732 --> 00:53:40.320 align:middle line:84% that the bn plus 1 was bigger than bn. 00:53:40.320 --> 00:53:43.520 align:middle line:84% So we now have that the sequence an 00:53:43.520 --> 00:53:46.320 align:middle line:84% defined this slightly cryptical way, 00:53:46.320 --> 00:53:50.280 align:middle line:84% but really, it sounds maybe cryptical, 00:53:50.280 --> 00:53:57.260 align:middle line:84% but it's just instead of using somehow digital numbers, 00:53:57.260 --> 00:54:02.360 align:middle line:84% you're just thinking about you want this and that many-- 00:54:02.360 --> 00:54:06.680 align:middle line:84% the n is referring to the number of digits, 00:54:06.680 --> 00:54:12.360 align:middle line:84% but including the digits that is before the point. 00:54:12.360 --> 00:54:14.460 align:middle line:84% So it's actually not very cryptical, 00:54:14.460 --> 00:54:17.290 align:middle line:84% but it maybe sounds a little formal here. 00:54:17.290 --> 00:54:23.890 align:middle line:84% OK, so we have an is an increasing sequence. 00:54:23.890 --> 00:54:25.950 align:middle line:90% Now, why is an bounded? 00:54:25.950 --> 00:54:28.890 align:middle line:90% 00:54:28.890 --> 00:54:33.570 align:middle line:84% But the an is defined from the bn. 00:54:33.570 --> 00:54:38.370 align:middle line:84% So in order to show that an is bounded, 00:54:38.370 --> 00:54:42.210 align:middle line:84% you have to figure out what is a bound for the bn. 00:54:42.210 --> 00:54:48.830 align:middle line:84% But you have this here, bn is the largest such number. 00:54:48.830 --> 00:54:53.530 align:middle line:84% If I take 2 times 10 to the power n minus 1, 00:54:53.530 --> 00:54:57.610 align:middle line:84% suppose I take this number here and I square it. 00:54:57.610 --> 00:54:59.610 align:middle line:90% This is definitely an integer. 00:54:59.610 --> 00:55:01.010 align:middle line:90% I square it. 00:55:01.010 --> 00:55:06.290 align:middle line:84% Then I'm getting 4 times 10 to the power 2n minus 2. 00:55:06.290 --> 00:55:10.810 align:middle line:84% So this in here, this here is the same as over here. 00:55:10.810 --> 00:55:12.570 align:middle line:90% But it's now 4. 00:55:12.570 --> 00:55:16.880 align:middle line:84% So this number here would not be allowed as a bn. 00:55:16.880 --> 00:55:20.120 align:middle line:84% In fact, it would be larger than bn. 00:55:20.120 --> 00:55:26.360 align:middle line:84% So bn is less than 2 to the power, 10 to the n minus 1. 00:55:26.360 --> 00:55:30.415 align:middle line:90% 00:55:30.415 --> 00:55:39.140 align:middle line:84% So this means that an, which is equal to bn over 10 to the power 00:55:39.140 --> 00:55:44.180 align:middle line:84% n minus 1, but this is now, you see, less than or equal to 2. 00:55:44.180 --> 00:55:47.060 align:middle line:90% 00:55:47.060 --> 00:55:50.700 align:middle line:84% So the sequence an that we have defined 00:55:50.700 --> 00:55:54.860 align:middle line:84% is, as you would have expected, they 00:55:54.860 --> 00:55:58.680 align:middle line:84% are increasing because, if you think about it, 00:55:58.680 --> 00:56:03.620 align:middle line:84% it's really defined to take the number of digits, 00:56:03.620 --> 00:56:06.760 align:middle line:84% it's going to be like the number of digits in square root of 2, 00:56:06.760 --> 00:56:09.640 align:middle line:90% including the 1, the starting 1. 00:56:09.640 --> 00:56:16.110 align:middle line:90% 00:56:16.110 --> 00:56:18.470 align:middle line:84% So they all are going to be less than 2. 00:56:18.470 --> 00:56:56.590 align:middle line:90% 00:56:56.590 --> 00:57:00.910 align:middle line:84% So we have that an is an increasing sequence 00:57:00.910 --> 00:57:01.740 align:middle line:90% and is bounded. 00:57:01.740 --> 00:57:04.670 align:middle line:90% 00:57:04.670 --> 00:57:21.320 align:middle line:84% So an, here, is convergent by the monotone convergence 00:57:21.320 --> 00:57:21.820 align:middle line:90% theorem. 00:57:21.820 --> 00:57:28.946 align:middle line:90% 00:57:28.946 --> 00:57:34.440 align:middle line:84% So the an is convergent by the monotone convergence theorem. 00:57:34.440 --> 00:57:40.000 align:middle line:84% But now, if I'm looking at an squared, an squared, 00:57:40.000 --> 00:57:44.440 align:middle line:84% well, an squared is the same as an times an. 00:57:44.440 --> 00:57:51.720 align:middle line:90% 00:57:51.720 --> 00:58:01.160 align:middle line:84% But an, by definition, was bn divided by 10 00:58:01.160 --> 00:58:06.640 align:middle line:90% to the power n minus 1. 00:58:06.640 --> 00:58:13.170 align:middle line:84% And so I can write it like this, 10 to the power n minus 1. 00:58:13.170 --> 00:58:16.370 align:middle line:84% And so you see that you can write this as bn squared. 00:58:16.370 --> 00:58:21.370 align:middle line:84% And then these are combined to 2n minus 2. 00:58:21.370 --> 00:58:26.090 align:middle line:84% And by assumption, if you divide by this here, 00:58:26.090 --> 00:58:28.710 align:middle line:84% this thing here is less than or equal to 2. 00:58:28.710 --> 00:58:34.410 align:middle line:90% 00:58:34.410 --> 00:58:38.633 align:middle line:84% So you have that an converges to a. 00:58:38.633 --> 00:58:40.050 align:middle line:84% And if you want to think about it, 00:58:40.050 --> 00:58:43.410 align:middle line:84% you think about this sequence again. 00:58:43.410 --> 00:58:49.970 align:middle line:84% And so now you know that an, by the product rule for sequences, 00:58:49.970 --> 00:58:53.330 align:middle line:84% an times an, the sequence times itself, 00:58:53.330 --> 00:58:57.850 align:middle line:90% is converging to a squared. 00:58:57.850 --> 00:59:03.250 align:middle line:84% And this thing here, because each of these squares 00:59:03.250 --> 00:59:06.820 align:middle line:84% here, each of them is less than or equal to 2, 00:59:06.820 --> 00:59:09.080 align:middle line:84% then the limit must be less than or equal to 2. 00:59:09.080 --> 00:59:14.780 align:middle line:90% 00:59:14.780 --> 00:59:19.440 align:middle line:84% What I want to prove now is that this a that we get as a limit, 00:59:19.440 --> 00:59:22.940 align:middle line:84% that the square is, in fact, equal to 2. 00:59:22.940 --> 00:59:33.020 align:middle line:84% I want to show that the square, when we're doing it, 00:59:33.020 --> 00:59:36.100 align:middle line:84% this is one more proof that square root of 2 00:59:36.100 --> 00:59:37.000 align:middle line:90% is a real number. 00:59:37.000 --> 00:59:39.420 align:middle line:84% But this is in a way how you think about it, 00:59:39.420 --> 00:59:44.328 align:middle line:90% in terms of the decimals. 00:59:44.328 --> 00:59:48.500 align:middle line:84% And so we have this sequence an, and we have proven 00:59:48.500 --> 00:59:50.720 align:middle line:90% that it's converging to some a. 00:59:50.720 --> 00:59:56.260 align:middle line:84% And we've shown that this a here has a property that a squared is 00:59:56.260 --> 00:59:59.640 align:middle line:90% less than or equal to 2. 00:59:59.640 --> 01:00:14.950 align:middle line:84% So I want to show that a squared is actually equal to 2. 01:00:14.950 --> 01:00:17.310 align:middle line:90% That's what I want to prove. 01:00:17.310 --> 01:00:19.430 align:middle line:84% But I've already proven that a squared 01:00:19.430 --> 01:00:23.755 align:middle line:84% is less than or equal to 2, so enough 01:00:23.755 --> 01:00:29.050 align:middle line:84% to show that a squared is bigger or equal to 2. 01:00:29.050 --> 01:00:32.010 align:middle line:90% 01:00:32.010 --> 01:00:33.370 align:middle line:90% So let's try to do that. 01:00:33.370 --> 01:00:39.230 align:middle line:90% 01:00:39.230 --> 01:00:41.670 align:middle line:84% And I'm going to use, this is going 01:00:41.670 --> 01:00:46.190 align:middle line:84% to be another illustration of the algebraic rules 01:00:46.190 --> 01:00:48.210 align:middle line:90% for sequences. 01:00:48.210 --> 01:00:55.710 align:middle line:90% 01:00:55.710 --> 01:01:00.190 align:middle line:84% Here, now, really to prove that is less than or equal to 2, 01:01:00.190 --> 01:01:05.920 align:middle line:84% I didn't use fully the definition of the an 01:01:05.920 --> 01:01:12.360 align:middle line:84% because I didn't really use that it was the largest such that an 01:01:12.360 --> 01:01:21.960 align:middle line:84% was defined in terms of bn, where bn was the largest 01:01:21.960 --> 01:01:22.630 align:middle line:90% such number. 01:01:22.630 --> 01:01:28.280 align:middle line:90% 01:01:28.280 --> 01:01:34.000 align:middle line:84% Suppose now I'm looking at bn and I'm adding 1 01:01:34.000 --> 01:01:35.260 align:middle line:90% and I'm squaring it. 01:01:35.260 --> 01:01:37.808 align:middle line:90% 01:01:37.808 --> 01:01:43.520 align:middle line:84% Well, this here cannot also be smaller than this here, 01:01:43.520 --> 01:01:50.280 align:middle line:84% because then the bn should have been this in here. 01:01:50.280 --> 01:01:54.760 align:middle line:84% So this here must actually be strictly larger than 2 10 01:01:54.760 --> 01:01:56.580 align:middle line:90% to the power 2n minus 2. 01:01:56.580 --> 01:02:00.180 align:middle line:90% 01:02:00.180 --> 01:02:06.810 align:middle line:84% And so now you just divide by this on both sides. 01:02:06.810 --> 01:02:12.270 align:middle line:90% So you have bn plus 1 squared. 01:02:12.270 --> 01:02:16.870 align:middle line:84% And you divide it by 10 to the power 2n minus 2. 01:02:16.870 --> 01:02:19.030 align:middle line:84% So you know that this thing is bigger than 2. 01:02:19.030 --> 01:02:21.690 align:middle line:90% 01:02:21.690 --> 01:02:24.770 align:middle line:84% But you can write this in here, this denominator, 01:02:24.770 --> 01:02:29.690 align:middle line:84% 10 to the power 2n minus 2, you can write as 10 to the power n 01:02:29.690 --> 01:02:34.090 align:middle line:84% minus 1 times 10 to the power n minus 1. 01:02:34.090 --> 01:02:36.430 align:middle line:84% So in this way, you can split this up. 01:02:36.430 --> 01:02:42.850 align:middle line:84% And you can write it as bn plus 1 over 10 01:02:42.850 --> 01:02:47.920 align:middle line:84% to the power n minus 1, and then times itself. 01:02:47.920 --> 01:02:50.570 align:middle line:90% 01:02:50.570 --> 01:02:57.850 align:middle line:84% This is just to make sure that it's bn, and then to the bn 01:02:57.850 --> 01:03:01.140 align:middle line:90% you're adding 1. 01:03:01.140 --> 01:03:06.780 align:middle line:84% So like that, you write it like that. 01:03:06.780 --> 01:03:16.140 align:middle line:84% But you see, this thing here, if you split the fraction up, 01:03:16.140 --> 01:03:23.340 align:middle line:84% you can write it as bn over 10 to the power n minus 1. 01:03:23.340 --> 01:03:28.220 align:middle line:84% And then you can add 1 over 10 to the power n minus 1. 01:03:28.220 --> 01:03:31.580 align:middle line:84% And then you can multiply that by 2. 01:03:31.580 --> 01:03:33.040 align:middle line:90% And you can write it like that. 01:03:33.040 --> 01:03:41.220 align:middle line:90% 01:03:41.220 --> 01:03:47.020 align:middle line:84% But now you see, now I can write this thing here. 01:03:47.020 --> 01:03:50.980 align:middle line:90% 01:03:50.980 --> 01:03:55.920 align:middle line:84% Now I can think about it in a slightly different way. 01:03:55.920 --> 01:03:58.030 align:middle line:90% So I can write this thing here. 01:03:58.030 --> 01:04:02.000 align:middle line:84% This thing here, that is the definition of an. 01:04:02.000 --> 01:04:07.030 align:middle line:90% 01:04:07.030 --> 01:04:13.270 align:middle line:84% So what I have here is that an plus 1 over 10 01:04:13.270 --> 01:04:17.130 align:middle line:84% to the power n minus 1 times itself, 01:04:17.130 --> 01:04:19.590 align:middle line:84% but I will just write it nicely out 01:04:19.590 --> 01:04:26.730 align:middle line:84% like this, that this thing here, remember the way it goes, 01:04:26.730 --> 01:04:30.070 align:middle line:84% that this is equal to that which is bigger than 2, 01:04:30.070 --> 01:04:31.690 align:middle line:90% so this is bigger than 2. 01:04:31.690 --> 01:04:35.430 align:middle line:90% 01:04:35.430 --> 01:04:41.430 align:middle line:84% Now, 1 over 10 to the n, you can think about that as a sequence. 01:04:41.430 --> 01:04:43.250 align:middle line:84% You can think about this as a sequence. 01:04:43.250 --> 01:04:45.230 align:middle line:90% That sequence goes to 0. 01:04:45.230 --> 01:04:47.002 align:middle line:90% This here goes to a. 01:04:47.002 --> 01:04:50.990 align:middle line:84% The sum of these two sequences, by the algebraic rules, 01:04:50.990 --> 01:04:53.190 align:middle line:90% go to the sum of the limits. 01:04:53.190 --> 01:04:55.730 align:middle line:84% So this is a plus 0, so that's a. 01:04:55.730 --> 01:04:58.790 align:middle line:90% So this thing here goes to a. 01:04:58.790 --> 01:05:02.040 align:middle line:84% And this is the same, so this goes to a. 01:05:02.040 --> 01:05:07.720 align:middle line:84% So you see that by the product rule for the limit, 01:05:07.720 --> 01:05:12.520 align:middle line:84% the limit here is, so you get that a squared here. 01:05:12.520 --> 01:05:14.560 align:middle line:84% And since it holds for each element, 01:05:14.560 --> 01:05:17.320 align:middle line:84% then it also holds for the limit. 01:05:17.320 --> 01:05:19.520 align:middle line:84% Except you have to be a little bit careful-- 01:05:19.520 --> 01:05:24.440 align:middle line:84% it may not be strict inequality, but just big or equal to. 01:05:24.440 --> 01:05:29.120 align:middle line:84% And of course, in fact, it is equal to because now you 01:05:29.120 --> 01:05:32.640 align:middle line:84% have established the other inequality that a squared was 01:05:32.640 --> 01:05:34.640 align:middle line:90% bigger or equal to 2. 01:05:34.640 --> 01:05:37.180 align:middle line:84% We had already proven that a squared was less than 01:05:37.180 --> 01:05:38.360 align:middle line:90% or equal to 2. 01:05:38.360 --> 01:05:42.440 align:middle line:84% So you conclude that a squared is equal to 2. 01:05:42.440 --> 01:05:47.080 align:middle line:84% If you just think about it, it sounds maybe a little 01:05:47.080 --> 01:05:48.560 align:middle line:90% bit convoluted. 01:05:48.560 --> 01:05:52.040 align:middle line:84% But it's really writing down precisely what 01:05:52.040 --> 01:05:53.930 align:middle line:84% you think about when you're thinking 01:05:53.930 --> 01:05:56.970 align:middle line:84% about a square root of 2 because you're thinking 01:05:56.970 --> 01:05:58.510 align:middle line:90% about the number of digits. 01:05:58.510 --> 01:06:04.510 align:middle line:84% And so including the first, the 1 in 1.4, 01:06:04.510 --> 01:06:07.410 align:middle line:84% blah, blah, blah, including that 1, 01:06:07.410 --> 01:06:12.090 align:middle line:84% if you think about the number of digits, like the an, 01:06:12.090 --> 01:06:16.850 align:middle line:84% then the an is really defined to make it precise. 01:06:16.850 --> 01:06:19.730 align:middle line:84% You would define it in terms of this other sequence bn. 01:06:19.730 --> 01:06:25.970 align:middle line:90% 01:06:25.970 --> 01:06:30.330 align:middle line:84% So this was just a little illustration 01:06:30.330 --> 01:06:32.920 align:middle line:84% of how the monotone convergence theorem works. 01:06:32.920 --> 01:06:36.298 align:middle line:90% 01:06:36.298 --> 01:06:41.610 align:middle line:84% And this will play an important role next time. 01:06:41.610 --> 01:06:42.650 align:middle line:90% Yeah? 01:06:42.650 --> 01:06:45.650 align:middle line:84% AUDIENCE: So the reason why at the top 01:06:45.650 --> 01:06:48.030 align:middle line:84% we have strictly greater than 2, and then at the bottom, 01:06:48.030 --> 01:06:50.730 align:middle line:84% we have greater or equal to 2 is because the a's 01:06:50.730 --> 01:06:53.380 align:middle line:84% and the a squared are limits, whereas on the top, 01:06:53.380 --> 01:06:56.260 align:middle line:84% we're considering each element in the sequence? 01:06:56.260 --> 01:06:59.417 align:middle line:84% TOBIAS COLDING: So this here, the an, 01:06:59.417 --> 01:07:01.000 align:middle line:84% you're talking about this here, right? 01:07:01.000 --> 01:07:01.860 align:middle line:90% AUDIENCE: Yeah. 01:07:01.860 --> 01:07:03.995 align:middle line:84% TOBIAS COLDING: So the an converges to a. 01:07:03.995 --> 01:07:04.620 align:middle line:90% AUDIENCE: Yeah. 01:07:04.620 --> 01:07:06.340 align:middle line:90% TOBIAS COLDING: Right? 01:07:06.340 --> 01:07:09.060 align:middle line:84% I think about this here, so I think about a sequence 01:07:09.060 --> 01:07:11.280 align:middle line:90% plus another sequence. 01:07:11.280 --> 01:07:14.180 align:middle line:84% This sequence here converges to 0. 01:07:14.180 --> 01:07:16.500 align:middle line:84% So the sum of these two sequences 01:07:16.500 --> 01:07:19.400 align:middle line:84% converges to 0, and likewise for this. 01:07:19.400 --> 01:07:23.660 align:middle line:84% And then I'm thinking about it as a product of this sequence 01:07:23.660 --> 01:07:26.260 align:middle line:90% with that sequence. 01:07:26.260 --> 01:07:31.700 align:middle line:84% But it's like each element in the product sequence 01:07:31.700 --> 01:07:33.780 align:middle line:90% is strictly bigger than 2. 01:07:33.780 --> 01:07:37.320 align:middle line:84% But it doesn't necessarily mean that the limit is strictly. 01:07:37.320 --> 01:07:40.140 align:middle line:84% But the limit has to be bigger than 2 because they 01:07:40.140 --> 01:07:41.640 align:middle line:90% could kind of approach 2. 01:07:41.640 --> 01:07:43.480 align:middle line:84% And that's exactly what happened, of course. 01:07:43.480 --> 01:07:46.940 align:middle line:90% 01:07:46.940 --> 01:07:51.390 align:middle line:84% So I also just want to say that this limsup and liminf-- 01:07:51.390 --> 01:07:53.790 align:middle line:84% well, we didn't actually define it. 01:07:53.790 --> 01:07:58.110 align:middle line:84% And it is how you define it usually. 01:07:58.110 --> 01:08:00.670 align:middle line:84% But we haven't talked about that concept yet. 01:08:00.670 --> 01:08:03.330 align:middle line:84% But this, I talked about with that, 01:08:03.330 --> 01:08:05.190 align:middle line:84% if you take a bounded sequence, then 01:08:05.190 --> 01:08:08.270 align:middle line:84% you could form two monotone sequences. 01:08:08.270 --> 01:08:10.550 align:middle line:84% This will play a really important role 01:08:10.550 --> 01:08:11.990 align:middle line:90% for several things. 01:08:11.990 --> 01:08:15.230 align:middle line:84% This is how one defines limsup and liminf. 01:08:15.230 --> 01:08:17.109 align:middle line:84% We haven't talked about those yet. 01:08:17.109 --> 01:08:22.930 align:middle line:84% It's also going to be used to prove an important theorem, 01:08:22.930 --> 01:08:24.229 align:middle line:90% really important theorem. 01:08:24.229 --> 01:08:27.649 align:middle line:84% And I will talk about that theorem next. 01:08:27.649 --> 01:08:30.430 align:middle line:84% But we won't prove it before next time. 01:08:30.430 --> 01:08:35.270 align:middle line:84% So what is that stuff going to be used for? 01:08:35.270 --> 01:08:41.262 align:middle line:84% And the monotone, what is another application? 01:08:41.262 --> 01:08:43.470 align:middle line:84% And in fact, there will be several other applications 01:08:43.470 --> 01:08:44.940 align:middle line:90% of monotone convergence theorem. 01:08:44.940 --> 01:08:49.279 align:middle line:90% 01:08:49.279 --> 01:08:52.080 align:middle line:84% Here's a really important concept. 01:08:52.080 --> 01:08:56.772 align:middle line:90% And so sometimes it's hard. 01:08:56.772 --> 01:08:59.200 align:middle line:84% If it was a monotone sequence, then 01:08:59.200 --> 01:09:02.899 align:middle line:84% you could write down explicitly what the limit was. 01:09:02.899 --> 01:09:06.600 align:middle line:84% And in all of the examples we saw so far, 01:09:06.600 --> 01:09:09.460 align:middle line:84% it was easy to write down the limits. 01:09:09.460 --> 01:09:13.660 align:middle line:84% But in general, you might not be able to write down the limit, 01:09:13.660 --> 01:09:16.200 align:middle line:84% but you want to prove that the limit exists. 01:09:16.200 --> 01:09:18.960 align:middle line:84% And so this is the following notion 01:09:18.960 --> 01:09:21.240 align:middle line:90% that then becomes really handy. 01:09:21.240 --> 01:09:23.130 align:middle line:84% And that's the notion of a Cauchy sequence. 01:09:23.130 --> 01:09:28.840 align:middle line:90% 01:09:28.840 --> 01:09:31.000 align:middle line:90% So what is a Cauchy sequence? 01:09:31.000 --> 01:09:45.120 align:middle line:84% So a Cauchy sequence, it's a sequence with the property 01:09:45.120 --> 01:09:54.410 align:middle line:84% that for all epsilon greater than 0, 01:09:54.410 --> 01:10:03.702 align:middle line:84% there exists N, just a natural number N, 01:10:03.702 --> 01:10:11.850 align:middle line:84% such that if n and m are bigger than this natural number, then 01:10:11.850 --> 01:10:19.220 align:middle line:84% the two elements an and am are less than epsilon apart. 01:10:19.220 --> 01:10:24.050 align:middle line:90% 01:10:24.050 --> 01:10:26.910 align:middle line:84% So this is a different notion than convergence. 01:10:26.910 --> 01:10:32.370 align:middle line:84% Convergence says that a sequence is convergent 01:10:32.370 --> 01:10:36.980 align:middle line:84% if everything kind of far out accumulates around the limit. 01:10:36.980 --> 01:10:44.500 align:middle line:90% 01:10:44.500 --> 01:10:50.540 align:middle line:84% So this here says that a Cauchy sequence, that's a sequence so 01:10:50.540 --> 01:10:54.700 align:middle line:84% that if you're far out, then things bunch together, 01:10:54.700 --> 01:10:56.060 align:middle line:90% basically. 01:10:56.060 --> 01:10:58.580 align:middle line:90% Now, of course, we will prove. 01:10:58.580 --> 01:11:07.890 align:middle line:84% And so we will prove on R. We will prove the Cauchy sequence. 01:11:07.890 --> 01:11:16.380 align:middle line:90% 01:11:16.380 --> 01:11:18.330 align:middle line:90% So we'll prove the following. 01:11:18.330 --> 01:11:21.525 align:middle line:90% 01:11:21.525 --> 01:11:22.800 align:middle line:90% Prove the following theorem-- 01:11:22.800 --> 01:11:26.460 align:middle line:90% 01:11:26.460 --> 01:11:28.600 align:middle line:84% for now, everything is in R at the moment-- 01:11:28.600 --> 01:11:31.660 align:middle line:90% 01:11:31.660 --> 01:11:48.312 align:middle line:84% that a sequence is a Cauchy sequence 01:11:48.312 --> 01:11:53.330 align:middle line:90% if and only if it is convergent. 01:11:53.330 --> 01:11:56.630 align:middle line:90% 01:11:56.630 --> 01:12:01.030 align:middle line:84% I won't prove this now, but I'll prove this next time. 01:12:01.030 --> 01:12:05.550 align:middle line:84% And again, the proof of this involved this stuff 01:12:05.550 --> 01:12:08.030 align:middle line:84% where we defined these two sequences 01:12:08.030 --> 01:12:09.610 align:middle line:90% from that bounded sequence. 01:12:09.610 --> 01:12:12.130 align:middle line:90% So it would be useful there. 01:12:12.130 --> 01:12:15.630 align:middle line:90% 01:12:15.630 --> 01:12:18.260 align:middle line:84% Now, let me give you some examples. 01:12:18.260 --> 01:12:25.150 align:middle line:90% 01:12:25.150 --> 01:12:33.350 align:middle line:84% One example, and we'll get to that later, but at the moment, 01:12:33.350 --> 01:12:36.590 align:middle line:84% we talk, everything is in R. And then we 01:12:36.590 --> 01:12:38.530 align:middle line:90% will later generalize this. 01:12:38.530 --> 01:12:42.960 align:middle line:84% And so to generalize this, a lot of these concepts 01:12:42.960 --> 01:12:45.320 align:middle line:90% work much more general. 01:12:45.320 --> 01:12:49.052 align:middle line:90% And that's extremely useful. 01:12:49.052 --> 01:12:53.600 align:middle line:84% And it can be used to find solutions 01:12:53.600 --> 01:12:56.600 align:middle line:90% to differential equations. 01:12:56.600 --> 01:13:01.080 align:middle line:84% But then you have to abstract these concepts 01:13:01.080 --> 01:13:04.200 align:middle line:84% to where sequences are not any more real numbers, 01:13:04.200 --> 01:13:06.140 align:middle line:84% but they are of some other space. 01:13:06.140 --> 01:13:09.320 align:middle line:84% And then you have to talk about convergence, et cetera. 01:13:09.320 --> 01:13:13.040 align:middle line:84% And so you can use this to define 01:13:13.040 --> 01:13:18.440 align:middle line:84% solutions to show that various differential equations have 01:13:18.440 --> 01:13:20.286 align:middle line:90% solutions. 01:13:20.286 --> 01:13:23.640 align:middle line:84% And so that often goes over using 01:13:23.640 --> 01:13:25.610 align:middle line:84% what's called the contracting mapping theorem. 01:13:25.610 --> 01:13:33.263 align:middle line:90% 01:13:33.263 --> 01:13:33.930 align:middle line:90% Mapping theorem. 01:13:33.930 --> 01:13:39.850 align:middle line:90% 01:13:39.850 --> 01:13:44.290 align:middle line:84% And the contraction mapping theorem have all kinds of forms. 01:13:44.290 --> 01:13:46.810 align:middle line:90% 01:13:46.810 --> 01:13:51.410 align:middle line:84% But they're all, it's just a matter of what space 01:13:51.410 --> 01:13:54.530 align:middle line:90% you're looking at, and so on. 01:13:54.530 --> 01:13:57.350 align:middle line:84% So we're looking at R here in this case. 01:13:57.350 --> 01:13:59.490 align:middle line:84% And so the contracting mapping theorem, 01:13:59.490 --> 01:14:05.850 align:middle line:84% so a map is just a map here from R to R. 01:14:05.850 --> 01:14:09.730 align:middle line:84% And that the map is contracting means that if you 01:14:09.730 --> 01:14:14.190 align:middle line:90% take two elements in R-- 01:14:14.190 --> 01:14:17.290 align:middle line:90% 01:14:17.290 --> 01:14:22.780 align:middle line:90% sorry, T is a contracting map. 01:14:22.780 --> 01:14:35.610 align:middle line:90% 01:14:35.610 --> 01:14:38.832 align:middle line:84% And this is really important-- c is positive. 01:14:38.832 --> 01:14:40.040 align:middle line:90% I mean, that's not important. 01:14:40.040 --> 01:14:41.920 align:middle line:90% It had to be positive. 01:14:41.920 --> 01:14:45.460 align:middle line:84% But the important point is that c 01:14:45.460 --> 01:14:47.320 align:middle line:90% has to be strictly less than 1. 01:14:47.320 --> 01:14:50.980 align:middle line:90% That's a crucial point. 01:14:50.980 --> 01:15:00.460 align:middle line:84% And then it has a property that when you take x and y in R 01:15:00.460 --> 01:15:05.400 align:middle line:84% and you're looking at how close x and y are together, 01:15:05.400 --> 01:15:14.540 align:middle line:84% so how close that the image under T of x and y, how close 01:15:14.540 --> 01:15:19.240 align:middle line:84% they are together, then those have to be some factor. 01:15:19.240 --> 01:15:21.990 align:middle line:84% And this is the factor, the c is the factor. 01:15:21.990 --> 01:15:25.760 align:middle line:90% 01:15:25.760 --> 01:15:27.720 align:middle line:90% So they have to come closer. 01:15:27.720 --> 01:15:30.560 align:middle line:84% This is why c has to be strictly less than 1 01:15:30.560 --> 01:15:32.320 align:middle line:84% because they have to come closer, 01:15:32.320 --> 01:15:36.670 align:middle line:84% if they come closer by a factor than the original one was. 01:15:36.670 --> 01:15:40.370 align:middle line:90% 01:15:40.370 --> 01:15:43.670 align:middle line:90% So this is a contracting map. 01:15:43.670 --> 01:15:47.430 align:middle line:84% So a contracting map is a map from R 01:15:47.430 --> 01:15:52.790 align:middle line:84% to R so that there exists some constant strictly less than 1, 01:15:52.790 --> 01:15:55.590 align:middle line:84% so that for all x and y, you have 01:15:55.590 --> 01:15:58.870 align:middle line:84% that the difference between the images 01:15:58.870 --> 01:16:02.590 align:middle line:84% is this factor that is strictly less than 1 01:16:02.590 --> 01:16:05.190 align:middle line:90% with what they were originally. 01:16:05.190 --> 01:16:07.770 align:middle line:84% And so now you get the contracting mapping theorem. 01:16:07.770 --> 01:16:09.710 align:middle line:84% And we'll talk about the proof next time. 01:16:09.710 --> 01:16:16.550 align:middle line:84% And again, once you know the more abstract concepts, 01:16:16.550 --> 01:16:18.470 align:middle line:90% then it's quite easy. 01:16:18.470 --> 01:16:21.230 align:middle line:84% The proof is going to be the same, et cetera, et cetera. 01:16:21.230 --> 01:16:23.470 align:middle line:84% And it will give you existence of solutions 01:16:23.470 --> 01:16:25.510 align:middle line:84% to a lot of differential equation. 01:16:25.510 --> 01:16:27.420 align:middle line:84% So this is the contracting mapping theorem. 01:16:27.420 --> 01:16:37.520 align:middle line:90% 01:16:37.520 --> 01:16:47.440 align:middle line:84% So it's saying the following, that if T from R to R 01:16:47.440 --> 01:17:04.350 align:middle line:84% is a contracting map, then T has a fixed point. 01:17:04.350 --> 01:17:08.260 align:middle line:90% 01:17:08.260 --> 01:17:09.690 align:middle line:90% It has a fixed point. 01:17:09.690 --> 01:17:13.720 align:middle line:90% 01:17:13.720 --> 01:17:19.160 align:middle line:84% And this means that T of x is equal to x. 01:17:19.160 --> 01:17:23.080 align:middle line:84% And when you apply this theorem, so this 01:17:23.080 --> 01:17:27.280 align:middle line:84% is the contracting mapping, and again it's crucial 01:17:27.280 --> 01:17:30.140 align:middle line:90% that this c is less than 1. 01:17:30.140 --> 01:17:32.370 align:middle line:90% That's what makes everything go. 01:17:32.370 --> 01:17:35.930 align:middle line:84% If it was equal to 1, you wouldn't get anything. 01:17:35.930 --> 01:17:38.650 align:middle line:84% It has to be strictly less than 1. 01:17:38.650 --> 01:17:42.890 align:middle line:84% And this thing here, if you set it up more generally, 01:17:42.890 --> 01:17:47.330 align:middle line:84% if the spaces are more general than just the real numbers, 01:17:47.330 --> 01:17:51.730 align:middle line:84% then this here will be that you have a solution-- 01:17:51.730 --> 01:17:53.590 align:middle line:84% it could be applied to many other things-- 01:17:53.590 --> 01:17:56.970 align:middle line:84% but it could be that you have a solution to a differential 01:17:56.970 --> 01:17:57.630 align:middle line:90% equation. 01:17:57.630 --> 01:18:02.330 align:middle line:84% This statement here is like a standard application 01:18:02.330 --> 01:18:05.190 align:middle line:84% to get solutions to differential equations. 01:18:05.190 --> 01:18:07.510 align:middle line:84% And you can use for all kinds of other stuff. 01:18:07.510 --> 01:18:10.450 align:middle line:90% 01:18:10.450 --> 01:18:12.120 align:middle line:84% So we'll talk about that next time. 01:18:12.120 --> 01:18:15.988 align:middle line:90% 01:18:15.988 --> 01:18:18.030 align:middle line:84% We'll talk about the contracting mapping theorem. 01:18:18.030 --> 01:18:22.130 align:middle line:84% We won't talk quite yet about the applications. 01:18:22.130 --> 01:18:25.090 align:middle line:90% Any questions? 01:18:25.090 --> 01:18:27.120 align:middle line:90% OK, great. 01:18:27.120 --> 01:18:31.000 align:middle line:90%