WEBVTT 00:00:00.000 --> 00:00:01.461 align:middle line:90% [SQUEAKING] 00:00:01.461 --> 00:00:02.435 align:middle line:90% 00:00:02.435 --> 00:00:03.409 align:middle line:90% [RUSTLING] 00:00:03.409 --> 00:00:04.383 align:middle line:90% 00:00:04.383 --> 00:00:05.357 align:middle line:90% [CLICKING] 00:00:05.357 --> 00:00:12.592 align:middle line:90% 00:00:12.592 --> 00:00:14.300 align:middle line:84% TOBIAS COLDING: Last time we talked apart 00:00:14.300 --> 00:00:17.880 align:middle line:84% the basic algebraic operation for limit of sequences. 00:00:17.880 --> 00:00:21.330 align:middle line:84% We talked about the monotone convergence theorem. 00:00:21.330 --> 00:00:23.090 align:middle line:84% So the monotone convergence theorem 00:00:23.090 --> 00:00:27.530 align:middle line:84% was that you had a sequence that was increasing 00:00:27.530 --> 00:00:29.070 align:middle line:90% or it could be decreasing. 00:00:29.070 --> 00:00:31.190 align:middle line:90% So either it was increasing-- 00:00:31.190 --> 00:00:33.740 align:middle line:84% this meant that the next element was bigger 00:00:33.740 --> 00:00:35.400 align:middle line:90% or equal to the previous. 00:00:35.400 --> 00:00:39.830 align:middle line:84% And then they were all assumed to be bounded. 00:00:39.830 --> 00:00:42.980 align:middle line:84% If you had such a sequence, or similarly 00:00:42.980 --> 00:00:45.820 align:middle line:84% if you had a sequence-- so this was the increasing case. 00:00:45.820 --> 00:00:49.610 align:middle line:90% 00:00:49.610 --> 00:00:52.490 align:middle line:84% Then there was a similar decreasing case 00:00:52.490 --> 00:00:59.240 align:middle line:84% where the previous is bigger or equal to the next one. 00:00:59.240 --> 00:01:03.740 align:middle line:84% And then they are assumed to be bounded from below. 00:01:03.740 --> 00:01:06.770 align:middle line:90% So this was decreasing. 00:01:06.770 --> 00:01:09.380 align:middle line:84% So monotone convergence theorem was 00:01:09.380 --> 00:01:11.720 align:middle line:84% that if you had a bounded increasing 00:01:11.720 --> 00:01:17.850 align:middle line:84% sequence or a bounded decreasing sequence, in either case, 00:01:17.850 --> 00:01:21.830 align:middle line:84% the sequence is convergent and it's converging, 00:01:21.830 --> 00:01:24.620 align:middle line:84% in the first case, in this case here, 00:01:24.620 --> 00:01:34.110 align:middle line:84% a, it's converged to the sup of the xn, so this is in case a. 00:01:34.110 --> 00:01:37.190 align:middle line:90% 00:01:37.190 --> 00:01:49.610 align:middle line:84% And in this case b, it converges to the nth in case b. 00:01:49.610 --> 00:01:53.480 align:middle line:84% So that was the monotone convergence theorem. 00:01:53.480 --> 00:01:55.790 align:middle line:84% And the third thing we talked about, 00:01:55.790 --> 00:01:58.270 align:middle line:84% and that's where we will continue today, 00:01:58.270 --> 00:02:01.900 align:middle line:90% is Cauchy sequence. 00:02:01.900 --> 00:02:05.830 align:middle line:84% So we talked about Cauchy sequence. 00:02:05.830 --> 00:02:09.640 align:middle line:84% So let me-- and we proved a few things about Cauchy sequence 00:02:09.640 --> 00:02:12.710 align:middle line:84% from what I examined that for today. 00:02:12.710 --> 00:02:16.160 align:middle line:84% So let me remind you what a Cauchy sequence is. 00:02:16.160 --> 00:02:38.800 align:middle line:84% So a sequence xn is said to be a Cauchy sequence 00:02:38.800 --> 00:02:53.190 align:middle line:84% if, for all epsilon greater than 0, there exists n such that. 00:02:53.190 --> 00:02:56.170 align:middle line:90% 00:02:56.170 --> 00:03:01.780 align:middle line:84% if little n is big or equal to the capital N, then-- 00:03:01.780 --> 00:03:05.080 align:middle line:90% sorry, if little n and little m. 00:03:05.080 --> 00:03:07.870 align:middle line:90% So if n and m-- 00:03:07.870 --> 00:03:11.770 align:middle line:84% m and n are bigger or equal, then 00:03:11.770 --> 00:03:24.220 align:middle line:84% they are epsilon close together xn and xm distance between them 00:03:24.220 --> 00:03:25.390 align:middle line:90% is a distribution. 00:03:25.390 --> 00:03:27.490 align:middle line:90% So this was a Cauchy sequence. 00:03:27.490 --> 00:03:29.890 align:middle line:84% And it just meant-- the Cauchy sequence just 00:03:29.890 --> 00:03:33.315 align:middle line:84% meant that the tail of the sequence bunched together. 00:03:33.315 --> 00:03:37.640 align:middle line:90% 00:03:37.640 --> 00:03:41.710 align:middle line:90% Now let me just prove-- 00:03:41.710 --> 00:03:43.640 align:middle line:84% I think we maybe talk about this, but let me-- 00:03:43.640 --> 00:03:44.950 align:middle line:90% but I'm going to-- 00:03:44.950 --> 00:03:48.840 align:middle line:84% let me just do a little theorem here. 00:03:48.840 --> 00:03:53.890 align:middle line:90% 00:03:53.890 --> 00:04:11.835 align:middle line:84% If xn is a Cauchy sequence, then xn is bounded. 00:04:11.835 --> 00:04:16.329 align:middle line:90% 00:04:16.329 --> 00:04:18.960 align:middle line:84% And so the proof of this is [? the ?] proof. 00:04:18.960 --> 00:04:21.610 align:middle line:90% 00:04:21.610 --> 00:04:26.840 align:middle line:84% So now in the definition of it being a Cauchy sequence, 00:04:26.840 --> 00:04:29.600 align:middle line:84% you can take epsilon to be equal to 1. 00:04:29.600 --> 00:04:36.850 align:middle line:84% So if you take epsilon to be equal to 1 00:04:36.850 --> 00:04:52.660 align:middle line:84% in the definition of being a Cauchy sequence, 00:04:52.660 --> 00:04:55.390 align:middle line:90% so this means that they exist. 00:04:55.390 --> 00:04:55.980 align:middle line:90% They exist. 00:04:55.980 --> 00:04:59.890 align:middle line:90% 00:04:59.890 --> 00:05:10.150 align:middle line:84% They exist some n such that if little n and little m 00:05:10.150 --> 00:05:16.960 align:middle line:84% are bigger than this n, then the difference between these two 00:05:16.960 --> 00:05:20.050 align:middle line:90% is less than 1. 00:05:20.050 --> 00:05:22.840 align:middle line:84% This is the definition where we just 00:05:22.840 --> 00:05:25.960 align:middle line:90% apply it to epsilon equal to 1. 00:05:25.960 --> 00:05:27.850 align:middle line:90% So you have this. 00:05:27.850 --> 00:05:40.660 align:middle line:84% But of course, in particular, we can set say m equal to this. 00:05:40.660 --> 00:05:41.440 align:middle line:90% Right. 00:05:41.440 --> 00:05:44.080 align:middle line:90% And then you have that xn-- 00:05:44.080 --> 00:05:45.760 align:middle line:90% all the other ones. 00:05:45.760 --> 00:05:53.760 align:middle line:84% This means that for all the [INAUDIBLE] for all little n 00:05:53.760 --> 00:06:00.120 align:middle line:84% bigger than capital N, if you're looking at this difference here, 00:06:00.120 --> 00:06:03.150 align:middle line:90% then this is less than 1. 00:06:03.150 --> 00:06:10.390 align:middle line:84% I'm only applying this now and just choosing just one single m. 00:06:10.390 --> 00:06:13.950 align:middle line:84% And I'm choosing it to be equal to that capital. 00:06:13.950 --> 00:06:15.180 align:middle line:90% Do you have that? 00:06:15.180 --> 00:06:22.290 align:middle line:84% And so this means that now you see that if you take-- 00:06:22.290 --> 00:06:26.640 align:middle line:90% so if you take x-- 00:06:26.640 --> 00:06:30.240 align:middle line:90% so now suppose that little n-- 00:06:30.240 --> 00:06:36.510 align:middle line:84% so for little n bigger than capital N, 00:06:36.510 --> 00:06:43.320 align:middle line:84% if you look at little n, then you have-- 00:06:43.320 --> 00:06:51.000 align:middle line:84% [INAUDIBLE] for little n bigger or equal to capital N, 00:06:51.000 --> 00:06:54.760 align:middle line:84% then you have that norm of little n 00:06:54.760 --> 00:07:02.610 align:middle line:84% where you can write little n as the difference. 00:07:02.610 --> 00:07:05.500 align:middle line:90% You can write little n. 00:07:05.500 --> 00:07:07.990 align:middle line:84% You can subtract it first and then add it again. 00:07:07.990 --> 00:07:11.430 align:middle line:90% So this has stayed the same. 00:07:11.430 --> 00:07:14.160 align:middle line:90% And now you can just-- 00:07:14.160 --> 00:07:16.750 align:middle line:84% so you have the sum of two things. 00:07:16.750 --> 00:07:19.140 align:middle line:84% You can think about this thing here is one number 00:07:19.140 --> 00:07:20.890 align:middle line:90% and this is another number. 00:07:20.890 --> 00:07:25.260 align:middle line:84% So this is now less than xn minus x capital 00:07:25.260 --> 00:07:33.090 align:middle line:84% N plus the norm of x capital N. And this here 00:07:33.090 --> 00:07:35.740 align:middle line:84% was because little n was bigger than capital N. 00:07:35.740 --> 00:07:37.450 align:middle line:90% This is bounded by 1. 00:07:37.450 --> 00:07:42.810 align:middle line:84% So this is less than 1 plus the norm. 00:07:42.810 --> 00:07:43.860 align:middle line:90% This. 00:07:43.860 --> 00:07:45.030 align:middle line:90% Right. 00:07:45.030 --> 00:07:47.790 align:middle line:84% So you see that for all the little n's 00:07:47.790 --> 00:07:50.620 align:middle line:84% that are bigger than capital, all of those, 00:07:50.620 --> 00:07:53.340 align:middle line:90% there is a uniform branch. 00:07:53.340 --> 00:07:56.880 align:middle line:84% And so this means that if you're now just taking 00:07:56.880 --> 00:08:06.870 align:middle line:84% the maximum of the norm of x1, the norm of x2 00:08:06.870 --> 00:08:11.440 align:middle line:90% up to the norm of xn minus-- 00:08:11.440 --> 00:08:20.490 align:middle line:84% sorry, xn minus 1, and then x capital n plus 1, 00:08:20.490 --> 00:08:26.340 align:middle line:84% if you take the maximum of this and call this m, then, clearly, 00:08:26.340 --> 00:08:28.200 align:middle line:90% for all n-- 00:08:28.200 --> 00:08:37.590 align:middle line:84% for all n, clearly, you have for all n, 00:08:37.590 --> 00:08:41.490 align:middle line:84% you have that xn is less than or equal to n 00:08:41.490 --> 00:08:47.720 align:middle line:84% because if little n is bigger than capital N, 00:08:47.720 --> 00:08:52.760 align:middle line:84% then it's bounded by capital M because it's 00:08:52.760 --> 00:08:56.550 align:middle line:84% a maximum in particular of this and all the other ones. 00:08:56.550 --> 00:08:58.640 align:middle line:90% That's obvious. 00:08:58.640 --> 00:09:01.980 align:middle line:84% So it means that if you have a Cauchy sequence, 00:09:01.980 --> 00:09:05.810 align:middle line:90% it's automatically bounded. 00:09:05.810 --> 00:09:09.680 align:middle line:84% Now we have the Cauchy convergence theorem. 00:09:09.680 --> 00:09:20.630 align:middle line:84% So next, we will discuss is the Cauchy convergence. 00:09:20.630 --> 00:09:27.800 align:middle line:90% 00:09:27.800 --> 00:09:41.270 align:middle line:84% So this is the following that any Cauchy sequence 00:09:41.270 --> 00:09:42.480 align:middle line:90% is convergent. 00:09:42.480 --> 00:09:49.760 align:middle line:90% 00:09:49.760 --> 00:09:54.810 align:middle line:84% So remember that last time we proved the following theorem. 00:09:54.810 --> 00:10:06.280 align:middle line:84% So recall that last time we showed the converse. 00:10:06.280 --> 00:10:14.180 align:middle line:90% 00:10:14.180 --> 00:10:16.450 align:middle line:84% So namely, we proved the following theorem. 00:10:16.450 --> 00:10:19.490 align:middle line:90% 00:10:19.490 --> 00:10:22.040 align:middle line:90% We proved that if-- 00:10:22.040 --> 00:10:34.345 align:middle line:84% that any convergent sequence is a Cauchy sequence. 00:10:34.345 --> 00:10:42.500 align:middle line:90% 00:10:42.500 --> 00:10:44.450 align:middle line:84% And so the Cauchy convergence theorem 00:10:44.450 --> 00:10:47.810 align:middle line:84% is just a converse [? convergence. ?] 00:10:47.810 --> 00:10:55.310 align:middle line:84% Now, so we've come back to the proof in just a little bit. 00:10:55.310 --> 00:10:58.920 align:middle line:84% But before that, let me talk about some applications. 00:10:58.920 --> 00:11:07.970 align:middle line:90% 00:11:07.970 --> 00:11:13.290 align:middle line:84% And so the first one we started talking a little bit last time. 00:11:13.290 --> 00:11:16.050 align:middle line:84% And this is the contracting mapping theorem. 00:11:16.050 --> 00:11:17.300 align:middle line:90% So contracting. 00:11:17.300 --> 00:11:20.010 align:middle line:84% And that, in itself, have a number of applications. 00:11:20.010 --> 00:11:26.570 align:middle line:90% So contracting mapping theorem. 00:11:26.570 --> 00:11:32.180 align:middle line:90% 00:11:32.180 --> 00:11:34.550 align:middle line:90% And for that, I need some-- 00:11:34.550 --> 00:11:38.570 align:middle line:84% I need to recall some definitions. 00:11:38.570 --> 00:11:51.200 align:middle line:84% So a map-- at map T from R into R 00:11:51.200 --> 00:12:11.840 align:middle line:84% is said to be a contracting map if there exists 00:12:11.840 --> 00:12:15.560 align:middle line:90% some constant c positive. 00:12:15.560 --> 00:12:16.920 align:middle line:90% c is positive. 00:12:16.920 --> 00:12:20.700 align:middle line:84% And the key point is it will have to be positive. 00:12:20.700 --> 00:12:23.450 align:middle line:84% But the key point is that this constant 00:12:23.450 --> 00:12:26.360 align:middle line:90% has to be strictly less than 1. 00:12:26.360 --> 00:12:29.480 align:middle line:84% And then contracting mapping just 00:12:29.480 --> 00:12:33.680 align:middle line:84% means that if you are looking at the difference 00:12:33.680 --> 00:12:39.750 align:middle line:84% between the images, then they are closer together. 00:12:39.750 --> 00:12:45.550 align:middle line:84% They're strictly closer together than the point 00:12:45.550 --> 00:12:48.610 align:middle line:90% that they are images of. 00:12:48.610 --> 00:12:49.720 align:middle line:90% OK. 00:12:49.720 --> 00:12:51.380 align:middle line:90% So this is a contracting map. 00:12:51.380 --> 00:12:54.170 align:middle line:90% 00:12:54.170 --> 00:12:58.090 align:middle line:84% So the contracting mapping again, this here inequality is-- 00:12:58.090 --> 00:13:00.940 align:middle line:84% it's saying that, under the map, the images 00:13:00.940 --> 00:13:04.480 align:middle line:90% are squeezed more together. 00:13:04.480 --> 00:13:09.860 align:middle line:90% So now that's one concept. 00:13:09.860 --> 00:13:13.250 align:middle line:84% The other one we need is that for fixed point. 00:13:13.250 --> 00:13:19.990 align:middle line:90% 00:13:19.990 --> 00:13:22.010 align:middle line:84% The other concept is fixed point. 00:13:22.010 --> 00:13:26.200 align:middle line:90% 00:13:26.200 --> 00:13:31.275 align:middle line:90% So x here is a fixed point. 00:13:31.275 --> 00:13:37.780 align:middle line:90% 00:13:37.780 --> 00:13:52.820 align:middle line:84% For a map T, if it is under the map, the point remain unchanged. 00:13:52.820 --> 00:13:56.090 align:middle line:90% So T of x is equal to x. 00:13:56.090 --> 00:13:57.640 align:middle line:90% So this is what it needs to be. 00:13:57.640 --> 00:13:58.360 align:middle line:90% Yeah. 00:13:58.360 --> 00:14:01.683 align:middle line:84% STUDENT: But for that, it's not the same c for all x and y. 00:14:01.683 --> 00:14:03.350 align:middle line:84% TOBIAS COLDING: It is the same C. It is. 00:14:03.350 --> 00:14:03.850 align:middle line:90% Yeah. 00:14:03.850 --> 00:14:04.880 align:middle line:90% It's the same thing. 00:14:04.880 --> 00:14:05.470 align:middle line:90% Yeah. 00:14:05.470 --> 00:14:07.700 align:middle line:84% And we'll come back to this in just a second. 00:14:07.700 --> 00:14:13.780 align:middle line:84% But it is supposed to be the same c. 00:14:13.780 --> 00:14:16.660 align:middle line:84% You can weaken that a little bit. 00:14:16.660 --> 00:14:19.575 align:middle line:84% But essentially, I mean you sort of, you know, [INAUDIBLE]. 00:14:19.575 --> 00:14:21.200 align:middle line:84% We'll talk about this in just a second. 00:14:21.200 --> 00:14:23.120 align:middle line:90% But it's important. 00:14:23.120 --> 00:14:25.870 align:middle line:84% Just think about it, that it's a requirement 00:14:25.870 --> 00:14:28.010 align:middle line:90% that it's the same c for all. 00:14:28.010 --> 00:14:32.940 align:middle line:84% And the key point is that this c is strictly less than 1. 00:14:32.940 --> 00:14:33.440 align:middle line:90% OK. 00:14:33.440 --> 00:14:34.650 align:middle line:90% So this is a fixed point. 00:14:34.650 --> 00:14:41.020 align:middle line:90% 00:14:41.020 --> 00:14:47.895 align:middle line:84% So now the first [? theorem ?] is that-- 00:14:47.895 --> 00:14:51.100 align:middle line:90% 00:14:51.100 --> 00:14:52.605 align:middle line:90% so we have a Lemma here. 00:14:52.605 --> 00:14:58.150 align:middle line:90% 00:14:58.150 --> 00:15:21.160 align:middle line:84% If T here is a contracting map, then there's 00:15:21.160 --> 00:15:22.670 align:middle line:90% at most one fixed point. 00:15:22.670 --> 00:15:32.050 align:middle line:90% 00:15:32.050 --> 00:15:34.240 align:middle line:84% And there cannot be more than one fixed point. 00:15:34.240 --> 00:15:37.570 align:middle line:90% So let's prove that. 00:15:37.570 --> 00:15:38.180 align:middle line:90% So proof. 00:15:38.180 --> 00:15:41.500 align:middle line:90% 00:15:41.500 --> 00:15:42.290 align:middle line:90% So yeah. 00:15:42.290 --> 00:15:49.250 align:middle line:84% So I'm going to-- so suppose x and y are fixed points. 00:15:49.250 --> 00:15:52.780 align:middle line:90% 00:15:52.780 --> 00:15:58.265 align:middle line:84% And I want to show that x is equal to y. 00:15:58.265 --> 00:16:03.160 align:middle line:90% So I'm looking at the images. 00:16:03.160 --> 00:16:09.890 align:middle line:90% So let's continue on. 00:16:09.890 --> 00:16:30.490 align:middle line:90% 00:16:30.490 --> 00:16:33.620 align:middle line:84% So I'm looking at the images under f. 00:16:33.620 --> 00:16:35.180 align:middle line:90% So I'm looking at-- 00:16:35.180 --> 00:16:38.580 align:middle line:90% so I take T of x. 00:16:38.580 --> 00:16:40.440 align:middle line:84% And I'm looking at the difference 00:16:40.440 --> 00:16:44.160 align:middle line:90% between T of x and T of y. 00:16:44.160 --> 00:16:48.300 align:middle line:84% And I want to estimate that difference. 00:16:48.300 --> 00:16:52.780 align:middle line:84% Well, on one hand, x is supposed to be a fixed point. 00:16:52.780 --> 00:16:54.670 align:middle line:90% So T of x is the same as x. 00:16:54.670 --> 00:16:56.940 align:middle line:90% And likewise, for y. 00:16:56.940 --> 00:16:58.660 align:middle line:90% So this here is the same. 00:16:58.660 --> 00:17:00.360 align:middle line:90% So T of x is the same as x. 00:17:00.360 --> 00:17:03.910 align:middle line:84% T of y is supposed to be the same as y. 00:17:03.910 --> 00:17:05.700 align:middle line:90% So I have this. 00:17:05.700 --> 00:17:08.410 align:middle line:84% On the other hand, it's a contracting map. 00:17:08.410 --> 00:17:12.210 align:middle line:84% So this here is less than or equal to c 00:17:12.210 --> 00:17:16.829 align:middle line:90% times the norm of x minus y. 00:17:16.829 --> 00:17:19.150 align:middle line:90% But now c here-- 00:17:19.150 --> 00:17:22.349 align:middle line:84% remember c is strictly less than 1. 00:17:22.349 --> 00:17:26.020 align:middle line:84% So you can only have an inequality like this, right, 00:17:26.020 --> 00:17:31.030 align:middle line:84% if this number here is 0, right, when c is strictly less than 1. 00:17:31.030 --> 00:17:39.240 align:middle line:84% So this here implies that x minus y, the distance is zero. 00:17:39.240 --> 00:17:43.550 align:middle line:84% So of course, this means that x is equal to y. 00:17:43.550 --> 00:17:44.050 align:middle line:90% OK. 00:17:44.050 --> 00:17:45.720 align:middle line:90% So that's what we want to prove. 00:17:45.720 --> 00:17:46.684 align:middle line:90% Yeah? 00:17:46.684 --> 00:17:49.645 align:middle line:90% STUDENT: [INAUDIBLE] 00:17:49.645 --> 00:17:50.520 align:middle line:90% TOBIAS COLDING: Yeah. 00:17:50.520 --> 00:17:51.330 align:middle line:90% I'm sorry. 00:17:51.330 --> 00:17:53.400 align:middle line:90% That was-- yeah. 00:17:53.400 --> 00:17:58.350 align:middle line:84% It should have been this sequence because otherwise it 00:17:58.350 --> 00:17:59.280 align:middle line:90% wouldn't be-- 00:17:59.280 --> 00:18:02.610 align:middle line:90% if x is equal to y equal to z. 00:18:02.610 --> 00:18:04.540 align:middle line:90% So it should be the same. 00:18:04.540 --> 00:18:07.770 align:middle line:90% 00:18:07.770 --> 00:18:13.650 align:middle line:84% So our constructing maps have at most one fixed point. 00:18:13.650 --> 00:18:14.910 align:middle line:90% Okay. 00:18:14.910 --> 00:18:19.920 align:middle line:90% Now, let's-- so let me just-- 00:18:19.920 --> 00:18:21.640 align:middle line:90% so you're going to prove this. 00:18:21.640 --> 00:18:31.500 align:middle line:84% So I have harassed the graders to give you back the homework. 00:18:31.500 --> 00:18:32.910 align:middle line:90% It should be back today. 00:18:32.910 --> 00:18:35.310 align:middle line:84% I mean otherwise they're in trouble. 00:18:35.310 --> 00:18:36.670 align:middle line:90% So hopefully. 00:18:36.670 --> 00:18:41.370 align:middle line:84% And the assignment for next week is already posted. 00:18:41.370 --> 00:18:46.545 align:middle line:90% And here is-- here's-- 00:18:46.545 --> 00:18:49.590 align:middle line:90% 00:18:49.590 --> 00:18:51.880 align:middle line:90% so this is on the next-- 00:18:51.880 --> 00:18:59.810 align:middle line:84% so I will-- so here's something on P set 4. 00:18:59.810 --> 00:19:03.090 align:middle line:90% 00:19:03.090 --> 00:19:05.910 align:middle line:90% And I think it's-- 00:19:05.910 --> 00:19:06.580 align:middle line:90% OK. 00:19:06.580 --> 00:19:07.080 align:middle line:90% Yeah. 00:19:07.080 --> 00:19:08.440 align:middle line:90% I forget which one it is. 00:19:08.440 --> 00:19:11.640 align:middle line:84% But anyway, on the third P set, you 00:19:11.640 --> 00:19:14.910 align:middle line:84% will be asked to prove the following, 00:19:14.910 --> 00:19:28.410 align:middle line:84% that if T here is a contracting map, 00:19:28.410 --> 00:19:34.910 align:middle line:84% and x here is any point, any point 00:19:34.910 --> 00:19:46.400 align:middle line:84% in the domain of this map, then if you form the sequence a n-- 00:19:46.400 --> 00:19:52.250 align:middle line:90% sorry, a1, that's just x, a2-- 00:19:52.250 --> 00:19:53.750 align:middle line:90% let's call it a-- sorry. 00:19:53.750 --> 00:20:04.190 align:middle line:84% Let's call it a0 is x, a1 here is T of x, T of x is 0. 00:20:04.190 --> 00:20:05.500 align:middle line:90% So this is T of x. 00:20:05.500 --> 00:20:09.680 align:middle line:90% 00:20:09.680 --> 00:20:14.360 align:middle line:90% And a2 is T of a1. 00:20:14.360 --> 00:20:18.240 align:middle line:84% So this is the same as T of this. 00:20:18.240 --> 00:20:24.445 align:middle line:84% So this is T times T of x, which I will write like this. 00:20:24.445 --> 00:20:27.260 align:middle line:90% 00:20:27.260 --> 00:20:34.670 align:middle line:84% And a3-- so we continue this way. a3 is T of a2 so 00:20:34.670 --> 00:20:43.220 align:middle line:84% that is T applied three times, so T T T of x. 00:20:43.220 --> 00:20:50.430 align:middle line:84% And I'm going to write this as T cubed of x and so on. 00:20:50.430 --> 00:20:56.490 align:middle line:90% So a n is going to be T of a. 00:20:56.490 --> 00:21:07.200 align:middle line:84% Sorry, it's T of a n plus 1 is going to be T of a n. 00:21:07.200 --> 00:21:14.420 align:middle line:84% So that's the same as t to the power n plus 1 of x. 00:21:14.420 --> 00:21:17.150 align:middle line:90% 00:21:17.150 --> 00:21:23.210 align:middle line:84% So if you have a contracting map and you start with any point 00:21:23.210 --> 00:21:26.450 align:middle line:84% you want, and now you form a sequence that 00:21:26.450 --> 00:21:31.190 align:middle line:84% is based on that point and the n-th element 00:21:31.190 --> 00:21:33.620 align:middle line:84% is just where you have applied the map 00:21:33.620 --> 00:21:36.410 align:middle line:90% n times to the original point. 00:21:36.410 --> 00:21:42.020 align:middle line:84% So if you form that sequence, then you will prove-- 00:21:42.020 --> 00:21:44.253 align:middle line:90% you will show that-- 00:21:44.253 --> 00:21:47.450 align:middle line:90% 00:21:47.450 --> 00:21:54.740 align:middle line:84% in the homework, you will show that this sequence, so P63-- 00:21:54.740 --> 00:22:00.050 align:middle line:84% and I will just do a special case of it 00:22:00.050 --> 00:22:01.670 align:middle line:90% that you will start with. 00:22:01.670 --> 00:22:07.100 align:middle line:84% But we'll just do this now and then the rest you'll do-- 00:22:07.100 --> 00:22:08.670 align:middle line:90% I'll leave for the homework. 00:22:08.670 --> 00:22:15.505 align:middle line:84% But on P set 3, you prove that this a n is a Cauchy sequence. 00:22:15.505 --> 00:22:22.100 align:middle line:90% 00:22:22.100 --> 00:22:25.400 align:middle line:84% And what is the key starting point? 00:22:25.400 --> 00:22:30.530 align:middle line:90% So key starting point for this. 00:22:30.530 --> 00:22:38.990 align:middle line:84% So now, because you hear, you're already well started on P set 3, 00:22:38.990 --> 00:22:40.790 align:middle line:84% There's a key starting point for this 00:22:40.790 --> 00:22:43.430 align:middle line:84% is to prove that it's a Cauchy sequence. 00:22:43.430 --> 00:22:46.490 align:middle line:90% So I want to estimate. 00:22:46.490 --> 00:22:48.320 align:middle line:90% So this is not enough. 00:22:48.320 --> 00:22:50.060 align:middle line:90% I mean I need to prove-- 00:22:50.060 --> 00:22:52.460 align:middle line:90% what I need to show is-- 00:22:52.460 --> 00:22:58.160 align:middle line:84% remember I need to prove that when little m and little n 00:22:58.160 --> 00:23:00.560 align:middle line:84% are sufficiently large, then I want 00:23:00.560 --> 00:23:03.080 align:middle line:84% to estimate the distance between these two 00:23:03.080 --> 00:23:05.190 align:middle line:84% and show that they are bunched together. 00:23:05.190 --> 00:23:08.210 align:middle line:84% That's what we need to prove this Cauchy sequence. 00:23:08.210 --> 00:23:12.230 align:middle line:84% I won't quite do this, but I'll do a special case 00:23:12.230 --> 00:23:14.450 align:middle line:90% that is a key starting point. 00:23:14.450 --> 00:23:18.300 align:middle line:84% And that is the way I'm looking at this thing here. 00:23:18.300 --> 00:23:21.960 align:middle line:84% I'm looking at two consecutive elements in the sequence. 00:23:21.960 --> 00:23:24.590 align:middle line:84% And I'm estimating the distance between them. 00:23:24.590 --> 00:23:27.320 align:middle line:84% That's not enough to prove that's a Cauchy sequence, 00:23:27.320 --> 00:23:30.770 align:middle line:84% right, because I need to prove that for all n 00:23:30.770 --> 00:23:32.510 align:middle line:90% and m sufficiently large. 00:23:32.510 --> 00:23:36.870 align:middle line:84% And I'm just proving here that for any n sufficiently large, 00:23:36.870 --> 00:23:39.530 align:middle line:84% if I'm looking at the neighbor, then 00:23:39.530 --> 00:23:41.450 align:middle line:90% this distance is very small. 00:23:41.450 --> 00:23:45.570 align:middle line:84% You can use the estimates that, we will now 00:23:45.570 --> 00:23:50.090 align:middle line:84% prove to prove that it is, in fact, a Cauchy sequence. 00:23:50.090 --> 00:23:53.400 align:middle line:84% So in order to estimate this right, 00:23:53.400 --> 00:24:01.790 align:middle line:84% I can think about this thing here as a to the n plus 1. 00:24:01.790 --> 00:24:09.500 align:middle line:84% This here is just T of a to the n and a n itself. 00:24:09.500 --> 00:24:13.890 align:middle line:84% This is T of the previous element, right? 00:24:13.890 --> 00:24:16.220 align:middle line:90% This is how we define them. 00:24:16.220 --> 00:24:19.310 align:middle line:84% So this means that this thing here-- so again, I 00:24:19.310 --> 00:24:21.120 align:middle line:90% can just write this in here. 00:24:21.120 --> 00:24:27.915 align:middle line:84% So this thing is equal to a n minus T of a n minus 1. 00:24:27.915 --> 00:24:31.720 align:middle line:90% 00:24:31.720 --> 00:24:35.800 align:middle line:84% And now I can use that contracting map because now I 00:24:35.800 --> 00:24:37.790 align:middle line:90% have these two T's here. 00:24:37.790 --> 00:24:39.950 align:middle line:84% So I can use this contracting map. 00:24:39.950 --> 00:24:46.780 align:middle line:84% So I get less than or equal to and then a n minus a n minus 1. 00:24:46.780 --> 00:24:50.930 align:middle line:84% And so you see that that's the next-- 00:24:50.930 --> 00:24:53.800 align:middle line:84% the distance between the next pair 00:24:53.800 --> 00:24:56.900 align:middle line:84% is because c is strictly less than 1. 00:24:56.900 --> 00:24:59.740 align:middle line:84% It has decreased with some factor. 00:24:59.740 --> 00:25:00.760 align:middle line:90% Right? 00:25:00.760 --> 00:25:03.910 align:middle line:84% And so now the key starting point 00:25:03.910 --> 00:25:06.750 align:middle line:84% is that if you just iterate this-- 00:25:06.750 --> 00:25:14.560 align:middle line:90% 00:25:14.560 --> 00:25:22.300 align:middle line:84% so now if I iterate this, so I'm looking at a n plus 1 minus a n. 00:25:22.300 --> 00:25:24.940 align:middle line:84% And I now have that this thing here 00:25:24.940 --> 00:25:30.370 align:middle line:84% is less than c a n minus a n minus 1. 00:25:30.370 --> 00:25:32.330 align:middle line:90% But now I'm writing-- 00:25:32.330 --> 00:25:34.190 align:middle line:90% forget about the c for a second. 00:25:34.190 --> 00:25:35.840 align:middle line:84% I'm just writing this thing here, 00:25:35.840 --> 00:25:38.270 align:middle line:90% so I do the same thing again. 00:25:38.270 --> 00:25:42.760 align:middle line:84% I write this here as the image of a n minus 1. 00:25:42.760 --> 00:25:49.340 align:middle line:84% And I write this n minus 1 as the image of the previous one, 00:25:49.340 --> 00:25:51.340 align:middle line:90% so n minus 2. 00:25:51.340 --> 00:25:52.010 align:middle line:90% Right? 00:25:52.010 --> 00:25:54.080 align:middle line:90% And now so I write this. 00:25:54.080 --> 00:25:57.740 align:middle line:84% So c here I just have to carry along for the ride. 00:25:57.740 --> 00:26:06.430 align:middle line:84% And then I have T a n minus 1 minus T a n minus 2. 00:26:06.430 --> 00:26:08.150 align:middle line:90% And now forget about this. 00:26:08.150 --> 00:26:10.520 align:middle line:84% Just use this contracting mapping. 00:26:10.520 --> 00:26:13.570 align:middle line:84% So then this is less than or equal to. 00:26:13.570 --> 00:26:16.220 align:middle line:84% And then this c here I have to carry along. 00:26:16.220 --> 00:26:19.440 align:middle line:84% But then there's another c that pops out. 00:26:19.440 --> 00:26:19.940 align:middle line:90% Right. 00:26:19.940 --> 00:26:23.820 align:middle line:84% So this times c, that pops out because of the contracting 00:26:23.820 --> 00:26:24.700 align:middle line:90% mapping. 00:26:24.700 --> 00:26:30.420 align:middle line:84% And then a n minus 1 minus a n minus 2. 00:26:30.420 --> 00:26:32.130 align:middle line:84% And so you see that you have written this 00:26:32.130 --> 00:26:39.210 align:middle line:84% as c squared a n minus 2 minus 1 minus a n minus 2. 00:26:39.210 --> 00:26:41.380 align:middle line:90% And so now you continue it. 00:26:41.380 --> 00:26:44.800 align:middle line:84% And so you see that the further out you get-- 00:26:44.800 --> 00:26:47.520 align:middle line:84% so you continue this all the way down 00:26:47.520 --> 00:26:52.030 align:middle line:84% to that you're getting that this element is the initial point 00:26:52.030 --> 00:26:55.260 align:middle line:84% and this one is the first, your a1. 00:26:55.260 --> 00:26:57.900 align:middle line:90% This is a0 and this is a1. 00:26:57.900 --> 00:26:59.970 align:middle line:90% Right. 00:26:59.970 --> 00:27:02.640 align:middle line:84% And so you see that you're getting something 00:27:02.640 --> 00:27:09.690 align:middle line:84% which is like ac to some power n times the distance 00:27:09.690 --> 00:27:11.130 align:middle line:90% that you started with. 00:27:11.130 --> 00:27:18.060 align:middle line:84% So this is bounded by something like c to the n times a to 1 00:27:18.060 --> 00:27:22.890 align:middle line:90% minus a0, something like that. 00:27:22.890 --> 00:27:24.940 align:middle line:84% And so that's a key starting point. 00:27:24.940 --> 00:27:27.420 align:middle line:84% So if you are going very, very far out, 00:27:27.420 --> 00:27:31.260 align:middle line:84% remember that this number is smaller than 1 so 00:27:31.260 --> 00:27:35.700 align:middle line:84% this here will become incredibly small when n is large. 00:27:35.700 --> 00:27:36.900 align:middle line:90% Right. 00:27:36.900 --> 00:27:41.010 align:middle line:84% And so this is the key starting point 00:27:41.010 --> 00:27:44.760 align:middle line:84% to prove that if you take a contracting map, 00:27:44.760 --> 00:27:49.800 align:middle line:84% you start with any point, and you form this sequence here, 00:27:49.800 --> 00:27:51.290 align:middle line:90% then it's a Cauchy sequence. 00:27:51.290 --> 00:27:54.960 align:middle line:90% 00:27:54.960 --> 00:27:56.810 align:middle line:84% And I leave the rest for the homework. 00:27:56.810 --> 00:28:00.810 align:middle line:90% 00:28:00.810 --> 00:28:05.220 align:middle line:84% So now let's prove the contracting mapping 00:28:05.220 --> 00:28:09.450 align:middle line:84% theorem using the Cauchy convergence theorem. 00:28:09.450 --> 00:28:27.390 align:middle line:84% So using the Cauchy convergence theorem now 00:28:27.390 --> 00:28:29.740 align:middle line:84% through the contracting mapping theorem. 00:28:29.740 --> 00:28:45.330 align:middle line:90% 00:28:45.330 --> 00:28:48.180 align:middle line:84% So what is a contracting mapping theorem? 00:28:48.180 --> 00:28:50.040 align:middle line:84% The contracting mapping theorem is 00:28:50.040 --> 00:29:02.400 align:middle line:84% that if T here is a contracting map, so T from T from R to R, 00:29:02.400 --> 00:29:17.830 align:middle line:84% maps R into R is a contracting map, then T has a fixed point. 00:29:17.830 --> 00:29:23.460 align:middle line:90% 00:29:23.460 --> 00:29:26.760 align:middle line:84% We already know that such a fixed point is unique. 00:29:26.760 --> 00:29:30.810 align:middle line:84% We already proved that it can and must be one fixed point. 00:29:30.810 --> 00:29:31.330 align:middle line:90% OK. 00:29:31.330 --> 00:29:33.030 align:middle line:90% So let's try to prove that. 00:29:33.030 --> 00:29:50.475 align:middle line:84% So we're going to use this result from P set 3 to prove it. 00:29:50.475 --> 00:30:00.390 align:middle line:90% 00:30:00.390 --> 00:30:04.200 align:middle line:90% So I start with any point, x. 00:30:04.200 --> 00:30:06.690 align:middle line:90% You pick any point you want. 00:30:06.690 --> 00:30:12.690 align:middle line:84% And then we form the sequence, where you apply this here, 00:30:12.690 --> 00:30:15.160 align:middle line:90% the n time to x. 00:30:15.160 --> 00:30:19.560 align:middle line:90% 00:30:19.560 --> 00:30:23.660 align:middle line:90% So then we have-- 00:30:23.660 --> 00:30:33.700 align:middle line:84% So this from P set 2 that a n is a Cauchy sequence. 00:30:33.700 --> 00:30:44.420 align:middle line:90% 00:30:44.420 --> 00:31:05.600 align:middle line:84% But then from Cauchy convergence theorem, 00:31:05.600 --> 00:31:12.200 align:middle line:84% we have that a n is converging, that this here 00:31:12.200 --> 00:31:13.950 align:middle line:90% is a Cauchy sequence. 00:31:13.950 --> 00:31:17.810 align:middle line:84% But the convergence theorems say that any Cauchy sequence 00:31:17.810 --> 00:31:19.290 align:middle line:90% is actually convergent. 00:31:19.290 --> 00:31:22.910 align:middle line:90% So this here converge to some a. 00:31:22.910 --> 00:31:26.850 align:middle line:84% And now the idea is that we want to show-- 00:31:26.850 --> 00:31:38.730 align:middle line:84% so we want to show that a is a fixed point. 00:31:38.730 --> 00:31:46.040 align:middle line:90% 00:31:46.040 --> 00:31:46.980 align:middle line:90% So that's the idea. 00:31:46.980 --> 00:31:53.660 align:middle line:90% 00:31:53.660 --> 00:31:58.130 align:middle line:84% The contracting mapping theorem is actually extremely powerful. 00:31:58.130 --> 00:32:05.180 align:middle line:84% It, again, can be used to solve what's 00:32:05.180 --> 00:32:08.960 align:middle line:84% called ODE, Ordinary Differential Equations. 00:32:08.960 --> 00:32:11.660 align:middle line:84% And we'll talk a little bit about that at some later point 00:32:11.660 --> 00:32:12.480 align:middle line:90% in the class. 00:32:12.480 --> 00:32:18.840 align:middle line:84% But it can also be used to solve PDEs, which is more complicated, 00:32:18.840 --> 00:32:21.050 align:middle line:84% Partial Differential Equations, and in all kinds 00:32:21.050 --> 00:32:22.450 align:middle line:90% of other areas of math. 00:32:22.450 --> 00:32:27.700 align:middle line:90% 00:32:27.700 --> 00:32:28.200 align:middle line:90% OK. 00:32:28.200 --> 00:32:32.470 align:middle line:84% So now we want to prove that this a here is a fixed point. 00:32:32.470 --> 00:32:41.510 align:middle line:90% 00:32:41.510 --> 00:32:42.090 align:middle line:90% All right. 00:32:42.090 --> 00:32:44.960 align:middle line:90% 00:32:44.960 --> 00:32:47.330 align:middle line:84% And we'll come back in just a minute 00:32:47.330 --> 00:32:52.260 align:middle line:84% to prove something related to this. 00:32:52.260 --> 00:32:53.720 align:middle line:90% But let me just-- 00:32:53.720 --> 00:32:55.130 align:middle line:90% very simple. 00:32:55.130 --> 00:32:57.260 align:middle line:90% So I'm talking about a n here. 00:32:57.260 --> 00:32:59.810 align:middle line:84% I know that it's converging to a. 00:32:59.810 --> 00:33:05.070 align:middle line:84% I then look at the sequence T of a n. 00:33:05.070 --> 00:33:10.340 align:middle line:84% Well, T of a n, that was a n plus 1. 00:33:10.340 --> 00:33:14.210 align:middle line:84% It was just 1 further out of the sequence. 00:33:14.210 --> 00:33:20.240 align:middle line:84% But since a n converge, this means, of course-- 00:33:20.240 --> 00:33:24.710 align:middle line:84% since a n converge to a, this means that these here 00:33:24.710 --> 00:33:27.080 align:middle line:90% will also converge to a-- 00:33:27.080 --> 00:33:32.330 align:middle line:84% so since this here, a n plus 1, as n becomes very large, 00:33:32.330 --> 00:33:35.624 align:middle line:90% will converge to a. 00:33:35.624 --> 00:33:36.124 align:middle line:90% OK. 00:33:36.124 --> 00:33:39.198 align:middle line:90% 00:33:39.198 --> 00:33:45.290 align:middle line:84% Now If you're looking at T of a n-- 00:33:45.290 --> 00:33:48.350 align:middle line:90% so you want to estimate now. 00:33:48.350 --> 00:33:51.290 align:middle line:90% So what I want to prove is-- 00:33:51.290 --> 00:33:53.690 align:middle line:84% so now I'm looking at this distance. 00:33:53.690 --> 00:33:57.680 align:middle line:90% 00:33:57.680 --> 00:34:06.712 align:middle line:84% So I want to show that T of a is equal to a. 00:34:06.712 --> 00:34:07.920 align:middle line:90% That's what we want to prove. 00:34:07.920 --> 00:34:10.449 align:middle line:90% 00:34:10.449 --> 00:34:18.730 align:middle line:84% And so if I'm looking at T of a n, T of a n, 00:34:18.730 --> 00:34:21.159 align:middle line:90% this here was a n plus 1. 00:34:21.159 --> 00:34:28.960 align:middle line:84% And so if I'm looking at T of a n minus T 00:34:28.960 --> 00:34:34.120 align:middle line:84% of a, because it is a contracting map, 00:34:34.120 --> 00:34:40.810 align:middle line:84% this thing here is bounded or equal to c a n minus a. 00:34:40.810 --> 00:34:43.900 align:middle line:90% 00:34:43.900 --> 00:34:51.790 align:middle line:84% And this is also equal to a n plus 1 minus T of a. 00:34:51.790 --> 00:34:57.760 align:middle line:84% That was just the definition of a n plus 1. 00:34:57.760 --> 00:35:02.050 align:middle line:84% So now I have that a n converges to a. 00:35:02.050 --> 00:35:07.090 align:middle line:84% So this means that this here converges to 0. 00:35:07.090 --> 00:35:16.870 align:middle line:84% So now since a n converts to a, then you have that a n minus a-- 00:35:16.870 --> 00:35:21.850 align:middle line:84% you can think about this as a sequence that converges to 0. 00:35:21.850 --> 00:35:26.290 align:middle line:84% But this inequality here says that since this goes to 0, 00:35:26.290 --> 00:35:28.720 align:middle line:84% this thing here must also go to 0. 00:35:28.720 --> 00:35:41.620 align:middle line:84% So you see that therefore, a n plus 1 minus T of a goes to 0. 00:35:41.620 --> 00:35:44.800 align:middle line:90% 00:35:44.800 --> 00:35:47.200 align:middle line:84% But that this here goes to 0, this just 00:35:47.200 --> 00:35:53.110 align:middle line:84% means that a n plus 1, this must converge to T of a. 00:35:53.110 --> 00:35:56.290 align:middle line:90% 00:35:56.290 --> 00:36:00.310 align:middle line:84% But this sequence already converts to a. 00:36:00.310 --> 00:36:02.110 align:middle line:90% There can only be one limit. 00:36:02.110 --> 00:36:08.710 align:middle line:84% So we already have that a n plus 1 converge to a. 00:36:08.710 --> 00:36:11.860 align:middle line:84% But then these two numbers must be the same. 00:36:11.860 --> 00:36:17.270 align:middle line:84% So the conclusion here is that a must be equal to T of a. 00:36:17.270 --> 00:36:24.310 align:middle line:90% 00:36:24.310 --> 00:36:28.580 align:middle line:84% So from the Cauchy convergence theorem, 00:36:28.580 --> 00:36:32.950 align:middle line:84% then you get the contracting mapping theorem. 00:36:32.950 --> 00:36:37.840 align:middle line:84% And we already proved that you get a fixed point. 00:36:37.840 --> 00:36:43.480 align:middle line:84% And we already knew that there was at most one fixed point. 00:36:43.480 --> 00:36:48.010 align:middle line:84% And so this is a very powerful idea. 00:36:48.010 --> 00:36:52.390 align:middle line:84% Because for many things, it's almost impossible 00:36:52.390 --> 00:36:55.360 align:middle line:90% to find the actual fixed point. 00:36:55.360 --> 00:36:59.320 align:middle line:84% But what you do is that you're starting with some-- 00:36:59.320 --> 00:37:02.650 align:middle line:84% so in this case, you just said you pick any x. 00:37:02.650 --> 00:37:06.150 align:middle line:84% And then you just apply the map enough times. 00:37:06.150 --> 00:37:08.910 align:middle line:84% Sometimes you would start with a-- 00:37:08.910 --> 00:37:12.210 align:middle line:84% you can't find a fixed point, but you 00:37:12.210 --> 00:37:15.840 align:middle line:84% can find something pretty close to the fixed point. 00:37:15.840 --> 00:37:19.500 align:middle line:90% And then you iterate from there. 00:37:19.500 --> 00:37:23.050 align:middle line:90% And we'll see just in a second. 00:37:23.050 --> 00:37:26.520 align:middle line:84% So I already talked about the [INAUDIBLE] existence 00:37:26.520 --> 00:37:31.120 align:middle line:84% of solutions to ODEs and to PDEs. 00:37:31.120 --> 00:37:37.800 align:middle line:84% And now let me illustrate this last point in another example. 00:37:37.800 --> 00:37:42.660 align:middle line:84% And this comes back to your question about, 00:37:42.660 --> 00:37:44.350 align:middle line:90% is this the same? 00:37:44.350 --> 00:37:48.010 align:middle line:84% But do think about it as being the same for all x and y. 00:37:48.010 --> 00:37:50.910 align:middle line:84% But you could have some kind of local statement 00:37:50.910 --> 00:37:54.670 align:middle line:84% where, it was true for x and y in some small range. 00:37:54.670 --> 00:37:58.180 align:middle line:84% And that would be enough to guarantee you a fixed point. 00:37:58.180 --> 00:38:01.410 align:middle line:90% But in general, it's the same. 00:38:01.410 --> 00:38:06.510 align:middle line:84% The definition is supposed to be the same for all [? next ?] one. 00:38:06.510 --> 00:38:12.245 align:middle line:84% So let me explain that in what's called Newton method. 00:38:12.245 --> 00:38:16.650 align:middle line:90% 00:38:16.650 --> 00:38:18.050 align:middle line:90% So this is the Newton method. 00:38:18.050 --> 00:38:26.730 align:middle line:90% 00:38:26.730 --> 00:38:29.820 align:middle line:84% And so the idea is that you have a function. 00:38:29.820 --> 00:38:32.640 align:middle line:90% You have some function here. 00:38:32.640 --> 00:38:35.640 align:middle line:84% Maybe it doesn't look like a function, but it's supposed to. 00:38:35.640 --> 00:38:38.520 align:middle line:90% This is f of x. 00:38:38.520 --> 00:38:49.320 align:middle line:84% And you want to find the point where this function vanishes. 00:38:49.320 --> 00:38:57.185 align:middle line:84% And so the method is a numerical method to find such a point. 00:38:57.185 --> 00:39:03.854 align:middle line:90% 00:39:03.854 --> 00:39:07.540 align:middle line:84% If this function was very simple, 00:39:07.540 --> 00:39:10.690 align:middle line:84% like a linear function or a quadratic function, then, 00:39:10.690 --> 00:39:12.520 align:middle line:84% of course, it would be easy to find. 00:39:12.520 --> 00:39:15.690 align:middle line:84% But maybe this is a very complicated function. 00:39:15.690 --> 00:39:18.360 align:middle line:84% And so you wouldn't necessarily be 00:39:18.360 --> 00:39:21.930 align:middle line:84% able to say exactly what the point is. 00:39:21.930 --> 00:39:33.660 align:middle line:84% And so the idea is that Newton's methods require that you 00:39:33.660 --> 00:39:36.510 align:middle line:90% start with a good guess. 00:39:36.510 --> 00:39:39.420 align:middle line:84% But this is the idea of the Newton method. 00:39:39.420 --> 00:39:47.550 align:middle line:84% So here's the idea, that we're looking at a function like this. 00:39:47.550 --> 00:39:52.620 align:middle line:84% I'm going to assume that is not zero. 00:39:52.620 --> 00:39:54.330 align:middle line:90% So it's actually monotone. 00:39:54.330 --> 00:39:56.230 align:middle line:84% It's either increasing or decreasing. 00:39:56.230 --> 00:39:59.850 align:middle line:90% 00:39:59.850 --> 00:40:03.420 align:middle line:84% So I'm assuming that there's nowhere where 00:40:03.420 --> 00:40:05.520 align:middle line:90% the differential vanishes. 00:40:05.520 --> 00:40:10.020 align:middle line:84% And then I defined a map, T of x, 00:40:10.020 --> 00:40:17.255 align:middle line:84% to be equal to x minus f of x, f prime of x. 00:40:17.255 --> 00:40:22.740 align:middle line:90% 00:40:22.740 --> 00:40:25.740 align:middle line:84% This seems maybe far away from contracting now. 00:40:25.740 --> 00:40:30.510 align:middle line:84% But if you define it like this, now 00:40:30.510 --> 00:40:36.300 align:middle line:84% let me take the derivative of this. 00:40:36.300 --> 00:40:38.910 align:middle line:84% If I take the derivative of this map, 00:40:38.910 --> 00:40:41.790 align:middle line:84% then I get that this thing here, the derivative of x, that's 00:40:41.790 --> 00:40:43.350 align:middle line:90% just 1. 00:40:43.350 --> 00:40:49.240 align:middle line:84% The derivative of this in here, this is a quotient. 00:40:49.240 --> 00:40:56.730 align:middle line:84% So it's f prime times its derivative of this times 00:40:56.730 --> 00:41:03.050 align:middle line:84% f prime and then f prime squared minus 00:41:03.050 --> 00:41:07.940 align:middle line:84% and then f f double prime, like that. 00:41:07.940 --> 00:41:13.220 align:middle line:84% That's the quotient for taking the derivative. 00:41:13.220 --> 00:41:15.510 align:middle line:84% And so you see that you can write this-- 00:41:15.510 --> 00:41:17.910 align:middle line:90% if you split this up into two-- 00:41:17.910 --> 00:41:19.700 align:middle line:90% this here is one. 00:41:19.700 --> 00:41:21.030 align:middle line:90% You split it up to two. 00:41:21.030 --> 00:41:26.270 align:middle line:84% This here is f prime squared divided by f prime squared. 00:41:26.270 --> 00:41:28.700 align:middle line:90% And then this here becomes plus. 00:41:28.700 --> 00:41:34.250 align:middle line:84% And then I'm writing it as f f double prime 00:41:34.250 --> 00:41:38.480 align:middle line:90% over [INAUDIBLE] squared. 00:41:38.480 --> 00:41:41.880 align:middle line:84% And so you see that this thing here cancel with that. 00:41:41.880 --> 00:41:43.550 align:middle line:90% So these here cancel. 00:41:43.550 --> 00:41:48.260 align:middle line:84% So you have that the derivative of this map 00:41:48.260 --> 00:41:56.990 align:middle line:84% is just f of x, f prime, double prime of x, f prime 00:41:56.990 --> 00:41:59.095 align:middle line:90% of x, and then squared. 00:41:59.095 --> 00:42:02.210 align:middle line:90% That's the derivative. 00:42:02.210 --> 00:42:17.840 align:middle line:84% And now, of course, if I'm using the mean value theorem here, 00:42:17.840 --> 00:42:22.080 align:middle line:84% then I have that the derivative-- 00:42:22.080 --> 00:42:27.830 align:middle line:90% 00:42:27.830 --> 00:42:33.500 align:middle line:84% then I have that T of x minus T of y-- 00:42:33.500 --> 00:42:37.100 align:middle line:84% the mean value theorem say that this thing here 00:42:37.100 --> 00:42:41.240 align:middle line:90% is the derivative. 00:42:41.240 --> 00:42:43.530 align:middle line:84% It's actually Wallis theorem or whatever. 00:42:43.530 --> 00:42:46.250 align:middle line:84% But you can also think about this mean value theorem. 00:42:46.250 --> 00:42:51.800 align:middle line:84% It's a derivative at somewhere in between x and y times 00:42:51.800 --> 00:42:53.665 align:middle line:90% the difference between x and y. 00:42:53.665 --> 00:42:55.990 align:middle line:90% 00:42:55.990 --> 00:42:57.240 align:middle line:90% That's the mean value theorem. 00:42:57.240 --> 00:43:02.360 align:middle line:90% 00:43:02.360 --> 00:43:07.520 align:middle line:84% So you see that if we already knew-- so if we can show-- 00:43:07.520 --> 00:43:17.850 align:middle line:84% if we knew that the derivative was bounded everywhere, 00:43:17.850 --> 00:43:22.670 align:middle line:84% was bounded by some c, little c that was less than 1-- 00:43:22.670 --> 00:43:25.350 align:middle line:84% if you had that, then, of course, 00:43:25.350 --> 00:43:27.720 align:middle line:84% then you just take the absolute value of this. 00:43:27.720 --> 00:43:31.820 align:middle line:84% So then you have that T of x minus T of y, 00:43:31.820 --> 00:43:36.140 align:middle line:84% the absolute value was equal to T 00:43:36.140 --> 00:43:41.090 align:middle line:84% prime in this intermediate point, x minus y. 00:43:41.090 --> 00:43:43.130 align:middle line:84% But in this intermediate point, we're 00:43:43.130 --> 00:43:45.230 align:middle line:84% assuming that everywhere in the derivative 00:43:45.230 --> 00:43:46.860 align:middle line:90% is less than or equal to c. 00:43:46.860 --> 00:43:52.040 align:middle line:84% So now this is less than or equal to c, so c x minus y. 00:43:52.040 --> 00:43:55.160 align:middle line:84% And we're assuming that c is strictly less than 1. 00:43:55.160 --> 00:43:59.740 align:middle line:84% So you see that this here, if we knew that the derivative of T 00:43:59.740 --> 00:44:02.690 align:middle line:84% was strictly less than 1 everywhere, 00:44:02.690 --> 00:44:08.200 align:middle line:84% then we would know that this map is a contracting map. 00:44:08.200 --> 00:44:08.780 align:middle line:90% OK. 00:44:08.780 --> 00:44:12.880 align:middle line:84% But, of course, what we have for the derivative is, 00:44:12.880 --> 00:44:14.265 align:middle line:90% we have this in here. 00:44:14.265 --> 00:44:19.420 align:middle line:90% 00:44:19.420 --> 00:44:24.150 align:middle line:84% But now the idea is that if you start with a really good guess-- 00:44:24.150 --> 00:44:34.660 align:middle line:90% 00:44:34.660 --> 00:44:42.375 align:middle line:84% so the idea with [INAUDIBLE] is, start with a good guess. 00:44:42.375 --> 00:44:49.300 align:middle line:90% 00:44:49.300 --> 00:44:53.920 align:middle line:84% And then I'm not going to make that too precise, 00:44:53.920 --> 00:44:55.700 align:middle line:84% as I'm putting it in quotation marks. 00:44:55.700 --> 00:44:59.600 align:middle line:84% But the idea is that you want to-- 00:44:59.600 --> 00:45:00.650 align:middle line:90% so good guesses. 00:45:00.650 --> 00:45:05.650 align:middle line:84% Remember that what we want is, we're interested in this point 00:45:05.650 --> 00:45:09.400 align:middle line:90% here, where this function here-- 00:45:09.400 --> 00:45:13.426 align:middle line:84% we're interested in the point where the function vanishes. 00:45:13.426 --> 00:45:16.000 align:middle line:84% So starting with a good guess means 00:45:16.000 --> 00:45:20.690 align:middle line:84% that you may not start where the function vanishes, 00:45:20.690 --> 00:45:24.800 align:middle line:84% but you're starting somewhere where the value is really small. 00:45:24.800 --> 00:45:27.980 align:middle line:90% That would be a good guess. 00:45:27.980 --> 00:45:38.770 align:middle line:84% And so start with a guess, meaning that f at that point 00:45:38.770 --> 00:45:43.365 align:middle line:90% is small, is very small. 00:45:43.365 --> 00:45:45.880 align:middle line:90% 00:45:45.880 --> 00:45:47.380 align:middle line:84% Now, of course, you're going to ask, 00:45:47.380 --> 00:45:49.570 align:middle line:90% what do you mean by very small? 00:45:49.570 --> 00:45:55.600 align:middle line:84% So very small here is going to mean that the value-- 00:45:55.600 --> 00:45:58.990 align:middle line:90% so now we have, from this here-- 00:45:58.990 --> 00:46:05.110 align:middle line:84% so we have that the derivative here of this map 00:46:05.110 --> 00:46:10.160 align:middle line:84% here was equal to-- these two terms cancel out. 00:46:10.160 --> 00:46:14.230 align:middle line:84% So this is f of x, f double prime 00:46:14.230 --> 00:46:20.260 align:middle line:90% of x, f prime of x squared. 00:46:20.260 --> 00:46:24.370 align:middle line:84% So starting with a good guess means that this thing here 00:46:24.370 --> 00:46:26.180 align:middle line:90% is really, really small. 00:46:26.180 --> 00:46:29.630 align:middle line:84% And, in fact, I want it to be small compared with this. 00:46:29.630 --> 00:46:32.990 align:middle line:84% So if this here is really small compared with that, 00:46:32.990 --> 00:46:35.830 align:middle line:84% then I can guarantee that the absolute value of this 00:46:35.830 --> 00:46:38.626 align:middle line:90% is smaller than 1. 00:46:38.626 --> 00:46:42.370 align:middle line:84% And so you see that if you start with a really good guess, 00:46:42.370 --> 00:46:45.730 align:middle line:84% then it would be a contracting mapping. 00:46:45.730 --> 00:46:49.100 align:middle line:84% And then, of course, you need to prove that. 00:46:49.100 --> 00:46:51.230 align:middle line:90% So you're forming the sequence. 00:46:51.230 --> 00:46:53.500 align:middle line:90% And you need to prove-- 00:46:53.500 --> 00:46:56.260 align:middle line:84% somehow, you need to estimate, that you 00:46:56.260 --> 00:47:01.240 align:middle line:84% need to be able to say that once you form the sequence, 00:47:01.240 --> 00:47:03.730 align:middle line:84% the next thing that it's spitting out 00:47:03.730 --> 00:47:07.420 align:middle line:84% is also a good guess, so that you stay 00:47:07.420 --> 00:47:09.170 align:middle line:90% in this range of good guess. 00:47:09.170 --> 00:47:12.970 align:middle line:84% And if it has this property, then it 00:47:12.970 --> 00:47:15.400 align:middle line:84% would be a contracting map, and you would 00:47:15.400 --> 00:47:17.110 align:middle line:90% guarantee to get a fixed point. 00:47:17.110 --> 00:47:21.230 align:middle line:84% But that's all the Newton methods say. 00:47:21.230 --> 00:47:27.260 align:middle line:84% So you need to justify why, if you start with a good guess, 00:47:27.260 --> 00:47:30.170 align:middle line:84% why, as you're iterating this process, 00:47:30.170 --> 00:47:34.780 align:middle line:84% why you keep in the range of good guesses. 00:47:34.780 --> 00:47:40.610 align:middle line:84% But the Newton method doesn't tell you anything about that. 00:47:40.610 --> 00:47:43.455 align:middle line:84% But it require that to be applied. 00:47:43.455 --> 00:47:46.730 align:middle line:90% 00:47:46.730 --> 00:47:47.230 align:middle line:90% OK. 00:47:47.230 --> 00:47:48.880 align:middle line:90% So it's a contracting map. 00:47:48.880 --> 00:47:52.390 align:middle line:90% This T is a contracting map. 00:47:52.390 --> 00:47:54.790 align:middle line:84% If I can start with a good guess-- 00:47:54.790 --> 00:47:57.300 align:middle line:84% and when I do this, when I form-- so I 00:47:57.300 --> 00:47:59.080 align:middle line:90% start with this good guess. 00:47:59.080 --> 00:48:04.440 align:middle line:84% And then I'm looking at that the next thing is this T of x 00:48:04.440 --> 00:48:08.010 align:middle line:90% and then a T squared of x. 00:48:08.010 --> 00:48:11.710 align:middle line:84% By then, I want all of these to also be good guesses. 00:48:11.710 --> 00:48:15.900 align:middle line:84% And then I'm saying that as n goes to infinity, 00:48:15.900 --> 00:48:20.770 align:middle line:84% this here is constant, is going to converge to this fixed point. 00:48:20.770 --> 00:48:22.890 align:middle line:90% I guess I called it x0. 00:48:22.890 --> 00:48:34.890 align:middle line:84% But this is the fixed point, fixed point for T. 00:48:34.890 --> 00:48:39.130 align:middle line:84% So now we started with a function. 00:48:39.130 --> 00:48:44.760 align:middle line:84% And what we wanted to find a root, so at x0, 00:48:44.760 --> 00:48:47.216 align:middle line:90% where this function vanishes. 00:48:47.216 --> 00:48:50.680 align:middle line:84% And I constructed a contracting map. 00:48:50.680 --> 00:48:54.810 align:middle line:84% But now I will need to relate the fixed point 00:48:54.810 --> 00:48:57.780 align:middle line:84% for the contracting map to the original problem. 00:48:57.780 --> 00:49:02.280 align:middle line:84% So what is a fixed point for the contracting map? 00:49:02.280 --> 00:49:07.590 align:middle line:84% So the map here, the map we defined 00:49:07.590 --> 00:49:15.365 align:middle line:84% was T of x is equal to x minus f of x over f prime of x. 00:49:15.365 --> 00:49:17.970 align:middle line:90% 00:49:17.970 --> 00:49:20.320 align:middle line:90% But a fixed point-- 00:49:20.320 --> 00:49:24.810 align:middle line:84% so if x0 is a fixed point, then this 00:49:24.810 --> 00:49:32.070 align:middle line:84% means that this here is the same as x0 minus f of x0, f 00:49:32.070 --> 00:49:34.440 align:middle line:90% prime of x0. 00:49:34.440 --> 00:49:37.300 align:middle line:84% And now you can just write-- so therefore, 00:49:37.300 --> 00:49:45.210 align:middle line:84% this implies that f of x0 minus f of x0 over f prime of x0, this 00:49:45.210 --> 00:49:45.870 align:middle line:90% here. 00:49:45.870 --> 00:49:48.430 align:middle line:84% Then you would subtract x0 0 on both sides. 00:49:48.430 --> 00:49:50.520 align:middle line:90% And this is equal to 0. 00:49:50.520 --> 00:49:53.710 align:middle line:84% But that, of course, means the denominator must be 0. 00:49:53.710 --> 00:49:56.400 align:middle line:90% 00:49:56.400 --> 00:50:02.880 align:middle line:84% So a fixed point is indeed a point where f of x at 0 00:50:02.880 --> 00:50:04.150 align:middle line:90% is equal to 0. 00:50:04.150 --> 00:50:07.785 align:middle line:90% 00:50:07.785 --> 00:50:08.285 align:middle line:90% OK. 00:50:08.285 --> 00:50:12.090 align:middle line:90% 00:50:12.090 --> 00:50:17.850 align:middle line:84% So now we want to prove the Cauchy convergence theorem. 00:50:17.850 --> 00:50:20.370 align:middle line:84% But we're going to cheat just a little bit, because we 00:50:20.370 --> 00:50:23.640 align:middle line:84% are going to deduce it from another theorem 00:50:23.640 --> 00:50:26.790 align:middle line:90% that I will prove next time. 00:50:26.790 --> 00:50:34.920 align:middle line:84% So to prove the Cauchy convergence 00:50:34.920 --> 00:50:38.760 align:middle line:84% for the Bolzano-Weierstrass theorem-- 00:50:38.760 --> 00:50:41.020 align:middle line:84% and that theorem, we'll prove next time. 00:50:41.020 --> 00:50:48.300 align:middle line:90% 00:50:48.300 --> 00:50:49.685 align:middle line:90% So what is the Bolzano-- 00:50:49.685 --> 00:51:05.550 align:middle line:90% 00:51:05.550 --> 00:51:07.870 align:middle line:84% so the Bolzano-Weierstrass theorem. 00:51:07.870 --> 00:51:09.630 align:middle line:84% So the Bolzano-Weierstrass theorem 00:51:09.630 --> 00:51:30.480 align:middle line:84% says that every bounded sequence has a convergent subsequence. 00:51:30.480 --> 00:51:34.920 align:middle line:90% 00:51:34.920 --> 00:51:36.940 align:middle line:90% So that's a Bolzano-Weierstrass. 00:51:36.940 --> 00:51:39.852 align:middle line:90% 00:51:39.852 --> 00:51:42.330 align:middle line:90% Now let's try to-- 00:51:42.330 --> 00:51:43.570 align:middle line:90% we'll assume this. 00:51:43.570 --> 00:51:46.470 align:middle line:84% Again, I'll prove this next time. 00:51:46.470 --> 00:51:51.510 align:middle line:84% But let's use this theorem to prove the Cauchy convergence 00:51:51.510 --> 00:51:53.630 align:middle line:90% theorem. 00:51:53.630 --> 00:52:11.750 align:middle line:84% So from the Bolzano-Weierstrass theorem, 00:52:11.750 --> 00:52:28.470 align:middle line:84% we get the Cauchy convergence theorem as follows. 00:52:28.470 --> 00:52:35.480 align:middle line:90% 00:52:35.480 --> 00:52:38.090 align:middle line:84% So, remember, the Cauchy convergence theorem 00:52:38.090 --> 00:52:42.260 align:middle line:84% say that if you have a Cauchy sequence, then it is convergent. 00:52:42.260 --> 00:52:56.060 align:middle line:84% So suppose xn this [? position, ?] OK? 00:52:56.060 --> 00:52:59.000 align:middle line:84% I want to prove that it's convergent. 00:52:59.000 --> 00:53:10.320 align:middle line:84% But we already shown that any Cauchy sequence-- 00:53:10.320 --> 00:53:16.400 align:middle line:90% 00:53:16.400 --> 00:53:20.950 align:middle line:84% shown that xn, any Cauchy sequence is bounded. 00:53:20.950 --> 00:53:24.950 align:middle line:90% 00:53:24.950 --> 00:53:27.222 align:middle line:90% This is this [INAUDIBLE]. 00:53:27.222 --> 00:53:31.410 align:middle line:90% 00:53:31.410 --> 00:53:44.990 align:middle line:84% Therefore, by the Bolzano-Weierstrass theorem, 00:53:44.990 --> 00:53:51.600 align:middle line:84% there's a convergent subsequence. 00:53:51.600 --> 00:53:59.090 align:middle line:90% 00:53:59.090 --> 00:54:01.820 align:middle line:84% So there's a convergent subsequent xnk. 00:54:01.820 --> 00:54:23.690 align:middle line:90% 00:54:23.690 --> 00:54:32.600 align:middle line:84% So you have that this subsequence 00:54:32.600 --> 00:54:37.250 align:middle line:90% here converge to some x. 00:54:37.250 --> 00:54:42.200 align:middle line:84% And now I want to show that the original sequence converge to x. 00:54:42.200 --> 00:54:46.630 align:middle line:90% 00:54:46.630 --> 00:55:02.800 align:middle line:84% So since xn is a Cauchy sequence, 00:55:02.800 --> 00:55:14.600 align:middle line:84% given epsilon greater than 0, there exists a capital N 00:55:14.600 --> 00:55:20.500 align:middle line:84% so that if m and n are bigger than capital N, 00:55:20.500 --> 00:55:28.990 align:middle line:84% then xn minus xm is less than epsilon over 2. 00:55:28.990 --> 00:55:33.250 align:middle line:84% I'm using the epsilon in the definition 00:55:33.250 --> 00:55:34.640 align:middle line:90% of being a Cauchy sequence. 00:55:34.640 --> 00:55:37.780 align:middle line:90% 00:55:37.780 --> 00:55:41.145 align:middle line:84% And let's call this-- this is the N. Let's called it N1. 00:55:41.145 --> 00:55:44.590 align:middle line:90% 00:55:44.590 --> 00:55:59.860 align:middle line:84% So now since xnk converge to x, but that epsilon, 00:55:59.860 --> 00:56:15.190 align:middle line:84% there exist N2 so that if k here is bigger than N2, 00:56:15.190 --> 00:56:23.470 align:middle line:84% then xnk minus x is less than epsilon over 2. 00:56:23.470 --> 00:56:45.640 align:middle line:90% 00:56:45.640 --> 00:57:04.491 align:middle line:84% So now we take n to be equal to the maximum of N1 and nN2. 00:57:04.491 --> 00:57:07.690 align:middle line:90% 00:57:07.690 --> 00:57:10.730 align:middle line:84% This is the one that corresponds to y. 00:57:10.730 --> 00:57:14.680 align:middle line:84% So this is where k is taking equal to this N2. 00:57:14.680 --> 00:57:17.830 align:middle line:90% 00:57:17.830 --> 00:57:25.510 align:middle line:84% So if we take n like that and you're looking at n bigger equal 00:57:25.510 --> 00:57:29.860 align:middle line:84% to N and you now want to estimate this thing here, 00:57:29.860 --> 00:57:43.480 align:middle line:84% then you can think about this difference here as x minus xn, 00:57:43.480 --> 00:58:02.655 align:middle line:84% like this, plus x, xN minus xn [INAUDIBLE] like that. 00:58:02.655 --> 00:58:10.190 align:middle line:90% 00:58:10.190 --> 00:58:14.090 align:middle line:84% So this is now in this subsequent. 00:58:14.090 --> 00:58:18.290 align:middle line:84% And so one of them-- this is less than epsilon over 2. 00:58:18.290 --> 00:58:21.400 align:middle line:84% And this here is less than epsilon over 2. 00:58:21.400 --> 00:58:23.350 align:middle line:84% And so you get that this difference here 00:58:23.350 --> 00:58:24.280 align:middle line:90% was less than epsilon. 00:58:24.280 --> 00:58:28.780 align:middle line:90% 00:58:28.780 --> 00:58:32.770 align:middle line:84% So this is why the Bolzano-Weierstrass theorem 00:58:32.770 --> 00:58:37.400 align:middle line:84% implies the Cauchy convergence theorem. 00:58:37.400 --> 00:58:37.900 align:middle line:90% OK. 00:58:37.900 --> 00:58:41.280 align:middle line:84% So now the next thing is that I want 00:58:41.280 --> 00:58:48.360 align:middle line:84% to give another application of the Cauchy convergence here. 00:58:48.360 --> 00:58:52.360 align:middle line:84% So here's another application of the Cauchy convergence theorem. 00:58:52.360 --> 00:59:08.250 align:middle line:90% 00:59:08.250 --> 00:59:12.070 align:middle line:84% So one application was to the contracting mapping theorem. 00:59:12.070 --> 00:59:17.180 align:middle line:84% But here's another application of the Cauchy convergence 00:59:17.180 --> 00:59:17.680 align:middle line:90% theorem. 00:59:17.680 --> 00:59:26.880 align:middle line:90% 00:59:26.880 --> 00:59:29.260 align:middle line:84% So this is the extreme value theorem. 00:59:29.260 --> 00:59:38.700 align:middle line:90% 00:59:38.700 --> 00:59:47.430 align:middle line:84% Now, before I explain that, let me 00:59:47.430 --> 00:59:51.690 align:middle line:84% introduce this notion of a function being 00:59:51.690 --> 00:59:56.340 align:middle line:90% continuous in a rigorous way. 00:59:56.340 --> 01:00:02.010 align:middle line:84% So what does it mean for a function to be continuous? 01:00:02.010 --> 01:00:13.882 align:middle line:84% So a function f, say, from R to R could be a subset of R, 01:00:13.882 --> 01:00:31.320 align:middle line:84% could be all of R, is said to be continuous at some point x0 01:00:31.320 --> 01:00:39.090 align:middle line:84% if, for all epsilon greater than 0, 01:00:39.090 --> 01:00:53.280 align:middle line:84% there exist delta greater than 0 such that if x minus x0 01:00:53.280 --> 01:00:58.860 align:middle line:84% is smaller than delta, then the images are 01:00:58.860 --> 01:01:01.985 align:middle line:90% within epsilon region. 01:01:01.985 --> 01:01:06.210 align:middle line:90% 01:01:06.210 --> 01:01:08.910 align:middle line:90% So if you think about it-- 01:01:08.910 --> 01:01:13.890 align:middle line:84% so if you have the function-- this is a function. 01:01:13.890 --> 01:01:20.190 align:middle line:84% You should say that it's continuous at some x0, means 01:01:20.190 --> 01:01:23.520 align:middle line:90% that if this thing here is-- 01:01:23.520 --> 01:01:27.390 align:middle line:90% this here is now f of x0. 01:01:27.390 --> 01:01:29.850 align:middle line:84% If you take a little interval around here-- 01:01:29.850 --> 01:01:32.060 align:middle line:90% this is a epsilon interval-- 01:01:32.060 --> 01:01:36.530 align:middle line:84% around here, then there should be some delta interval 01:01:36.530 --> 01:01:39.710 align:middle line:84% around here so that if you're looking at images here, 01:01:39.710 --> 01:01:43.140 align:middle line:90% then it will map into here. 01:01:43.140 --> 01:01:46.895 align:middle line:84% That's what it means to be continuous. 01:01:46.895 --> 01:01:50.240 align:middle line:84% OK, so this is continuous at a point. 01:01:50.240 --> 01:01:53.270 align:middle line:84% And a function is said to be continuous 01:01:53.270 --> 01:01:55.300 align:middle line:84% if it is continuous at all points. 01:01:55.300 --> 01:01:59.780 align:middle line:90% 01:01:59.780 --> 01:02:23.300 align:middle line:84% So a function is said to be continuous 01:02:23.300 --> 01:02:33.730 align:middle line:84% if it is continuous at all points. 01:02:33.730 --> 01:02:42.050 align:middle line:90% 01:02:42.050 --> 01:02:45.890 align:middle line:84% And so now we have a little theorem here. 01:02:45.890 --> 01:02:53.000 align:middle line:84% And this is often useful way of thinking about continuity. 01:02:53.000 --> 01:02:58.590 align:middle line:90% So if f is continuous-- 01:02:58.590 --> 01:03:02.000 align:middle line:90% 01:03:02.000 --> 01:03:05.430 align:middle line:84% so let's just say it doesn't need to be all of R, 01:03:05.430 --> 01:03:08.030 align:middle line:84% but you just have to be careful that everything that you 01:03:08.030 --> 01:03:09.930 align:middle line:90% talk about is in the domain. 01:03:09.930 --> 01:03:13.550 align:middle line:84% But let's just say that f is a function from R to R, 01:03:13.550 --> 01:03:14.626 align:middle line:90% is continuous. 01:03:14.626 --> 01:03:19.880 align:middle line:90% 01:03:19.880 --> 01:03:29.960 align:middle line:84% So if f is continuous and xn is a sequence that converges 01:03:29.960 --> 01:03:35.900 align:middle line:90% to x, then the images-- 01:03:35.900 --> 01:03:38.460 align:middle line:90% so f of xn-- 01:03:38.460 --> 01:03:40.340 align:middle line:90% this is another sequence-- 01:03:40.340 --> 01:03:42.480 align:middle line:90% those converge to f of x. 01:03:42.480 --> 01:03:46.580 align:middle line:90% 01:03:46.580 --> 01:03:51.200 align:middle line:84% This is a useful way of thinking about being continuous. 01:03:51.200 --> 01:03:53.810 align:middle line:90% 01:03:53.810 --> 01:03:56.830 align:middle line:84% And so now, why is this the case? 01:03:56.830 --> 01:04:01.280 align:middle line:90% 01:04:01.280 --> 01:04:13.826 align:middle line:84% So proof-- since f is continuous-- 01:04:13.826 --> 01:04:18.380 align:middle line:90% 01:04:18.380 --> 01:04:22.670 align:middle line:84% you only need that is continuous at each x. 01:04:22.670 --> 01:04:25.490 align:middle line:84% So since it's continuous at x, this 01:04:25.490 --> 01:04:28.585 align:middle line:84% means that for all epsilon that are given-- 01:04:28.585 --> 01:04:31.100 align:middle line:90% 01:04:31.100 --> 01:04:34.280 align:middle line:84% given epsilon greater than 0, there 01:04:34.280 --> 01:04:50.210 align:middle line:84% exist delta greater than 0 such that if x minus y-- 01:04:50.210 --> 01:04:52.970 align:middle line:84% this is this x here that you think 01:04:52.970 --> 01:04:56.660 align:middle line:84% about-- is fixed, if you have another point that 01:04:56.660 --> 01:05:08.540 align:middle line:84% is within delta, then the images f of x minus f of y [INAUDIBLE]. 01:05:08.540 --> 01:05:13.610 align:middle line:84% This is just that it's continuous at that one point, x. 01:05:13.610 --> 01:05:26.200 align:middle line:84% Now, since xn converges to x, this 01:05:26.200 --> 01:05:29.590 align:middle line:84% means that if you choose this delta here, 01:05:29.590 --> 01:05:33.310 align:middle line:84% in the definition of convergence, 01:05:33.310 --> 01:05:45.170 align:middle line:84% then there exist capital N such that 01:05:45.170 --> 01:05:47.840 align:middle line:84% if little n is bigger than capital N, 01:05:47.840 --> 01:05:54.670 align:middle line:84% then xn minus x is less than delta. 01:05:54.670 --> 01:05:58.910 align:middle line:84% But now you see that if little n is bigger than capital N, 01:05:58.910 --> 01:06:03.410 align:middle line:90% then these lies this close to x. 01:06:03.410 --> 01:06:06.920 align:middle line:84% But then this say that it is-- so therefore, 01:06:06.920 --> 01:06:18.940 align:middle line:84% the images f of xn minus f of x is less than epsilon. 01:06:18.940 --> 01:06:21.640 align:middle line:84% And this was just as long as little n was bigger 01:06:21.640 --> 01:06:23.530 align:middle line:84% than capital N. And that's exactly 01:06:23.530 --> 01:06:27.040 align:middle line:84% proving that this sequence converge to that. 01:06:27.040 --> 01:06:28.070 align:middle line:90% So that's the claim. 01:06:28.070 --> 01:06:33.670 align:middle line:90% 01:06:33.670 --> 01:06:37.490 align:middle line:84% Now let's prove the extreme value theorem. 01:06:37.490 --> 01:06:38.850 align:middle line:90% So I need to state that first. 01:06:38.850 --> 01:06:48.100 align:middle line:90% 01:06:48.100 --> 01:06:52.840 align:middle line:90% And there's two versions of it. 01:06:52.840 --> 01:06:54.890 align:middle line:90% So extreme value theorem-- 01:06:54.890 --> 01:07:04.870 align:middle line:90% 01:07:04.870 --> 01:07:08.777 align:middle line:84% so you take a function f on some interval. a 01:07:08.777 --> 01:07:12.070 align:middle line:90% and b are real number. 01:07:12.070 --> 01:07:14.560 align:middle line:90% They're both included. 01:07:14.560 --> 01:07:16.760 align:middle line:90% And f here is continuous. 01:07:16.760 --> 01:07:19.930 align:middle line:90% 01:07:19.930 --> 01:07:23.140 align:middle line:90% It's continuous at all points. 01:07:23.140 --> 01:07:26.600 align:middle line:84% It needs to be continuous at all points. 01:07:26.600 --> 01:07:31.300 align:middle line:84% And this interval needs to contain the endpoints also. 01:07:31.300 --> 01:07:37.420 align:middle line:84% So if this is continuous, then the sup 01:07:37.420 --> 01:07:53.860 align:middle line:84% of f is achieved at some x, meaning 01:07:53.860 --> 01:08:03.763 align:middle line:84% that sup of f over this interval is equal to the value-- 01:08:03.763 --> 01:08:07.210 align:middle line:90% 01:08:07.210 --> 01:08:13.030 align:middle line:84% let's call it x capital M, capital M for Max-- 01:08:13.030 --> 01:08:15.130 align:middle line:90% is achieved at some point. 01:08:15.130 --> 01:08:17.740 align:middle line:84% In particular, the sup is finite. 01:08:17.740 --> 01:08:19.149 align:middle line:90% The sup is finite. 01:08:19.149 --> 01:08:21.290 align:middle line:90% And it's not really a sup. 01:08:21.290 --> 01:08:22.689 align:middle line:90% It's a max. 01:08:22.689 --> 01:08:28.240 align:middle line:84% And so, likewise, it also says that, moreover-- 01:08:28.240 --> 01:08:29.540 align:middle line:90% just the same thing. 01:08:29.540 --> 01:08:33.279 align:middle line:84% And in the usual way, you could obtain it from this 01:08:33.279 --> 01:08:36.200 align:middle line:84% by looking at the function minus f if you wanted. 01:08:36.200 --> 01:08:38.830 align:middle line:84% Or you could just prove it directly with the same people 01:08:38.830 --> 01:08:40.310 align:middle line:90% that we're going to give here. 01:08:40.310 --> 01:08:48.250 align:middle line:84% Moreover, the inf of f over this interval 01:08:48.250 --> 01:08:52.370 align:middle line:84% is equal to the value at some point. 01:08:52.370 --> 01:08:54.890 align:middle line:90% So it's not just the inf. 01:08:54.890 --> 01:08:59.513 align:middle line:84% It actually is a minimum where x little m is-- 01:08:59.513 --> 01:09:06.957 align:middle line:90% 01:09:06.957 --> 01:09:09.200 align:middle line:84% and so now let's try to prove that. 01:09:09.200 --> 01:09:14.620 align:middle line:84% And so this is going to use this theorem that we proved, 01:09:14.620 --> 01:09:15.520 align:middle line:90% this theorem. 01:09:15.520 --> 01:09:19.899 align:middle line:84% And then it's going to use the Cauchy convergence theorem. 01:09:19.899 --> 01:09:45.580 align:middle line:90% 01:09:45.580 --> 01:09:48.450 align:middle line:84% And I'm just going to prove, just 01:09:48.450 --> 01:09:54.149 align:middle line:84% because there's two parts to the extreme value theorem-- that's 01:09:54.149 --> 01:09:56.560 align:middle line:84% this with the sup, and that's this is with the inf. 01:09:56.560 --> 01:10:02.970 align:middle line:84% But the proof, with obvious modification, they're the same. 01:10:02.970 --> 01:10:05.760 align:middle line:84% So I'm just going to focus on the sup. 01:10:05.760 --> 01:10:11.410 align:middle line:90% 01:10:11.410 --> 01:10:13.740 align:middle line:84% So I'm looking at the sup of this function. 01:10:13.740 --> 01:10:17.620 align:middle line:84% A priori, the sup could be infinity. 01:10:17.620 --> 01:10:19.960 align:middle line:84% So usually, we like sup to be finite. 01:10:19.960 --> 01:10:24.510 align:middle line:84% But let me allow this, that if there is no upper bound, then 01:10:24.510 --> 01:10:25.900 align:middle line:90% I will write infinity. 01:10:25.900 --> 01:10:28.500 align:middle line:90% 01:10:28.500 --> 01:10:40.590 align:middle line:84% So in any case, there exist xn so that-- 01:10:40.590 --> 01:10:43.200 align:middle line:90% 01:10:43.200 --> 01:10:52.500 align:middle line:84% there exists a sequence in this domain here so that the f of xn 01:10:52.500 --> 01:10:58.950 align:middle line:90% converge to the sup of it 01:10:58.950 --> 01:10:59.680 align:middle line:90% OK. 01:10:59.680 --> 01:11:03.430 align:middle line:84% I'm allowing here-- and we don't usually like this, 01:11:03.430 --> 01:11:05.710 align:middle line:84% but I allow this here to be infinite. 01:11:05.710 --> 01:11:08.010 align:middle line:84% If it's infinite, then I just mean 01:11:08.010 --> 01:11:11.790 align:middle line:84% that there's a sequence where the value of these 01:11:11.790 --> 01:11:14.390 align:middle line:84% become larger and larger without bound. 01:11:14.390 --> 01:11:18.990 align:middle line:90% 01:11:18.990 --> 01:11:20.070 align:middle line:90% So I have that. 01:11:20.070 --> 01:11:24.730 align:middle line:84% But now you see, this here is in this range. 01:11:24.730 --> 01:11:26.040 align:middle line:90% So this is a bounded. 01:11:26.040 --> 01:11:27.190 align:middle line:90% This is a bounded. 01:11:27.190 --> 01:11:33.920 align:middle line:84% So note that xn is a bounded sequence. 01:11:33.920 --> 01:11:40.140 align:middle line:90% 01:11:40.140 --> 01:11:43.110 align:middle line:84% So then I'm using not Cauchy convergence theorem. 01:11:43.110 --> 01:11:46.180 align:middle line:84% I'm using the Bolzano-Weierstrass theorem, 01:11:46.180 --> 01:11:48.090 align:middle line:84% which implies the Cauchy convergence theorem. 01:11:48.090 --> 01:11:51.510 align:middle line:84% But I will use the Bolzano-Weierstrass theorem. 01:11:51.510 --> 01:11:53.080 align:middle line:90% So it's a bounded sequence. 01:11:53.080 --> 01:12:04.398 align:middle line:84% So I know that there exists a convergent subsequence. 01:12:04.398 --> 01:12:10.020 align:middle line:90% 01:12:10.020 --> 01:12:15.590 align:middle line:84% It's a good exercise to see that this sequence in itself-- 01:12:15.590 --> 01:12:17.060 align:middle line:84% you can just draw a little picture, 01:12:17.060 --> 01:12:21.800 align:middle line:84% and you convince yourself that this sequence in itself 01:12:21.800 --> 01:12:23.280 align:middle line:90% may not be convergent. 01:12:23.280 --> 01:12:26.060 align:middle line:84% But after you pass to a subsequence, 01:12:26.060 --> 01:12:29.480 align:middle line:84% you can find a convergent subsequence. 01:12:29.480 --> 01:12:31.160 align:middle line:90% So it converge. 01:12:31.160 --> 01:12:34.190 align:middle line:90% So this here converge. 01:12:34.190 --> 01:12:38.520 align:middle line:84% And it converges to something in this interval. 01:12:38.520 --> 01:12:39.810 align:middle line:90% And that's important. 01:12:39.810 --> 01:12:43.740 align:middle line:90% So it converge to some x. 01:12:43.740 --> 01:12:50.400 align:middle line:84% And I call this x capital M. So it converging. 01:12:50.400 --> 01:12:53.660 align:middle line:84% And the key point here is that this thing here 01:12:53.660 --> 01:12:56.780 align:middle line:90% is lying in this interval. 01:12:56.780 --> 01:13:03.090 align:middle line:84% And the function is defined in particular at this point. 01:13:03.090 --> 01:13:07.910 align:middle line:84% So if this here has not included the endpoint, 01:13:07.910 --> 01:13:11.190 align:middle line:84% it converges to one of the endpoint, 01:13:11.190 --> 01:13:13.590 align:middle line:84% but the function would not be defined there. 01:13:13.590 --> 01:13:17.080 align:middle line:84% So you wouldn't be able to conclude the extreme value 01:13:17.080 --> 01:13:17.810 align:middle line:90% theorem. 01:13:17.810 --> 01:13:21.950 align:middle line:84% But anyway, since we include the endpoint here, 01:13:21.950 --> 01:13:29.670 align:middle line:84% then now we have that from this theorem that is right here, 01:13:29.670 --> 01:13:33.380 align:middle line:84% we have that since this sequence converge to that, we have 01:13:33.380 --> 01:13:49.070 align:middle line:84% that the images converge to the image of x capital N. 01:13:49.070 --> 01:13:51.800 align:middle line:84% But on the other hand, this sequence 01:13:51.800 --> 01:13:55.220 align:middle line:84% here was a subsequence of a sequence 01:13:55.220 --> 01:13:57.480 align:middle line:90% that converges to the sup. 01:13:57.480 --> 01:14:01.950 align:middle line:84% But if you take a sequence that converges to whatever number, 01:14:01.950 --> 01:14:05.030 align:middle line:84% then any subsequence will converge to the same number. 01:14:05.030 --> 01:14:15.740 align:middle line:84% So this here also converge to the sup of f minus. 01:14:15.740 --> 01:14:21.470 align:middle line:84% And so this means that these two numbers here must be the same. 01:14:21.470 --> 01:14:24.236 align:middle line:90% And so you should just know-- 01:14:24.236 --> 01:14:27.080 align:middle line:84% it would take you just a second to convince yourself 01:14:27.080 --> 01:14:29.000 align:middle line:90% that everything is fine. 01:14:29.000 --> 01:14:32.900 align:middle line:84% If a priori, I didn't know that the sup 01:14:32.900 --> 01:14:37.160 align:middle line:84% was a finite number, but infinite [INAUDIBLE] that 01:14:37.160 --> 01:14:38.570 align:middle line:90% is still fine. 01:14:38.570 --> 01:14:42.140 align:middle line:84% But that should only take a minute, 01:14:42.140 --> 01:14:44.480 align:middle line:90% to convince yourself about that. 01:14:44.480 --> 01:14:47.980 align:middle line:84% So that proves the extreme value theorem, 01:14:47.980 --> 01:14:53.630 align:middle line:84% that if you take a function that is continuous and defined 01:14:53.630 --> 01:14:56.390 align:middle line:84% on an interval, where the interval includes 01:14:56.390 --> 01:15:01.230 align:middle line:84% both endpoints, then the sup is achieved. 01:15:01.230 --> 01:15:02.480 align:middle line:90% So the sup is really a max. 01:15:02.480 --> 01:15:06.950 align:middle line:84% And the inf-- by a very similar proof, the infimum 01:15:06.950 --> 01:15:07.880 align:middle line:90% is also achieved. 01:15:07.880 --> 01:15:10.760 align:middle line:84% And so both the min and max, see? 01:15:10.760 --> 01:15:11.474 align:middle line:90% Yeah? 01:15:11.474 --> 01:15:16.005 align:middle line:90% STUDENT: [INAUDIBLE] 01:15:16.005 --> 01:15:17.213 align:middle line:90% TOBIAS COLDING: Sorry, sorry. 01:15:17.213 --> 01:15:18.572 align:middle line:90% STUDENT: How [INAUDIBLE]? 01:15:18.572 --> 01:15:21.290 align:middle line:90% 01:15:21.290 --> 01:15:23.550 align:middle line:90% TOBIAS COLDING: OK. 01:15:23.550 --> 01:15:24.690 align:middle line:90% So yeah, that's right. 01:15:24.690 --> 01:15:26.940 align:middle line:84% So you're saying-- so this here-- 01:15:26.940 --> 01:15:28.550 align:middle line:84% so it was clear enough that it was 01:15:28.550 --> 01:15:32.420 align:middle line:84% a bounded sequence because it's lying in this bounded interval. 01:15:32.420 --> 01:15:34.680 align:middle line:90% So it's a bounded sequence. 01:15:34.680 --> 01:15:37.580 align:middle line:84% Are you asking why such a bounded sequence exists? 01:15:37.580 --> 01:15:39.080 align:middle line:84% STUDENT: So how does [INAUDIBLE]? 01:15:39.080 --> 01:15:39.872 align:middle line:90% TOBIAS COLDING: OK. 01:15:39.872 --> 01:15:45.140 align:middle line:90% 01:15:45.140 --> 01:15:47.793 align:middle line:90% So you have this function. 01:15:47.793 --> 01:15:54.270 align:middle line:90% 01:15:54.270 --> 01:16:00.230 align:middle line:84% So you take the sup of f over this interval. 01:16:00.230 --> 01:16:05.480 align:middle line:84% And so this means that if you take any epsilon 01:16:05.480 --> 01:16:08.410 align:middle line:84% and you subtract it from the sup, 01:16:08.410 --> 01:16:14.150 align:middle line:84% then, actually, depending on this epsilon, 01:16:14.150 --> 01:16:18.190 align:middle line:84% there must exist so that the value at this x 01:16:18.190 --> 01:16:20.275 align:middle line:90% is larger than this. 01:16:20.275 --> 01:16:22.510 align:middle line:84% And so you could imagine, instead of 01:16:22.510 --> 01:16:26.480 align:middle line:84% epsilon here, looking at a sequence like this. 01:16:26.480 --> 01:16:29.000 align:middle line:84% And then this here gives you a sequence like that. 01:16:29.000 --> 01:16:31.863 align:middle line:90% 01:16:31.863 --> 01:16:33.530 align:middle line:84% So that would be how you construct this. 01:16:33.530 --> 01:16:35.920 align:middle line:90% And the same when you do the-- 01:16:35.920 --> 01:16:39.100 align:middle line:84% and so once you have it, it's a bounded sequence 01:16:39.100 --> 01:16:41.720 align:middle line:84% because it's lie between a and b. 01:16:41.720 --> 01:16:44.330 align:middle line:84% You're using the Bolzano-Weierstrass theorem 01:16:44.330 --> 01:16:47.840 align:middle line:84% that we'll prove next time to get a convergent subsequence. 01:16:47.840 --> 01:16:51.700 align:middle line:84% And then you apply this theorem here 01:16:51.700 --> 01:16:54.800 align:middle line:84% to get, in that case, the maximum-- 01:16:54.800 --> 01:16:56.320 align:middle line:90% and likewise for the minimum-- 01:16:56.320 --> 01:16:59.560 align:middle line:90% that they are achieved. 01:16:59.560 --> 01:17:02.260 align:middle line:90% Any questions? 01:17:02.260 --> 01:17:08.000 align:middle line:90%