WEBVTT 00:00:00.000 --> 00:00:01.960 align:middle line:90% [SQUEAKING] 00:00:01.960 --> 00:00:03.430 align:middle line:90% [RUSTLING] 00:00:03.430 --> 00:00:04.900 align:middle line:90% [CLICKING] 00:00:04.900 --> 00:00:11.208 align:middle line:90% 00:00:11.208 --> 00:00:12.000 align:middle line:90% TOBIAS COLDING: OK. 00:00:12.000 --> 00:00:17.220 align:middle line:84% So let me first remind you about-- 00:00:17.220 --> 00:00:25.560 align:middle line:84% we'll be looking-- so last time we were looking at-- we 00:00:25.560 --> 00:00:34.160 align:middle line:84% had a sequence of functions on an interval I. 00:00:34.160 --> 00:00:40.120 align:middle line:84% And we have a function f on that same interval. 00:00:40.120 --> 00:00:43.660 align:middle line:84% And then we talked about two notions of convergence. 00:00:43.660 --> 00:00:55.480 align:middle line:90% 00:00:55.480 --> 00:00:58.580 align:middle line:84% One was the pointwise convergence. 00:00:58.580 --> 00:01:06.590 align:middle line:90% 00:01:06.590 --> 00:01:08.410 align:middle line:90% So this meant that-- 00:01:08.410 --> 00:01:12.150 align:middle line:84% so we say that this sequence converges 00:01:12.150 --> 00:01:24.830 align:middle line:84% to f pointwise if, for all x fixed, 00:01:24.830 --> 00:01:28.990 align:middle line:84% you're looking at the sequence fn of x-- 00:01:28.990 --> 00:01:30.590 align:middle line:90% so x is fixed-- 00:01:30.590 --> 00:01:34.390 align:middle line:84% and that this here converges to f of x. 00:01:34.390 --> 00:01:37.510 align:middle line:84% So that was the pointwise convergence. 00:01:37.510 --> 00:01:43.430 align:middle line:84% And the second convergence that we talked about 00:01:43.430 --> 00:01:47.150 align:middle line:90% was uniform convergence. 00:01:47.150 --> 00:01:53.190 align:middle line:84% So this is 2, uniform convergence. 00:01:53.190 --> 00:01:56.690 align:middle line:90% 00:01:56.690 --> 00:01:58.630 align:middle line:84% And so we say that this sequence here 00:01:58.630 --> 00:02:11.270 align:middle line:84% converges to f uniformly if, for all epsilon greater than 0, 00:02:11.270 --> 00:02:18.870 align:middle line:84% there exists an N so that if you're further out than N, 00:02:18.870 --> 00:02:25.750 align:middle line:84% then the difference between fn of x and f of x 00:02:25.750 --> 00:02:27.210 align:middle line:90% is smaller than epsilon. 00:02:27.210 --> 00:02:33.780 align:middle line:84% And this here has to hold for all x. 00:02:33.780 --> 00:02:36.310 align:middle line:90% 00:02:36.310 --> 00:02:48.270 align:middle line:84% And so, again, uniform refers to that it's for all x. 00:02:48.270 --> 00:02:51.510 align:middle line:90% So it's uniform in x. 00:02:51.510 --> 00:02:54.630 align:middle line:84% And so there was one example that 00:02:54.630 --> 00:02:56.770 align:middle line:90% was particularly illuminating. 00:02:56.770 --> 00:03:00.220 align:middle line:90% 00:03:00.220 --> 00:03:03.620 align:middle line:84% And that was that we took-- so if I here 00:03:03.620 --> 00:03:08.740 align:middle line:84% was the integral from 0 to 1 and if we're 00:03:08.740 --> 00:03:14.860 align:middle line:84% looking at the function fn of x, that was just x to the power n. 00:03:14.860 --> 00:03:26.300 align:middle line:84% And f of x is then the function that is 0 unless-- 00:03:26.300 --> 00:03:31.980 align:middle line:84% so 0 if x here is between 0 and strictly less than 1, 00:03:31.980 --> 00:03:38.084 align:middle line:90% and 1 if x is equal to 1. 00:03:38.084 --> 00:03:40.670 align:middle line:90% And so the picture was that-- 00:03:40.670 --> 00:03:43.700 align:middle line:90% 00:03:43.700 --> 00:03:49.060 align:middle line:84% so the picture was that you have this sequence here. 00:03:49.060 --> 00:03:53.380 align:middle line:84% They are defined on the interval from 0 to 1. 00:03:53.380 --> 00:03:56.410 align:middle line:90% They all end up at 1. 00:03:56.410 --> 00:04:01.650 align:middle line:84% And then as n becomes larger and larger, 00:04:01.650 --> 00:04:06.870 align:middle line:84% these functions become more and more steep towards 1, like that. 00:04:06.870 --> 00:04:13.170 align:middle line:84% And so then we had that this sequence fn converges 00:04:13.170 --> 00:04:18.810 align:middle line:84% to f pointwise, because if you fixed 00:04:18.810 --> 00:04:24.570 align:middle line:84% x, then really these values here converge either to-- 00:04:24.570 --> 00:04:31.730 align:middle line:84% if x is not equal to 1, then as n becomes large, 00:04:31.730 --> 00:04:33.110 align:middle line:90% they converge to 0. 00:04:33.110 --> 00:04:40.290 align:middle line:84% If n is 1, then the fn of 1, that's always 1. 00:04:40.290 --> 00:04:43.970 align:middle line:84% So it's clear that it's converging pointwise. 00:04:43.970 --> 00:04:49.610 align:middle line:90% But fn does not converge. 00:04:49.610 --> 00:04:59.390 align:middle line:84% Does not converge to f uniformly. 00:04:59.390 --> 00:05:05.330 align:middle line:90% 00:05:05.330 --> 00:05:08.490 align:middle line:84% And the reason why it didn't converge uniformly 00:05:08.490 --> 00:05:12.330 align:middle line:90% was that we saw that for each-- 00:05:12.330 --> 00:05:14.390 align:middle line:84% by the intermediate value theorem, 00:05:14.390 --> 00:05:19.450 align:middle line:84% if we were looking at 1/2 here, then for each of the function 00:05:19.450 --> 00:05:27.650 align:middle line:84% there would be an xn so that the value at xn was 1. 00:05:27.650 --> 00:05:29.710 align:middle line:90% Was a half, sorry. 00:05:29.710 --> 00:05:35.930 align:middle line:84% So you define xn so that the value of this is a half. 00:05:35.930 --> 00:05:46.130 align:middle line:84% But now, if you're looking f of x minus f of xn 00:05:46.130 --> 00:05:50.770 align:middle line:84% like that and if this-- sorry-- is n, 00:05:50.770 --> 00:05:57.480 align:middle line:84% then this difference here is actually just a half. 00:05:57.480 --> 00:06:00.580 align:middle line:84% And this difference here should be small. 00:06:00.580 --> 00:06:05.480 align:middle line:84% If n was sufficiently large, it should be small for all x. 00:06:05.480 --> 00:06:06.980 align:middle line:90% But it's not. 00:06:06.980 --> 00:06:11.760 align:middle line:84% We've used one particular x to show that it wasn't. 00:06:11.760 --> 00:06:15.240 align:middle line:90% 00:06:15.240 --> 00:06:16.680 align:middle line:90% OK. 00:06:16.680 --> 00:06:20.240 align:middle line:84% That was just a quick review of some of the things 00:06:20.240 --> 00:06:22.720 align:middle line:90% we talked about last time. 00:06:22.720 --> 00:06:29.000 align:middle line:90% Now, let's get to theorem. 00:06:29.000 --> 00:06:32.160 align:middle line:84% And so maybe also one more thing about this example 00:06:32.160 --> 00:06:37.160 align:middle line:84% is that you see that these functions here, these functions 00:06:37.160 --> 00:06:41.600 align:middle line:90% fn, they're all continuous. 00:06:41.600 --> 00:06:43.920 align:middle line:90% But the limit is not continuous. 00:06:43.920 --> 00:06:45.560 align:middle line:90% So pointwise. 00:06:45.560 --> 00:07:04.670 align:middle line:84% So note that-- so note that pointwise convergence is not 00:07:04.670 --> 00:07:23.590 align:middle line:84% enough to guarantee that limits of continuous functions-- 00:07:23.590 --> 00:07:31.782 align:middle line:90% 00:07:31.782 --> 00:07:37.610 align:middle line:84% that the limit of continuous function is continuous. 00:07:37.610 --> 00:07:42.510 align:middle line:90% 00:07:42.510 --> 00:07:43.010 align:middle line:90% OK. 00:07:43.010 --> 00:07:45.770 align:middle line:90% 00:07:45.770 --> 00:07:48.610 align:middle line:84% So that's what this example illustrates. 00:07:48.610 --> 00:07:52.670 align:middle line:84% So now, instead, we have the following theorem. 00:07:52.670 --> 00:07:57.150 align:middle line:84% So we have this interval, and we have a sequence fn, 00:07:57.150 --> 00:08:02.030 align:middle line:84% and we have f all defined on this interval. 00:08:02.030 --> 00:08:17.430 align:middle line:84% And if we assume that all of these fn's are continuous, 00:08:17.430 --> 00:08:22.230 align:middle line:84% and b, we assume that the convergence here, 00:08:22.230 --> 00:08:29.230 align:middle line:84% the convergence of fn to f is uniform, 00:08:29.230 --> 00:08:37.850 align:middle line:84% then the claim is that then f here is also continuous. 00:08:37.850 --> 00:08:44.430 align:middle line:90% 00:08:44.430 --> 00:08:45.330 align:middle line:90% OK. 00:08:45.330 --> 00:08:46.730 align:middle line:90% So let's try to prove that. 00:08:46.730 --> 00:08:49.540 align:middle line:90% 00:08:49.540 --> 00:08:53.020 align:middle line:84% So pointwise convergence is not enough to guarantee 00:08:53.020 --> 00:08:55.300 align:middle line:90% that the limit is continuous. 00:08:55.300 --> 00:09:00.420 align:middle line:90% Uniform convergence is enough. 00:09:00.420 --> 00:09:02.680 align:middle line:90% Now, let's see. 00:09:02.680 --> 00:09:04.890 align:middle line:90% So let's prove that. 00:09:04.890 --> 00:09:07.980 align:middle line:90% 00:09:07.980 --> 00:09:26.260 align:middle line:84% So given epsilon greater than 0, there exists capital N such 00:09:26.260 --> 00:09:32.740 align:middle line:84% that if little n is bigger or equal to capital N, 00:09:32.740 --> 00:09:41.860 align:middle line:84% then fn of dx minus f of x is smaller than epsilon over 3 00:09:41.860 --> 00:09:42.840 align:middle line:90% for all x. 00:09:42.840 --> 00:09:48.480 align:middle line:90% 00:09:48.480 --> 00:09:52.940 align:middle line:84% This just follows that the sequence converge uniformly. 00:09:52.940 --> 00:09:57.940 align:middle line:90% So this is the first thing. 00:09:57.940 --> 00:10:00.420 align:middle line:90% And the second thing is-- 00:10:00.420 --> 00:10:07.520 align:middle line:84% so this was, since fn converges to f uniformly. 00:10:07.520 --> 00:10:11.900 align:middle line:90% 00:10:11.900 --> 00:10:14.460 align:middle line:90% The second thing is that-- 00:10:14.460 --> 00:10:18.260 align:middle line:90% so now we have n. 00:10:18.260 --> 00:10:22.140 align:middle line:84% This here holds for all n bigger or equal to capital N. 00:10:22.140 --> 00:10:25.940 align:middle line:84% But in particular, it holds for capital N. So now, 00:10:25.940 --> 00:10:31.600 align:middle line:84% since f capital N is continuous-- 00:10:31.600 --> 00:10:33.460 align:middle line:84% so this is a fixed function now-- 00:10:33.460 --> 00:10:40.580 align:middle line:90% 00:10:40.580 --> 00:11:13.570 align:middle line:84% then for any fixed x0, we have that there exists a delta 00:11:13.570 --> 00:11:22.130 align:middle line:84% greater than 0, such that if x minus x0 is smaller than delta, 00:11:22.130 --> 00:11:27.890 align:middle line:84% then this function-- the value at these two points differ 00:11:27.890 --> 00:11:29.970 align:middle line:90% at most-- 00:11:29.970 --> 00:11:33.490 align:middle line:90% so this is for the fn. 00:11:33.490 --> 00:11:38.210 align:middle line:84% The value at these two points differ at most 00:11:38.210 --> 00:11:42.970 align:middle line:90% by epsilon over 3. 00:11:42.970 --> 00:11:47.600 align:middle line:84% So again, we have a sequence, fn, 00:11:47.600 --> 00:11:50.360 align:middle line:84% that converges uniformly to some f. 00:11:50.360 --> 00:11:53.400 align:middle line:84% We want to prove that this f is continuous. 00:11:53.400 --> 00:11:57.080 align:middle line:84% So we just need to prove that for each x0 fixed, 00:11:57.080 --> 00:11:59.960 align:middle line:90% it's continuous at x0. 00:11:59.960 --> 00:12:02.120 align:middle line:84% First, I'm using that the sequence 00:12:02.120 --> 00:12:04.520 align:middle line:90% is converging uniformly. 00:12:04.520 --> 00:12:08.080 align:middle line:84% So given this epsilon, there exists this capital N 00:12:08.080 --> 00:12:11.120 align:middle line:84% so that if you're further out, then you have this. 00:12:11.120 --> 00:12:14.700 align:middle line:84% I'm not even going to use the part that is further out. 00:12:14.700 --> 00:12:18.160 align:middle line:84% I'm just going to pick this capital N. 00:12:18.160 --> 00:12:24.640 align:middle line:84% And so then I'm using that this f capital N is continuous, 00:12:24.640 --> 00:12:26.580 align:middle line:90% and it's continuous at x0. 00:12:26.580 --> 00:12:28.600 align:middle line:90% So I have this. 00:12:28.600 --> 00:12:32.160 align:middle line:90% And so now we need to estimate-- 00:12:32.160 --> 00:12:36.760 align:middle line:84% we now need to estimate the difference. 00:12:36.760 --> 00:12:39.280 align:middle line:84% So what we're really interested is 00:12:39.280 --> 00:12:42.920 align:middle line:84% in f, but the f capital N, that's 00:12:42.920 --> 00:12:47.120 align:middle line:84% going to help us estimate the difference between these two 00:12:47.120 --> 00:12:48.880 align:middle line:90% numbers. 00:12:48.880 --> 00:12:50.580 align:middle line:90% And so these two numbers-- 00:12:50.580 --> 00:12:56.120 align:middle line:84% so x here, x0 was this fixed point 00:12:56.120 --> 00:13:02.240 align:middle line:84% that we want to show that f is continuous at that point. 00:13:02.240 --> 00:13:09.240 align:middle line:84% And the delta, well, the delta came from f capital N. 00:13:09.240 --> 00:13:12.760 align:middle line:84% So we're looking at this x, this [? closed, ?] 00:13:12.760 --> 00:13:17.240 align:middle line:84% where the delta was given by f capital N. 00:13:17.240 --> 00:13:21.240 align:middle line:84% And now we use the triangle inequality. 00:13:21.240 --> 00:13:23.360 align:middle line:90% So this difference here is-- 00:13:23.360 --> 00:13:26.200 align:middle line:84% so you may as well think about this one here. 00:13:26.200 --> 00:13:31.720 align:middle line:84% And you could compare it with the value at the same point. 00:13:31.720 --> 00:13:33.120 align:middle line:90% So I'm just writing down-- 00:13:33.120 --> 00:13:38.480 align:middle line:84% I'm writing this difference down as a sum of things. 00:13:38.480 --> 00:13:41.990 align:middle line:84% And then afterwards, I use the triangle inequality. 00:13:41.990 --> 00:13:47.710 align:middle line:84% So here, minus f capital N of x0. 00:13:47.710 --> 00:13:53.710 align:middle line:84% And then I'm adding back this here again f capital N of x0. 00:13:53.710 --> 00:13:58.050 align:middle line:84% And I'm comparing this with f of x0. 00:13:58.050 --> 00:14:02.150 align:middle line:84% So you see that here, this is really the same because here I'm 00:14:02.150 --> 00:14:04.210 align:middle line:84% subtracting this thing that I didn't have, 00:14:04.210 --> 00:14:06.870 align:middle line:84% but I'm adding it back, subtracting this thing that I 00:14:06.870 --> 00:14:08.650 align:middle line:84% didn't have, but I'm adding it back. 00:14:08.650 --> 00:14:12.910 align:middle line:84% So I'm just left with those two things, which is that. 00:14:12.910 --> 00:14:16.670 align:middle line:84% And so now we use the triangle inequality, 00:14:16.670 --> 00:14:19.910 align:middle line:84% that this is now bounded by f of x 00:14:19.910 --> 00:14:32.270 align:middle line:84% minus fN of x plus fN of x minus f capital N of x0, like that, 00:14:32.270 --> 00:14:39.350 align:middle line:84% plus, and then f capital N of x0 minus f of x0, 00:14:39.350 --> 00:14:43.460 align:middle line:84% with an absolute value sign like that. 00:14:43.460 --> 00:14:48.740 align:middle line:84% Now, this thing here is smaller than epsilon over 3. 00:14:48.740 --> 00:14:54.140 align:middle line:84% And that came from that it's evaluated at the same point 00:14:54.140 --> 00:14:56.160 align:middle line:90% and the convergence was uniform. 00:14:56.160 --> 00:14:59.460 align:middle line:84% So it really was the first part here. 00:14:59.460 --> 00:15:04.620 align:middle line:84% So that is less than epsilon over 3. 00:15:04.620 --> 00:15:10.260 align:middle line:84% This one here comes from that the difference between x and x0 00:15:10.260 --> 00:15:14.940 align:middle line:84% is less than delta and that this function 00:15:14.940 --> 00:15:18.660 align:middle line:90% fN was continuous at x0. 00:15:18.660 --> 00:15:21.600 align:middle line:84% So this is another epsilon over delta. 00:15:21.600 --> 00:15:24.900 align:middle line:90% This is this second part here. 00:15:24.900 --> 00:15:27.260 align:middle line:84% And then the last here is, again, 00:15:27.260 --> 00:15:29.800 align:middle line:84% the same that gave us this bound, 00:15:29.800 --> 00:15:31.900 align:middle line:84% namely that the convergence was uniform 00:15:31.900 --> 00:15:34.420 align:middle line:84% and is evaluated at the same point. 00:15:34.420 --> 00:15:37.580 align:middle line:90% So you have that. 00:15:37.580 --> 00:15:44.300 align:middle line:84% And so what we've proven here is that for this given x0, 00:15:44.300 --> 00:15:49.980 align:middle line:84% if the difference between x and x0 is less than delta, 00:15:49.980 --> 00:15:53.480 align:middle line:84% then the difference in the images is less than epsilon. 00:15:53.480 --> 00:16:01.380 align:middle line:84% So this means that the function f is continuous at x0 and x0 00:16:01.380 --> 00:16:05.309 align:middle line:90% was arbitrary. 00:16:05.309 --> 00:16:17.890 align:middle line:84% Now, the first application of this is that suppose you take-- 00:16:17.890 --> 00:16:22.110 align:middle line:90% 00:16:22.110 --> 00:16:24.110 align:middle line:84% suppose you're looking at the following example. 00:16:24.110 --> 00:16:27.340 align:middle line:90% 00:16:27.340 --> 00:16:33.740 align:middle line:84% xn of x is the sum here from k equal to 0 00:16:33.740 --> 00:16:37.930 align:middle line:90% to n of xk over k factorial. 00:16:37.930 --> 00:16:43.330 align:middle line:84% Suppose you're looking at xn like this. 00:16:43.330 --> 00:16:53.850 align:middle line:84% Then the xn here of x, they are all continuous. 00:16:53.850 --> 00:16:55.770 align:middle line:90% They are sum of polynomials. 00:16:55.770 --> 00:16:57.570 align:middle line:84% They're just a polynomial, actually 00:16:57.570 --> 00:17:04.069 align:middle line:84% the sum of what's called homogeneous polynomial. 00:17:04.069 --> 00:17:06.410 align:middle line:84% But they are themselves polynomial. 00:17:06.410 --> 00:17:08.109 align:middle line:90% So they are all continuous. 00:17:08.109 --> 00:17:12.730 align:middle line:90% 00:17:12.730 --> 00:17:15.410 align:middle line:90% And Weierstrass M-test. 00:17:15.410 --> 00:17:27.089 align:middle line:84% So by Weierstrass M-test that we proved last time-- 00:17:27.089 --> 00:17:30.370 align:middle line:90% 00:17:30.370 --> 00:17:33.010 align:middle line:84% by Weierstrass M-test, the convergence 00:17:33.010 --> 00:17:40.900 align:middle line:84% here to E, where E here is the exponential function. 00:17:40.900 --> 00:17:44.880 align:middle line:84% So it's this power series sum from k 00:17:44.880 --> 00:17:53.060 align:middle line:84% equal to 0 to infinity of x to the power k over k factorial. 00:17:53.060 --> 00:17:55.480 align:middle line:84% This convergence here is uniform. 00:17:55.480 --> 00:18:02.100 align:middle line:84% It's uniform assuming that you do it on a compact interval. 00:18:02.100 --> 00:18:18.820 align:middle line:84% So it's uniform on any fixed interval of the form, 00:18:18.820 --> 00:18:24.600 align:middle line:90% say, minus L to L. 00:18:24.600 --> 00:18:27.120 align:middle line:84% So if you're fixing an interval like this, 00:18:27.120 --> 00:18:30.320 align:middle line:90% then the convergence is uniform. 00:18:30.320 --> 00:18:36.160 align:middle line:84% And so this means that since these guys here are continuous, 00:18:36.160 --> 00:18:40.380 align:middle line:84% then it implies that this one here is also continuous. 00:18:40.380 --> 00:18:47.088 align:middle line:90% 00:18:47.088 --> 00:18:50.610 align:middle line:84% So this is implied by the theorem that we just proved. 00:18:50.610 --> 00:18:55.840 align:middle line:90% 00:18:55.840 --> 00:18:59.760 align:middle line:90% So now this will-- 00:18:59.760 --> 00:19:02.480 align:middle line:90% uniform convergence. 00:19:02.480 --> 00:19:08.160 align:middle line:84% There's another way of thinking about uniform convergence that, 00:19:08.160 --> 00:19:13.490 align:middle line:90% in a way, makes it more natural. 00:19:13.490 --> 00:19:16.900 align:middle line:90% 00:19:16.900 --> 00:19:21.120 align:middle line:84% So remember that if we looked at-- 00:19:21.120 --> 00:19:22.920 align:middle line:84% so if we take an interval-- let's 00:19:22.920 --> 00:19:28.440 align:middle line:84% say that this is an interval from a to b. a and b 00:19:28.440 --> 00:19:31.200 align:middle line:90% are real numbers. 00:19:31.200 --> 00:19:35.750 align:middle line:84% Then we looked at the space of continuous functions 00:19:35.750 --> 00:19:38.750 align:middle line:90% on this interval. 00:19:38.750 --> 00:19:41.570 align:middle line:84% And this space had a natural metric. 00:19:41.570 --> 00:19:45.950 align:middle line:84% So I'm using here that both endpoints are included. 00:19:45.950 --> 00:19:50.380 align:middle line:84% This space here has a natural metric. 00:19:50.380 --> 00:19:56.830 align:middle line:90% 00:19:56.830 --> 00:20:02.270 align:middle line:84% And that metric is that if you take two functions-- 00:20:02.270 --> 00:20:06.070 align:middle line:84% two continuous functions-- on the interval, then 00:20:06.070 --> 00:20:09.310 align:middle line:84% the distance between these two functions 00:20:09.310 --> 00:20:21.270 align:middle line:84% is just the max of the difference between the two 00:20:21.270 --> 00:20:22.090 align:middle line:90% functions. 00:20:22.090 --> 00:20:30.510 align:middle line:90% 00:20:30.510 --> 00:20:33.780 align:middle line:84% The difference-- again, why is this? 00:20:33.780 --> 00:20:35.900 align:middle line:90% The difference is continuous. 00:20:35.900 --> 00:20:39.620 align:middle line:84% So this means that the maximum is-- 00:20:39.620 --> 00:20:41.040 align:middle line:90% the difference is continuous. 00:20:41.040 --> 00:20:42.660 align:middle line:84% So this is a continuous function, 00:20:42.660 --> 00:20:45.740 align:middle line:84% and it's still continuous if you take absolute value. 00:20:45.740 --> 00:20:49.060 align:middle line:84% And so this means that the maximum by the extreme value 00:20:49.060 --> 00:20:50.105 align:middle line:90% theorem is achieved. 00:20:50.105 --> 00:20:53.500 align:middle line:90% 00:20:53.500 --> 00:20:57.340 align:middle line:84% And so this here was a natural distance between them. 00:20:57.340 --> 00:21:02.300 align:middle line:84% Now to say that the sequence here-- 00:21:02.300 --> 00:21:08.500 align:middle line:84% to say that the sequence fn is converging uniformly, 00:21:08.500 --> 00:21:12.340 align:middle line:84% you can think about that in terms of this distance. 00:21:12.340 --> 00:21:16.080 align:middle line:84% So we have the following proposition. 00:21:16.080 --> 00:21:21.260 align:middle line:90% 00:21:21.260 --> 00:21:30.060 align:middle line:84% fn converge uniformly to f if and only 00:21:30.060 --> 00:21:53.210 align:middle line:84% if fn converges to f in the metric space with this metric. 00:21:53.210 --> 00:21:56.290 align:middle line:90% 00:21:56.290 --> 00:21:59.060 align:middle line:90% Now, why is this the case? 00:21:59.060 --> 00:22:01.420 align:middle line:90% So this is just that-- 00:22:01.420 --> 00:22:07.540 align:middle line:84% so proof of that is really just a one-liner because you see that 00:22:07.540 --> 00:22:08.620 align:middle line:90% to say that-- 00:22:08.620 --> 00:22:11.620 align:middle line:90% 00:22:11.620 --> 00:22:18.700 align:middle line:84% so that f of x minus fn of x is less than 00:22:18.700 --> 00:22:34.450 align:middle line:84% or equal to epsilon for all x if this here is the case, where 00:22:34.450 --> 00:22:42.250 align:middle line:84% this is equivalent to saying that the distance between fn 00:22:42.250 --> 00:22:48.270 align:middle line:84% and an f is less than or equal to epsilon for all x. 00:22:48.270 --> 00:22:51.930 align:middle line:90% 00:22:51.930 --> 00:22:54.850 align:middle line:84% Because if this is the case for all x, 00:22:54.850 --> 00:22:57.830 align:middle line:84% then the maximum is less than or equal to epsilon. 00:22:57.830 --> 00:22:59.410 align:middle line:90% This was the maximum. 00:22:59.410 --> 00:23:02.090 align:middle line:84% And on the other hand, if the maximum, well, then it 00:23:02.090 --> 00:23:04.090 align:middle line:90% holds for each x. 00:23:04.090 --> 00:23:06.130 align:middle line:90% So this is sort of a tautology. 00:23:06.130 --> 00:23:08.910 align:middle line:90% 00:23:08.910 --> 00:23:14.250 align:middle line:84% And so now we see, of course, that if the sequence-- so you 00:23:14.250 --> 00:23:15.730 align:middle line:90% have that. 00:23:15.730 --> 00:23:23.050 align:middle line:84% But this means-- so this means that if the sequence-- 00:23:23.050 --> 00:23:36.660 align:middle line:84% therefore, if the sequence fn converges to f uniformly, well, 00:23:36.660 --> 00:23:39.760 align:middle line:84% then it means that for all epsilon, 00:23:39.760 --> 00:23:44.040 align:middle line:84% if we're sufficiently far out, then this here can be-- 00:23:44.040 --> 00:23:47.180 align:middle line:84% for any given epsilon, if you're sufficiently far out, 00:23:47.180 --> 00:23:49.828 align:middle line:84% this here can be made less than or equal to epsilon, which 00:23:49.828 --> 00:23:51.620 align:middle line:84% means that this error can be made less than 00:23:51.620 --> 00:23:54.720 align:middle line:84% or equal to epsilon if you're sufficiently far out. 00:23:54.720 --> 00:23:57.600 align:middle line:84% So you see that if this converges uniformly, then 00:23:57.600 --> 00:24:04.680 align:middle line:84% it implies that this thing here converges to 0. 00:24:04.680 --> 00:24:06.800 align:middle line:84% But that's exactly what it means, 00:24:06.800 --> 00:24:14.010 align:middle line:84% that this sequence fn converges to f in the metric space. 00:24:14.010 --> 00:24:23.040 align:middle line:90% 00:24:23.040 --> 00:24:24.410 align:middle line:90% And the other way around. 00:24:24.410 --> 00:24:27.280 align:middle line:90% 00:24:27.280 --> 00:24:30.840 align:middle line:84% If this sequence converges, fn converges 00:24:30.840 --> 00:24:33.000 align:middle line:84% to f in the metric space, this is 00:24:33.000 --> 00:24:36.480 align:middle line:84% equivalent to saying that this here goes to 0. 00:24:36.480 --> 00:24:39.560 align:middle line:84% But if this here goes to 0, this means 00:24:39.560 --> 00:24:42.480 align:middle line:84% that if you're sufficiently far out-- for a given 00:24:42.480 --> 00:24:44.360 align:middle line:84% epsilon, if you're sufficiently far out, 00:24:44.360 --> 00:24:46.980 align:middle line:84% this here is less than or equal to epsilon. 00:24:46.980 --> 00:24:49.400 align:middle line:84% But it means that this here is less than epsilon. 00:24:49.400 --> 00:24:54.080 align:middle line:84% So that means it really converges uniformly. 00:24:54.080 --> 00:24:56.445 align:middle line:90% OK. 00:24:56.445 --> 00:25:01.920 align:middle line:84% And that's in a way that makes it much more natural. 00:25:01.920 --> 00:25:04.680 align:middle line:90% I mean, yeah. 00:25:04.680 --> 00:25:11.920 align:middle line:84% So it makes uniform convergence much more natural. 00:25:11.920 --> 00:25:16.840 align:middle line:84% So now let's prove the next term. 00:25:16.840 --> 00:25:21.040 align:middle line:84% So for this theorem, we're going to think 00:25:21.040 --> 00:25:27.570 align:middle line:84% about uniform convergence in terms of this metric 00:25:27.570 --> 00:25:31.510 align:middle line:84% and in terms of convergence in this metric space. 00:25:31.510 --> 00:25:36.210 align:middle line:84% So this theorem is the next thing. 00:25:36.210 --> 00:25:41.990 align:middle line:84% Let's say that if you take an interval from a to b 00:25:41.990 --> 00:25:45.350 align:middle line:84% and you're looking at the continuous function 00:25:45.350 --> 00:25:50.110 align:middle line:84% and you equip it with this metric that we talked about, 00:25:50.110 --> 00:25:54.290 align:middle line:84% then this here is this space, which is a metric space. 00:25:54.290 --> 00:26:03.300 align:middle line:84% So this metric space is a Cauchy complete. 00:26:03.300 --> 00:26:13.250 align:middle line:90% 00:26:13.250 --> 00:26:16.390 align:middle line:84% So remember, Cauchy complete in a metric space 00:26:16.390 --> 00:26:20.670 align:middle line:84% means that if you take a Cauchy sequence-- so just let 00:26:20.670 --> 00:26:22.950 align:middle line:90% me just recall here over here. 00:26:22.950 --> 00:26:24.590 align:middle line:90% So Cauchy. 00:26:24.590 --> 00:26:34.070 align:middle line:84% So if x comma d is a metric space, 00:26:34.070 --> 00:27:00.270 align:middle line:84% then Cauchy complete means that any Cauchy sequence 00:27:00.270 --> 00:27:01.450 align:middle line:90% is convergent. 00:27:01.450 --> 00:27:07.030 align:middle line:90% 00:27:07.030 --> 00:27:10.490 align:middle line:84% That's what it means to be a Cauchy complete. 00:27:10.490 --> 00:27:17.070 align:middle line:84% And of course, an example of a Cauchy complete metric space 00:27:17.070 --> 00:27:21.610 align:middle line:84% was just if we took R with the usual metric. 00:27:21.610 --> 00:27:26.532 align:middle line:90% 00:27:26.532 --> 00:27:29.240 align:middle line:90% So what we have here is-- 00:27:29.240 --> 00:27:30.520 align:middle line:90% so let's try to prove this. 00:27:30.520 --> 00:27:33.460 align:middle line:90% 00:27:33.460 --> 00:27:35.000 align:middle line:90% So we need to prove. 00:27:35.000 --> 00:27:38.340 align:middle line:90% So this here is now-- 00:27:38.340 --> 00:27:51.460 align:middle line:84% we need to prove that if this here is a Cauchy sequence, 00:27:51.460 --> 00:28:04.566 align:middle line:84% then there exists a function that is also continuous. 00:28:04.566 --> 00:28:07.580 align:middle line:90% 00:28:07.580 --> 00:28:22.050 align:middle line:84% And so this sequence here converging to f in the metric d. 00:28:22.050 --> 00:28:23.930 align:middle line:90% That's what we need to prove. 00:28:23.930 --> 00:28:31.890 align:middle line:84% So in particular, we need to first construct a function, f. 00:28:31.890 --> 00:28:36.090 align:middle line:90% 00:28:36.090 --> 00:28:40.320 align:middle line:84% So now let's try first to construct a function. 00:28:40.320 --> 00:28:47.890 align:middle line:90% 00:28:47.890 --> 00:29:01.400 align:middle line:84% So we first construct f as follows. 00:29:01.400 --> 00:29:07.090 align:middle line:90% 00:29:07.090 --> 00:29:11.570 align:middle line:90% Well, if you fix x-- 00:29:11.570 --> 00:29:16.490 align:middle line:84% so x is something, a point in the interval. 00:29:16.490 --> 00:29:20.590 align:middle line:84% And now look at this sequence fn. 00:29:20.590 --> 00:29:24.490 align:middle line:90% This is now a regular sequence. 00:29:24.490 --> 00:29:28.010 align:middle line:84% But if you're looking at the sequence, 00:29:28.010 --> 00:29:38.090 align:middle line:84% then we have that fn of x minus fm of x, if you have n and m 00:29:38.090 --> 00:29:41.650 align:middle line:84% like this, then the difference here 00:29:41.650 --> 00:29:47.530 align:middle line:84% is less than or equal to the max where 00:29:47.530 --> 00:29:59.570 align:middle line:84% this is for all z in the interval of fn 00:29:59.570 --> 00:30:11.386 align:middle line:90% of z minus fm of z, like that. 00:30:11.386 --> 00:30:17.520 align:middle line:84% Because of course, it's like in particular if z was x. 00:30:17.520 --> 00:30:18.830 align:middle line:90% So you have clearly this. 00:30:18.830 --> 00:30:21.720 align:middle line:90% 00:30:21.720 --> 00:30:23.680 align:middle line:90% But now you see that-- 00:30:23.680 --> 00:30:27.420 align:middle line:90% so I claim that this means-- 00:30:27.420 --> 00:30:43.040 align:middle line:84% so I claim that this implies that this sequence here-- 00:30:43.040 --> 00:30:45.720 align:middle line:90% sorry, this sequence here. 00:30:45.720 --> 00:30:50.160 align:middle line:84% fn of x, this sequence here, which 00:30:50.160 --> 00:30:53.000 align:middle line:90% is a sequence of real numbers-- 00:30:53.000 --> 00:30:55.480 align:middle line:90% these here are real numbers-- 00:30:55.480 --> 00:31:06.350 align:middle line:84% that this here is a Cauchy sequence of real numbers. 00:31:06.350 --> 00:31:13.120 align:middle line:90% 00:31:13.120 --> 00:31:16.610 align:middle line:90% Why is that the case? 00:31:16.610 --> 00:31:24.250 align:middle line:84% Well, it's the case because given-- 00:31:24.250 --> 00:31:28.150 align:middle line:90% so this is just because given-- 00:31:28.150 --> 00:31:43.070 align:middle line:84% so this is the case because given epsilon greater than 0, 00:31:43.070 --> 00:31:56.300 align:middle line:84% since fn is a Cauchy sequence in the metric space-- 00:31:56.300 --> 00:32:00.434 align:middle line:90% 00:32:00.434 --> 00:32:02.710 align:middle line:90% in this metric space-- 00:32:02.710 --> 00:32:14.590 align:middle line:84% then there exist capital N so that if little n and little m 00:32:14.590 --> 00:32:17.910 align:middle line:84% are bigger than capital N, then the difference 00:32:17.910 --> 00:32:19.530 align:middle line:90% between these guys-- 00:32:19.530 --> 00:32:23.410 align:middle line:84% sorry, the difference between this function here 00:32:23.410 --> 00:32:28.890 align:middle line:84% and that function is less than epsilon. 00:32:28.890 --> 00:32:32.230 align:middle line:84% This is because they were Cauchy sequence. 00:32:32.230 --> 00:32:35.450 align:middle line:84% And so now this thing here is, of course-- 00:32:35.450 --> 00:32:38.090 align:middle line:84% this thing here, that's what we just saw. 00:32:38.090 --> 00:32:39.990 align:middle line:90% This is the maximum. 00:32:39.990 --> 00:32:48.950 align:middle line:84% So this is bigger than when you evaluate it in just one point, 00:32:48.950 --> 00:32:51.390 align:middle line:90% like that. 00:32:51.390 --> 00:33:00.510 align:middle line:84% And so this proof that when x is fixed, 00:33:00.510 --> 00:33:04.870 align:middle line:84% these guys here is a Cauchy sequence of real numbers. 00:33:04.870 --> 00:33:06.270 align:middle line:90% So now we use-- 00:33:06.270 --> 00:33:20.740 align:middle line:84% since R here is Cauchy complete and fn 00:33:20.740 --> 00:33:29.340 align:middle line:84% of x is a Cauchy sequence, then it 00:33:29.340 --> 00:33:37.340 align:middle line:84% means that the sequence fn of x, that this is converging, 00:33:37.340 --> 00:33:42.860 align:middle line:84% and it's converging to some number, to just one number. 00:33:42.860 --> 00:33:49.540 align:middle line:84% And that number we can call f of x. 00:33:49.540 --> 00:33:55.420 align:middle line:84% So we just defined f to be the limit of the sequence. 00:33:55.420 --> 00:33:58.220 align:middle line:84% And that's well defined because there's just one number. 00:33:58.220 --> 00:34:02.140 align:middle line:84% So for each x it gives you a number. 00:34:02.140 --> 00:34:03.020 align:middle line:90% OK. 00:34:03.020 --> 00:34:04.970 align:middle line:90% So now what we have proven is-- 00:34:04.970 --> 00:34:08.340 align:middle line:90% 00:34:08.340 --> 00:34:11.139 align:middle line:90% so now we have a candidate. 00:34:11.139 --> 00:34:16.090 align:middle line:90% So namely, we have shown that-- 00:34:16.090 --> 00:34:18.889 align:middle line:84% so now what we have proven here is 00:34:18.889 --> 00:34:24.330 align:middle line:90% that we have found a function. 00:34:24.330 --> 00:34:29.610 align:middle line:84% So we have now proven that this sequence here is 00:34:29.610 --> 00:34:32.889 align:middle line:90% converging to some function f. 00:34:32.889 --> 00:34:35.830 align:middle line:84% The convergence is just pointwise. 00:34:35.830 --> 00:34:39.409 align:middle line:90% 00:34:39.409 --> 00:34:44.130 align:middle line:84% And we don't know whether this function is continuous or not. 00:34:44.130 --> 00:34:45.670 align:middle line:90% The convergence is pointwise. 00:34:45.670 --> 00:34:52.130 align:middle line:84% This is just because for each fixed x, f of x 00:34:52.130 --> 00:34:58.636 align:middle line:84% was defined so that it was a limit of these guys, fn of x. 00:34:58.636 --> 00:35:04.290 align:middle line:84% So now we need to promote this convergence that was pointwise, 00:35:04.290 --> 00:35:06.930 align:middle line:84% and we need to promote the convergence 00:35:06.930 --> 00:35:08.830 align:middle line:90% to uniform convergence. 00:35:08.830 --> 00:35:11.370 align:middle line:90% 00:35:11.370 --> 00:35:12.590 align:middle line:90% So now we need-- 00:35:12.590 --> 00:35:18.650 align:middle line:90% 00:35:18.650 --> 00:35:22.970 align:middle line:84% so given epsilon greater than 0, again, 00:35:22.970 --> 00:35:27.410 align:middle line:84% since these guys, fn, was a Cauchy 00:35:27.410 --> 00:35:38.130 align:middle line:90% sequence in the metric space-- 00:35:38.130 --> 00:35:43.410 align:middle line:90% in this metric space-- 00:35:43.410 --> 00:35:52.650 align:middle line:84% then as we saw over there, there exists capital N 00:35:52.650 --> 00:35:59.090 align:middle line:84% so that if little n and little m are bigger than this capital N, 00:35:59.090 --> 00:36:05.170 align:middle line:84% then the difference between these two, fn 00:36:05.170 --> 00:36:15.680 align:middle line:84% of x minus fm of x, this thing here, is less than or equal to-- 00:36:15.680 --> 00:36:18.320 align:middle line:90% and I'm using epsilon over 2-- 00:36:18.320 --> 00:36:19.380 align:middle line:90% for all x. 00:36:19.380 --> 00:36:26.456 align:middle line:90% 00:36:26.456 --> 00:36:30.120 align:middle line:90% So you have that. 00:36:30.120 --> 00:36:36.960 align:middle line:84% But now this here holds for all m and n. 00:36:36.960 --> 00:36:39.960 align:middle line:90% Let's just fix n. 00:36:39.960 --> 00:36:41.800 align:middle line:90% And now we can use that. 00:36:41.800 --> 00:36:43.360 align:middle line:90% It's still true. 00:36:43.360 --> 00:36:45.720 align:middle line:90% I can still let m vary. 00:36:45.720 --> 00:36:56.400 align:middle line:84% And so if you let m go-- so fixed n bigger than capital N. 00:36:56.400 --> 00:37:04.870 align:middle line:90% And let little m go to infinity. 00:37:04.870 --> 00:37:09.590 align:middle line:84% Well, this has just some fixed number. 00:37:09.590 --> 00:37:14.430 align:middle line:84% But as m goes to infinity, this here goes to f of x. 00:37:14.430 --> 00:37:17.510 align:middle line:84% And so since this difference is always less than or equal 00:37:17.510 --> 00:37:22.630 align:middle line:84% to epsilon over 2, it's also the case 00:37:22.630 --> 00:37:29.870 align:middle line:84% that the difference here of the limit 00:37:29.870 --> 00:37:33.930 align:middle line:84% is less than or equal to epsilon over 2. 00:37:33.930 --> 00:37:36.330 align:middle line:90% So you have this here for all x. 00:37:36.330 --> 00:37:41.110 align:middle line:90% 00:37:41.110 --> 00:37:48.270 align:middle line:84% This is just letting m go through to the limit. 00:37:48.270 --> 00:37:54.790 align:middle line:84% But this, of course, says that the max, if you will-- 00:37:54.790 --> 00:37:55.770 align:middle line:90% so you have that. 00:37:55.770 --> 00:38:01.270 align:middle line:84% So this implies that the max of fn 00:38:01.270 --> 00:38:07.190 align:middle line:84% of x minus f of x, that this thing here, 00:38:07.190 --> 00:38:12.390 align:middle line:84% where x is now in this interval, that this is less than or equal 00:38:12.390 --> 00:38:12.920 align:middle line:90% to epsilon. 00:38:12.920 --> 00:38:15.942 align:middle line:90% 00:38:15.942 --> 00:38:17.670 align:middle line:90% But this is the same. 00:38:17.670 --> 00:38:24.210 align:middle line:84% This here is just the distance between fn and f. 00:38:24.210 --> 00:38:28.790 align:middle line:84% So say that this distance here is less than-- 00:38:28.790 --> 00:38:31.210 align:middle line:90% I guess I had epsilon over 2. 00:38:31.210 --> 00:38:36.990 align:middle line:84% So it means that this thing here is strictly less than epsilon. 00:38:36.990 --> 00:38:39.290 align:middle line:84% It's actually less than or equal to epsilon over 2. 00:38:39.290 --> 00:38:41.870 align:middle line:84% So it's strictly less than epsilon. 00:38:41.870 --> 00:38:45.810 align:middle line:84% But that's exactly what it means for this sequence here. 00:38:45.810 --> 00:38:48.350 align:middle line:90% So this is the same. 00:38:48.350 --> 00:39:01.650 align:middle line:84% This is the same as saying that fn converges to f uniformly. 00:39:01.650 --> 00:39:10.040 align:middle line:90% 00:39:10.040 --> 00:39:13.180 align:middle line:84% Sorry, here, i should have taken the sup. 00:39:13.180 --> 00:39:16.380 align:middle line:84% The reason why I should take the sup 00:39:16.380 --> 00:39:21.200 align:middle line:84% here is that f is just a function. 00:39:21.200 --> 00:39:24.600 align:middle line:84% It's not necessarily continuous at this stage. 00:39:24.600 --> 00:39:27.420 align:middle line:84% But it was clearly the case with the sup. 00:39:27.420 --> 00:39:30.940 align:middle line:84% So the maximum a priori may not be achieved yet. 00:39:30.940 --> 00:39:34.340 align:middle line:84% But now you see that they converge. 00:39:34.340 --> 00:39:39.460 align:middle line:84% These fn converge to f uniformly. 00:39:39.460 --> 00:39:42.760 align:middle line:84% And these guys here was continuous. 00:39:42.760 --> 00:39:45.940 align:middle line:90% These are continuous. 00:39:45.940 --> 00:39:49.060 align:middle line:84% So this implies that by this theorem 00:39:49.060 --> 00:39:54.700 align:middle line:84% we just proved, that f is also continuous. 00:39:54.700 --> 00:39:57.060 align:middle line:84% And so you see that-- so now you have 00:39:57.060 --> 00:40:00.520 align:middle line:84% that the limit is indeed in the space. 00:40:00.520 --> 00:40:06.180 align:middle line:84% And we already proved here that the convergence to this limit 00:40:06.180 --> 00:40:11.600 align:middle line:84% is uniform, which means that it's in the metric. 00:40:11.600 --> 00:40:15.375 align:middle line:90% 00:40:15.375 --> 00:40:18.840 align:middle line:90% The next thing is that-- 00:40:18.840 --> 00:40:22.660 align:middle line:84% so this here was just a little bit different way 00:40:22.660 --> 00:40:24.660 align:middle line:84% of thinking about uniform convergence, 00:40:24.660 --> 00:40:28.780 align:middle line:84% that you think about it more abstractly as a convergence 00:40:28.780 --> 00:40:30.200 align:middle line:90% in a metric space. 00:40:30.200 --> 00:40:33.020 align:middle line:84% And it's a useful way of thinking about it. 00:40:33.020 --> 00:40:34.700 align:middle line:84% The next thing we want to look at 00:40:34.700 --> 00:40:41.740 align:middle line:84% is how this uniform convergence gives you 00:40:41.740 --> 00:40:44.640 align:middle line:90% a way of computing things. 00:40:44.640 --> 00:40:50.020 align:middle line:84% So in particular, if you have some power series. 00:40:50.020 --> 00:40:54.340 align:middle line:84% And we talked about that a little bit last time. 00:40:54.340 --> 00:40:56.580 align:middle line:90% We didn't prove anything. 00:40:56.580 --> 00:41:00.180 align:middle line:84% But one of the motivation for this 00:41:00.180 --> 00:41:03.570 align:middle line:84% is that suppose you take a power series, like this. 00:41:03.570 --> 00:41:08.220 align:middle line:90% 00:41:08.220 --> 00:41:11.050 align:middle line:84% And we'll return to this in just a second. 00:41:11.050 --> 00:41:13.930 align:middle line:84% But suppose you take a power series, like this, 00:41:13.930 --> 00:41:17.470 align:middle line:84% and you are inside the radius of convergence. 00:41:17.470 --> 00:41:21.790 align:middle line:90% So inside radius of convergence. 00:41:21.790 --> 00:41:26.850 align:middle line:90% 00:41:26.850 --> 00:41:29.250 align:middle line:84% So you know that this thing here makes sense, 00:41:29.250 --> 00:41:33.410 align:middle line:84% that this, for each x, gives you a number. 00:41:33.410 --> 00:41:35.070 align:middle line:90% And so it gives you a function. 00:41:35.070 --> 00:41:37.450 align:middle line:84% And so you're interested in, what 00:41:37.450 --> 00:41:39.770 align:middle line:84% is the derivative of that function? 00:41:39.770 --> 00:41:43.470 align:middle line:84% And what is the integral of that function? 00:41:43.470 --> 00:41:46.730 align:middle line:84% How do we integrate it, and how do we take the derivative? 00:41:46.730 --> 00:41:48.530 align:middle line:84% And the thing that worries you a bit 00:41:48.530 --> 00:41:52.750 align:middle line:84% is that if it was just a finite sum, it would be trivial. 00:41:52.750 --> 00:41:54.850 align:middle line:90% But it's an infinite sum. 00:41:54.850 --> 00:41:57.070 align:middle line:90% And so is it OK-- 00:41:57.070 --> 00:42:00.330 align:middle line:84% so the natural question is, is it OK? 00:42:00.330 --> 00:42:02.760 align:middle line:84% If you want to take the derivative, 00:42:02.760 --> 00:42:06.960 align:middle line:84% can you interchange this infinite sum 00:42:06.960 --> 00:42:08.920 align:middle line:90% with taking derivative? 00:42:08.920 --> 00:42:12.920 align:middle line:84% And likewise, if you want to integrate this infinite sum, 00:42:12.920 --> 00:42:18.640 align:middle line:84% is it OK to interchange the infinite sum taking 00:42:18.640 --> 00:42:19.920 align:middle line:90% the integral? 00:42:19.920 --> 00:42:23.840 align:middle line:90% And the answer is, it's OK. 00:42:23.840 --> 00:42:25.720 align:middle line:90% So let's see that. 00:42:25.720 --> 00:42:27.960 align:middle line:90% So we do this. 00:42:27.960 --> 00:42:29.880 align:middle line:84% So we have two versions of this, one 00:42:29.880 --> 00:42:32.680 align:middle line:84% that's dealing with taking integrals 00:42:32.680 --> 00:42:36.060 align:middle line:84% and another version that deals with taking derivatives. 00:42:36.060 --> 00:42:45.300 align:middle line:90% 00:42:45.300 --> 00:42:49.440 align:middle line:84% And so the first thing is that I take-- 00:42:49.440 --> 00:42:52.960 align:middle line:84% so the first theorem is the following, 00:42:52.960 --> 00:42:56.860 align:middle line:84% that I have some interval from a to b. 00:42:56.860 --> 00:43:02.200 align:middle line:90% 00:43:02.200 --> 00:43:08.000 align:middle line:84% And then I have a sequence of functions, fn, 00:43:08.000 --> 00:43:13.000 align:middle line:90% that are integrable. 00:43:13.000 --> 00:43:17.920 align:middle line:84% They may not be continuous, they're just integrable. 00:43:17.920 --> 00:43:23.320 align:middle line:84% So in particular, they are bounded. 00:43:23.320 --> 00:43:32.380 align:middle line:84% And then I have that fn here converges to f uniformly. 00:43:32.380 --> 00:43:37.720 align:middle line:90% 00:43:37.720 --> 00:43:41.060 align:middle line:90% And I want to prove that-- 00:43:41.060 --> 00:43:52.220 align:middle line:84% so the claim is that fn is also integrable. 00:43:52.220 --> 00:43:55.280 align:middle line:90% 00:43:55.280 --> 00:44:07.210 align:middle line:84% And the integral of f is equal to the limit of these integrals. 00:44:07.210 --> 00:44:20.950 align:middle line:90% 00:44:20.950 --> 00:44:24.390 align:middle line:84% Sorry, in particular, of course, those integrals 00:44:24.390 --> 00:44:26.230 align:middle line:90% here has a limit. 00:44:26.230 --> 00:44:27.530 align:middle line:90% It's part of the statement. 00:44:27.530 --> 00:44:51.690 align:middle line:90% 00:44:51.690 --> 00:44:52.190 align:middle line:90% OK. 00:44:52.190 --> 00:44:58.060 align:middle line:90% 00:44:58.060 --> 00:44:58.560 align:middle line:90% Right. 00:44:58.560 --> 00:45:02.540 align:middle line:90% 00:45:02.540 --> 00:45:05.710 align:middle line:84% So what do we want to-- what do we need to prove? 00:45:05.710 --> 00:45:06.210 align:middle line:90% So prove. 00:45:06.210 --> 00:45:10.280 align:middle line:90% 00:45:10.280 --> 00:45:12.220 align:middle line:90% So we want to show-- 00:45:12.220 --> 00:45:21.200 align:middle line:84% so we first want to show that f is integrable. 00:45:21.200 --> 00:45:26.420 align:middle line:90% 00:45:26.420 --> 00:45:28.900 align:middle line:84% And we had our criteria for doing that. 00:45:28.900 --> 00:45:31.840 align:middle line:90% So we're using the criteria. 00:45:31.840 --> 00:45:42.210 align:middle line:84% So we can use that all we need to show-- 00:45:42.210 --> 00:45:47.580 align:middle line:90% 00:45:47.580 --> 00:45:59.060 align:middle line:84% to show-- is the following, that given epsilon greater than 0, 00:45:59.060 --> 00:46:06.900 align:middle line:84% there exists a partition P of the interval from a 00:46:06.900 --> 00:46:18.900 align:middle line:84% to b, so that the upper sum of f with respect to this partition 00:46:18.900 --> 00:46:25.300 align:middle line:84% minus the lower sum of f with respect to the same partition, 00:46:25.300 --> 00:46:28.620 align:middle line:84% that this thing here is less than epsilon. 00:46:28.620 --> 00:46:30.000 align:middle line:90% That's all we need to prove. 00:46:30.000 --> 00:46:31.300 align:middle line:90% We need to prove it. 00:46:31.300 --> 00:46:33.540 align:middle line:84% For each epsilon, we need to prove 00:46:33.540 --> 00:46:35.240 align:middle line:90% that there exists a partition. 00:46:35.240 --> 00:46:38.820 align:middle line:84% So if we have that, then we already 00:46:38.820 --> 00:46:43.580 align:middle line:90% know that then f is integrable. 00:46:43.580 --> 00:46:46.480 align:middle line:84% So that's just achieved this first task. 00:46:46.480 --> 00:46:48.720 align:middle line:84% And then, afterwards, we need to prove that. 00:46:48.720 --> 00:47:05.923 align:middle line:90% 00:47:05.923 --> 00:47:06.423 align:middle line:90% Right. 00:47:06.423 --> 00:47:06.930 align:middle line:90% OK. 00:47:06.930 --> 00:47:09.970 align:middle line:90% So now, how do we prove that? 00:47:09.970 --> 00:47:19.530 align:middle line:84% So the first thing we observe is that if we take-- 00:47:19.530 --> 00:47:21.810 align:middle line:90% if we're looking at-- 00:47:21.810 --> 00:47:26.370 align:middle line:90% suppose that fn and f-- 00:47:26.370 --> 00:47:26.910 align:middle line:90% here. 00:47:26.910 --> 00:47:36.410 align:middle line:84% Suppose that fn of x minus f of x, 00:47:36.410 --> 00:47:45.050 align:middle line:84% suppose that this thing here was less than epsilon 00:47:45.050 --> 00:47:51.570 align:middle line:90% over 3 b minus a. 00:47:51.570 --> 00:47:54.720 align:middle line:90% Suppose you have that. 00:47:54.720 --> 00:47:57.230 align:middle line:84% And of course, this here can be achieved. 00:47:57.230 --> 00:47:59.740 align:middle line:90% 00:47:59.740 --> 00:48:02.606 align:middle line:90% So epsilon is now given. 00:48:02.606 --> 00:48:04.540 align:middle line:90% And so you have that. 00:48:04.540 --> 00:48:07.920 align:middle line:90% So maybe I should just say that. 00:48:07.920 --> 00:48:11.000 align:middle line:90% So epsilon is given. 00:48:11.000 --> 00:48:22.400 align:middle line:84% So since fn converges to f uniformly, 00:48:22.400 --> 00:48:30.280 align:middle line:84% this means that there exists capital N 00:48:30.280 --> 00:48:33.760 align:middle line:84% so that if you're further out than capital N, then 00:48:33.760 --> 00:48:37.240 align:middle line:84% actually you have this thing here. 00:48:37.240 --> 00:48:38.560 align:middle line:90% b minus b. 00:48:38.560 --> 00:48:39.820 align:middle line:90% b and a are fixed. 00:48:39.820 --> 00:48:41.480 align:middle line:90% This is just something fixed. 00:48:41.480 --> 00:48:43.420 align:middle line:90% So this is just another epsilon. 00:48:43.420 --> 00:48:45.960 align:middle line:90% 00:48:45.960 --> 00:48:46.580 align:middle line:90% OK. 00:48:46.580 --> 00:48:51.880 align:middle line:84% So if you have that, now suppose you take-- 00:48:51.880 --> 00:48:55.120 align:middle line:84% so remember, we defined, when we did 00:48:55.120 --> 00:48:59.160 align:middle line:84% the-- when we had the partition, the partition had 00:48:59.160 --> 00:49:11.040 align:middle line:84% the dividing point with dividing points xi. 00:49:11.040 --> 00:49:14.440 align:middle line:84% So we were looking at these small intervals 00:49:14.440 --> 00:49:18.560 align:middle line:90% from xi minus 1 to xi. 00:49:18.560 --> 00:49:22.750 align:middle line:84% And we're looking at the sup of the function. 00:49:22.750 --> 00:49:25.600 align:middle line:90% 00:49:25.600 --> 00:49:28.480 align:middle line:84% So if you're looking at the sup of-- suppose 00:49:28.480 --> 00:49:34.000 align:middle line:84% you're looking at-- so this here is on this little interval. 00:49:34.000 --> 00:49:34.960 align:middle line:90% But it's for f. 00:49:34.960 --> 00:49:37.480 align:middle line:90% 00:49:37.480 --> 00:49:40.580 align:middle line:84% But the function f here, you see the function. 00:49:40.580 --> 00:49:46.080 align:middle line:84% So you have that f of x, this function here-- sorry, 00:49:46.080 --> 00:49:47.740 align:middle line:84% I don't even need the absolute value. 00:49:47.740 --> 00:49:58.610 align:middle line:84% But f of x is less than or equal to fn of x plus, of course, 00:49:58.610 --> 00:50:02.490 align:middle line:90% f of x minus fn of x. 00:50:02.490 --> 00:50:07.270 align:middle line:90% 00:50:07.270 --> 00:50:12.630 align:middle line:84% And so this means that this thing here is-- 00:50:12.630 --> 00:50:18.430 align:middle line:84% this thing here is less than or equal to fn of x. 00:50:18.430 --> 00:50:22.290 align:middle line:84% This thing here is bounded by the absolute value of it. 00:50:22.290 --> 00:50:30.750 align:middle line:90% 00:50:30.750 --> 00:50:33.590 align:middle line:90% And so this thing here-- 00:50:33.590 --> 00:50:36.190 align:middle line:90% so this is fn of x. 00:50:36.190 --> 00:50:46.870 align:middle line:84% And this here is just epsilon over 3 b minus a, like that. 00:50:46.870 --> 00:50:49.660 align:middle line:90% 00:50:49.660 --> 00:50:54.100 align:middle line:84% So this means that if I take the sup on this little interval 00:50:54.100 --> 00:50:59.020 align:middle line:84% of f, then it's bounded by the sup of this function 00:50:59.020 --> 00:51:00.540 align:middle line:90% plus that number. 00:51:00.540 --> 00:51:02.300 align:middle line:90% So I have that-- 00:51:02.300 --> 00:51:21.609 align:middle line:90% 00:51:21.609 --> 00:51:28.000 align:middle line:84% so I have that, again, the sup of f on this little interval-- 00:51:28.000 --> 00:51:31.340 align:middle line:90% 00:51:31.340 --> 00:51:37.900 align:middle line:90% this is what we call Mi of f-- 00:51:37.900 --> 00:51:40.620 align:middle line:90% that this is less or equal to-- 00:51:40.620 --> 00:51:43.380 align:middle line:84% because of this inequality, it's less 00:51:43.380 --> 00:51:52.660 align:middle line:84% than the sup of fn of x on the same little interval 00:51:52.660 --> 00:51:56.940 align:middle line:90% plus epsilon over 3 b minus a. 00:51:56.940 --> 00:51:59.940 align:middle line:90% 00:51:59.940 --> 00:52:11.230 align:middle line:84% And likewise-- so likewise for the inf. 00:52:11.230 --> 00:52:14.980 align:middle line:90% 00:52:14.980 --> 00:52:19.500 align:middle line:84% So again, I let little mi of f, this 00:52:19.500 --> 00:52:22.990 align:middle line:84% is now the infimum on this little interval. 00:52:22.990 --> 00:52:25.780 align:middle line:90% 00:52:25.780 --> 00:52:26.940 align:middle line:90% Oops. 00:52:26.940 --> 00:52:29.540 align:middle line:90% On this little interval of f. 00:52:29.540 --> 00:52:34.060 align:middle line:90% 00:52:34.060 --> 00:52:35.340 align:middle line:90% Sorry. 00:52:35.340 --> 00:52:38.780 align:middle line:84% Here, I wanted to insert what this thing was. 00:52:38.780 --> 00:52:43.860 align:middle line:84% So this was M. The notation for this sup here is Mifn. 00:52:43.860 --> 00:52:53.850 align:middle line:90% 00:52:53.850 --> 00:52:59.770 align:middle line:84% And the infimum for f on this little interval, the notation 00:52:59.770 --> 00:53:04.890 align:middle line:84% for that is this little m, i referring to the interval, f 00:53:04.890 --> 00:53:06.810 align:middle line:90% to the function. 00:53:06.810 --> 00:53:14.750 align:middle line:84% And so similarly, if you take the infimum, 00:53:14.750 --> 00:53:19.570 align:middle line:84% and you compare it to the infimum 00:53:19.570 --> 00:53:26.570 align:middle line:84% of this fn on the same interval, then the same argument 00:53:26.570 --> 00:53:34.650 align:middle line:84% with obvious changes means that mif here 00:53:34.650 --> 00:53:47.300 align:middle line:84% is now bigger or equal to mifn minus epsilon over 3 b minus a. 00:53:47.300 --> 00:53:52.210 align:middle line:90% 00:53:52.210 --> 00:53:55.480 align:middle line:90% It's the same-- 00:53:55.480 --> 00:53:58.920 align:middle line:84% OK, maybe I'll just explain it, but it's the same sort of thing, 00:53:58.920 --> 00:54:01.400 align:middle line:90% that you take-- 00:54:01.400 --> 00:54:04.200 align:middle line:84% oh, maybe I shouldn't erase this one. 00:54:04.200 --> 00:54:08.200 align:middle line:84% So it's sort of the same thing, that you're looking at-- 00:54:08.200 --> 00:54:14.280 align:middle line:84% you have f of x again is equal to fn 00:54:14.280 --> 00:54:22.280 align:middle line:90% of x plus fn plus f of x minus. 00:54:22.280 --> 00:54:26.200 align:middle line:90% So you have obviously this. 00:54:26.200 --> 00:54:32.720 align:middle line:84% And so this means that f of x is bigger 00:54:32.720 --> 00:54:39.760 align:middle line:90% or equal to fn of x minus. 00:54:39.760 --> 00:54:44.960 align:middle line:84% If you think that this here becomes negative and the maximum 00:54:44.960 --> 00:54:48.640 align:middle line:84% it can become negative on this little interval here-- 00:54:48.640 --> 00:54:56.600 align:middle line:84% on this little interval, the worst this here can become 00:54:56.600 --> 00:55:01.360 align:middle line:90% negative is epsilon over-- 00:55:01.360 --> 00:55:06.200 align:middle line:84% actually, on any interval, it's this. 00:55:06.200 --> 00:55:07.160 align:middle line:90% Like that. 00:55:07.160 --> 00:55:10.000 align:middle line:90% So you always have this. 00:55:10.000 --> 00:55:12.040 align:middle line:90% OK. 00:55:12.040 --> 00:55:16.680 align:middle line:84% And so if you take now the infimum on this little interval, 00:55:16.680 --> 00:55:24.120 align:middle line:84% then you're getting mif bigger or equal to mifn minus 00:55:24.120 --> 00:55:29.240 align:middle line:90% epsilon over 3 b minus a. 00:55:29.240 --> 00:55:30.600 align:middle line:90% So you have that. 00:55:30.600 --> 00:55:32.240 align:middle line:84% So that's what we claim over there. 00:55:32.240 --> 00:55:35.040 align:middle line:90% 00:55:35.040 --> 00:55:37.080 align:middle line:90% OK. 00:55:37.080 --> 00:55:42.030 align:middle line:84% So we have this thing here, and we have the corresponding thing. 00:55:42.030 --> 00:55:45.190 align:middle line:84% This is less than or equal to the maximum of the function 00:55:45.190 --> 00:55:47.550 align:middle line:90% on the same interval. 00:55:47.550 --> 00:55:52.810 align:middle line:84% And then that's-- and then this equality here, 00:55:52.810 --> 00:55:59.070 align:middle line:84% this equality here was that it's bounded by this guy here plus 00:55:59.070 --> 00:56:04.390 align:middle line:90% epsilon over 3 b minus a. 00:56:04.390 --> 00:56:09.030 align:middle line:84% So this is for the little m's, the little mi's, and the capital 00:56:09.030 --> 00:56:09.790 align:middle line:90% Mi. 00:56:09.790 --> 00:56:12.030 align:middle line:84% And now we need to sum these guys, 00:56:12.030 --> 00:56:17.510 align:middle line:84% and we need to multiply those with a delta xi. 00:56:17.510 --> 00:56:20.150 align:middle line:84% So this means that if you're looking at-- so 00:56:20.150 --> 00:56:22.910 align:middle line:84% what we're interested in is we're 00:56:22.910 --> 00:56:27.870 align:middle line:90% looking at this times delta xi. 00:56:27.870 --> 00:56:31.710 align:middle line:90% This gives you the-- 00:56:31.710 --> 00:56:37.150 align:middle line:84% sorry, this gives you the lower sum of f with respect 00:56:37.150 --> 00:56:41.250 align:middle line:84% to this partition, where you're summing here over the i. 00:56:41.250 --> 00:56:43.830 align:middle line:90% 00:56:43.830 --> 00:56:54.470 align:middle line:84% If you're summing the sups, then you're 00:56:54.470 --> 00:57:00.150 align:middle line:84% getting the upper sum, like that. 00:57:00.150 --> 00:57:07.880 align:middle line:90% And so now-- so you take-- 00:57:07.880 --> 00:57:15.110 align:middle line:90% 00:57:15.110 --> 00:57:20.430 align:middle line:84% you take this inequality, this string of inequality, 00:57:20.430 --> 00:57:30.150 align:middle line:84% and you're just multiplying by delta xi on both sides. 00:57:30.150 --> 00:57:31.810 align:middle line:90% And then I'm summing. 00:57:31.810 --> 00:57:33.630 align:middle line:90% So if I multiply-- 00:57:33.630 --> 00:57:36.070 align:middle line:90% so let's call this here star. 00:57:36.070 --> 00:57:53.840 align:middle line:84% So multiply star by delta xi and sum over i. 00:57:53.840 --> 00:57:56.820 align:middle line:84% If you do that, then what you get is you're 00:57:56.820 --> 00:58:15.820 align:middle line:84% getting mifn minus epsilon over 3 b minus a times this, 00:58:15.820 --> 00:58:18.180 align:middle line:90% times delta xi. 00:58:18.180 --> 00:58:26.440 align:middle line:84% This is less than or equal to mif delta xi, which is less than 00:58:26.440 --> 00:58:28.480 align:middle line:90% or equal to this one here. 00:58:28.480 --> 00:58:31.460 align:middle line:90% 00:58:31.460 --> 00:58:31.980 align:middle line:90% Sorry. 00:58:31.980 --> 00:58:34.380 align:middle line:90% And I should also sum. 00:58:34.380 --> 00:58:40.490 align:middle line:90% Summing over i, like this. 00:58:40.490 --> 00:58:43.850 align:middle line:84% And then I have one more inequality, namely 00:58:43.850 --> 00:58:46.370 align:middle line:90% this right-hand side. 00:58:46.370 --> 00:58:49.210 align:middle line:90% So less than or equal to. 00:58:49.210 --> 00:58:50.670 align:middle line:90% I'm summing over i. 00:58:50.670 --> 00:58:53.890 align:middle line:90% 00:58:53.890 --> 00:59:06.050 align:middle line:84% And I have-- this is mif plus epsilon over 3 b minus a times 00:59:06.050 --> 00:59:07.890 align:middle line:90% delta xi. 00:59:07.890 --> 00:59:12.050 align:middle line:90% 00:59:12.050 --> 00:59:13.690 align:middle line:90% So we have that. 00:59:13.690 --> 00:59:16.020 align:middle line:84% Now let's try to figure out what this means. 00:59:16.020 --> 00:59:22.490 align:middle line:90% 00:59:22.490 --> 00:59:27.250 align:middle line:84% So this here-- so now, if you multiply 00:59:27.250 --> 00:59:34.730 align:middle line:84% this out, the first thing, the top left corner, 00:59:34.730 --> 00:59:39.210 align:middle line:84% that gives you-- when you multiply the bracket out, 00:59:39.210 --> 00:59:46.370 align:middle line:84% the first term you get is the mifn times delta xi. 00:59:46.370 --> 00:59:50.650 align:middle line:84% But that is just the lower sum with respect 00:59:50.650 --> 00:59:53.730 align:middle line:90% to the partition P for fn. 00:59:53.730 --> 00:59:56.610 align:middle line:90% 00:59:56.610 --> 01:00:01.230 align:middle line:84% But then you get another term, which is you get one other term. 01:00:01.230 --> 01:00:03.530 align:middle line:90% You get this minus. 01:00:03.530 --> 01:00:08.030 align:middle line:84% And then you're summing over the i of epsilon 3 01:00:08.030 --> 01:00:10.920 align:middle line:90% b minus a times xi. 01:00:10.920 --> 01:00:16.890 align:middle line:90% 01:00:16.890 --> 01:00:27.810 align:middle line:84% And then it's less than or equal to the term up there. 01:00:27.810 --> 01:00:34.000 align:middle line:84% That's just the lower sum of f with respect to the partition. 01:00:34.000 --> 01:00:38.080 align:middle line:84% THE next term is the upper sum of f 01:00:38.080 --> 01:00:40.520 align:middle line:90% with respect to the partition. 01:00:40.520 --> 01:00:44.100 align:middle line:84% And then the last inequality, the first term-- 01:00:44.100 --> 01:00:48.420 align:middle line:84% when you multiply this bracket out, the first term here is-- 01:00:48.420 --> 01:00:52.440 align:middle line:84% and unfortunately, there was a missing n. 01:00:52.440 --> 01:00:54.140 align:middle line:90% I copied it from here. 01:00:54.140 --> 01:00:56.800 align:middle line:90% Sorry. 01:00:56.800 --> 01:00:57.840 align:middle line:90% There was a missing n. 01:00:57.840 --> 01:00:59.340 align:middle line:90% I just copied it from this one. 01:00:59.340 --> 01:01:07.760 align:middle line:90% 01:01:07.760 --> 01:01:12.120 align:middle line:84% So now when you multiply this bracket out, 01:01:12.120 --> 01:01:16.800 align:middle line:90% the first term you get here is-- 01:01:16.800 --> 01:01:19.220 align:middle line:84% so you're getting less than or equal to the first term 01:01:19.220 --> 01:01:22.000 align:middle line:84% you get there when you're summing over the i. 01:01:22.000 --> 01:01:28.240 align:middle line:90% That's exactly u fn, p. 01:01:28.240 --> 01:01:35.590 align:middle line:84% And then the second term you get is this i epsilon over 3 b 01:01:35.590 --> 01:01:37.670 align:middle line:90% minus a delta xi. 01:01:37.670 --> 01:01:40.430 align:middle line:90% 01:01:40.430 --> 01:01:48.070 align:middle line:84% Now, this thing here, when I'm summing here, 01:01:48.070 --> 01:01:51.330 align:middle line:90% then this is just a constant. 01:01:51.330 --> 01:01:53.590 align:middle line:84% So I can just pull that constant out. 01:01:53.590 --> 01:01:55.550 align:middle line:84% And the sum of these differences, 01:01:55.550 --> 01:01:58.270 align:middle line:90% that just gives you b minus a. 01:01:58.270 --> 01:02:02.830 align:middle line:84% So if you put that in and it's exactly the same 01:02:02.830 --> 01:02:05.510 align:middle line:84% you get on this term, then you have 01:02:05.510 --> 01:02:11.830 align:middle line:84% that L fn, P minus epsilon over 3 01:02:11.830 --> 01:02:22.070 align:middle line:84% is less than or equal to L f, P, which is less than 01:02:22.070 --> 01:02:28.950 align:middle line:84% or equal to u f, P. And this is less than or equal 01:02:28.950 --> 01:02:38.090 align:middle line:90% to u fn, P plus epsilon over 3. 01:02:38.090 --> 01:02:41.270 align:middle line:90% 01:02:41.270 --> 01:02:42.070 align:middle line:90% OK. 01:02:42.070 --> 01:02:44.350 align:middle line:90% Now what have we used so far? 01:02:44.350 --> 01:02:47.470 align:middle line:90% So far, we have just used-- 01:02:47.470 --> 01:02:49.630 align:middle line:90% P was any partition. 01:02:49.630 --> 01:02:56.430 align:middle line:84% And all we have used is that the fn converges to f uniformly. 01:02:56.430 --> 01:02:59.070 align:middle line:84% And then, for this given epsilon, 01:02:59.070 --> 01:03:02.190 align:middle line:84% we chose this capital N sufficiently large. 01:03:02.190 --> 01:03:05.510 align:middle line:90% 01:03:05.510 --> 01:03:10.910 align:middle line:90% But now, now you see that-- 01:03:10.910 --> 01:03:12.850 align:middle line:90% now we can just fix-- 01:03:12.850 --> 01:03:17.510 align:middle line:90% so now we just let little n-- 01:03:17.510 --> 01:03:21.110 align:middle line:84% we just said, let this be equal to this capital N. We 01:03:21.110 --> 01:03:25.890 align:middle line:84% just need to use it for one little n. 01:03:25.890 --> 01:03:29.528 align:middle line:90% 01:03:29.528 --> 01:03:31.660 align:middle line:90% And then we can-- 01:03:31.660 --> 01:03:38.060 align:middle line:84% and now we have that since this particular function, this one 01:03:38.060 --> 01:03:49.980 align:middle line:84% function, was integrable, then there exists a partition. 01:03:49.980 --> 01:03:53.860 align:middle line:84% But this is now a partition that comes from, 01:03:53.860 --> 01:03:56.780 align:middle line:84% that is determined from this function here. 01:03:56.780 --> 01:04:04.100 align:middle line:84% But there exists a partition so that the L-- 01:04:04.100 --> 01:04:14.380 align:middle line:84% sorry, so that the difference between the upper sum 01:04:14.380 --> 01:04:20.500 align:middle line:84% for this function minus the lower sum for this function 01:04:20.500 --> 01:04:25.500 align:middle line:84% is less than, say, epsilon over 3. 01:04:25.500 --> 01:04:30.730 align:middle line:90% 01:04:30.730 --> 01:04:35.260 align:middle line:84% So you see that this number here-- 01:04:35.260 --> 01:04:38.210 align:middle line:90% 01:04:38.210 --> 01:04:43.250 align:middle line:90% so you see that the difference-- 01:04:43.250 --> 01:04:44.910 align:middle line:90% now I can erase this thing here. 01:04:44.910 --> 01:04:48.210 align:middle line:90% 01:04:48.210 --> 01:04:53.690 align:middle line:90% So now the first-- 01:04:53.690 --> 01:05:05.714 align:middle line:84% so we observe now that this number here, because of this, 01:05:05.714 --> 01:05:09.056 align:middle line:84% this number is always larger than that. 01:05:09.056 --> 01:05:15.750 align:middle line:84% And so this means that this thing here, by this inequality, 01:05:15.750 --> 01:05:19.130 align:middle line:84% it just tells you that this is less than-- 01:05:19.130 --> 01:05:23.050 align:middle line:84% that this thing here-- let me continue here below-- 01:05:23.050 --> 01:05:32.010 align:middle line:84% that this thing here is less than L fn, P plus epsilon 01:05:32.010 --> 01:05:33.250 align:middle line:90% over 3. 01:05:33.250 --> 01:05:35.870 align:middle line:84% This is coming from that this one here-- 01:05:35.870 --> 01:05:38.070 align:middle line:84% just moving this over on the other side. 01:05:38.070 --> 01:05:40.530 align:middle line:84% Then you have that this is bounded by that. 01:05:40.530 --> 01:05:42.990 align:middle line:84% And then you have another epsilon over 3. 01:05:42.990 --> 01:05:46.402 align:middle line:90% 01:05:46.402 --> 01:05:49.930 align:middle line:84% And so you see that those two things-- 01:05:49.930 --> 01:05:52.010 align:middle line:90% this here and that here-- 01:05:52.010 --> 01:05:56.890 align:middle line:84% is squeezed in an interval of length epsilon. 01:05:56.890 --> 01:06:00.690 align:middle line:84% So this means that the difference between these two 01:06:00.690 --> 01:06:03.650 align:middle line:90% is bounded by epsilon. 01:06:03.650 --> 01:06:12.148 align:middle line:84% So L f with respect to this partition and u f, 01:06:12.148 --> 01:06:29.220 align:middle line:84% P, they both lie in an interval of length epsilon. 01:06:29.220 --> 01:06:31.520 align:middle line:84% So it means that the difference between the two. 01:06:31.520 --> 01:06:33.420 align:middle line:90% So the difference between. 01:06:33.420 --> 01:06:37.240 align:middle line:84% So this here, which is always positive. 01:06:37.240 --> 01:06:41.280 align:middle line:84% But this here, the difference must be, 01:06:41.280 --> 01:06:42.620 align:middle line:90% I guess, less than or equal to. 01:06:42.620 --> 01:06:43.420 align:middle line:90% It doesn't matter. 01:06:43.420 --> 01:06:48.320 align:middle line:90% 01:06:48.320 --> 01:07:01.440 align:middle line:84% And this proves-- so this show that f here is integrable. 01:07:01.440 --> 01:07:02.340 align:middle line:90% It's integrable. 01:07:02.340 --> 01:07:08.440 align:middle line:90% 01:07:08.440 --> 01:07:11.080 align:middle line:84% Because to show that this function was integrable, 01:07:11.080 --> 01:07:12.320 align:middle line:90% we just needed to-- 01:07:12.320 --> 01:07:15.280 align:middle line:84% for a given epsilon, we needed to find a partition 01:07:15.280 --> 01:07:19.000 align:middle line:84% so that the difference between the upper and the lower sum 01:07:19.000 --> 01:07:22.120 align:middle line:84% was less than or equal to epsilon. 01:07:22.120 --> 01:07:28.590 align:middle line:90% So the function f is integrable. 01:07:28.590 --> 01:07:33.950 align:middle line:84% Now, I claim that it also proves that-- 01:07:33.950 --> 01:07:41.890 align:middle line:84% it also shows that the limit is what it should be. 01:07:41.890 --> 01:07:51.910 align:middle line:90% 01:07:51.910 --> 01:07:58.350 align:middle line:84% Because you see that we now know that the function-- 01:07:58.350 --> 01:08:15.145 align:middle line:84% so we have now that this integral of f dx, this exists, 01:08:15.145 --> 01:08:16.770 align:middle line:84% because we proved that it's integrable. 01:08:16.770 --> 01:08:18.590 align:middle line:90% So it means that this exists. 01:08:18.590 --> 01:08:23.510 align:middle line:84% And this is always squeezed between-- 01:08:23.510 --> 01:08:25.550 align:middle line:84% for any partition, it's always squeezed 01:08:25.550 --> 01:08:26.620 align:middle line:90% between these two things. 01:08:26.620 --> 01:08:31.990 align:middle line:90% 01:08:31.990 --> 01:08:33.970 align:middle line:90% But now you see that-- 01:08:33.970 --> 01:08:40.430 align:middle line:84% so this means that it would have to lie-- 01:08:40.430 --> 01:08:43.990 align:middle line:84% in this string of inequality, it would 01:08:43.990 --> 01:08:47.630 align:middle line:84% have to lie here, between these two numbers. 01:08:47.630 --> 01:08:55.729 align:middle line:84% But this thing here, if you make this-- 01:08:55.729 --> 01:08:58.390 align:middle line:90% so we have also-- 01:08:58.390 --> 01:09:01.670 align:middle line:90% so what do we have? 01:09:01.670 --> 01:09:14.390 align:middle line:84% We have that-- so we have that this integral here, 01:09:14.390 --> 01:09:18.410 align:middle line:84% from a to b of f dx, is less than 01:09:18.410 --> 01:09:25.140 align:middle line:84% or equal to u f with respect to any partition. 01:09:25.140 --> 01:09:27.300 align:middle line:90% The same thing here. 01:09:27.300 --> 01:09:33.140 align:middle line:90% And now, if we let-- 01:09:33.140 --> 01:09:36.740 align:middle line:84% if we looked at-- if we use this-- 01:09:36.740 --> 01:09:40.700 align:middle line:84% if we use, for this given epsilon, 01:09:40.700 --> 01:09:45.750 align:middle line:84% that n here is bigger than capital N, bigger or equal to, 01:09:45.750 --> 01:09:49.060 align:middle line:84% then we have that this thing here was 01:09:49.060 --> 01:09:52.540 align:middle line:84% bounded below from this inequality 01:09:52.540 --> 01:10:00.780 align:middle line:90% L fn, P minus epsilon over 3. 01:10:00.780 --> 01:10:04.220 align:middle line:84% And we had that it was bounded from above. 01:10:04.220 --> 01:10:11.380 align:middle line:84% So this is just this inequality u fn, P plus epsilon over 3. 01:10:11.380 --> 01:10:18.620 align:middle line:84% And so now we made use of-- and so we do for this capital N. 01:10:18.620 --> 01:10:23.330 align:middle line:84% So in particular, it holds for capital N. 01:10:23.330 --> 01:10:27.250 align:middle line:90% And now we made use of that. 01:10:27.250 --> 01:10:29.330 align:middle line:84% So now all I want to do is here I 01:10:29.330 --> 01:10:36.130 align:middle line:84% want to estimate the difference between this integral here 01:10:36.130 --> 01:10:38.730 align:middle line:90% and this integral here. 01:10:38.730 --> 01:10:48.210 align:middle line:84% And the integral from a to b of f capital N. And I 01:10:48.210 --> 01:10:51.530 align:middle line:84% want to prove that if I'm very far out, 01:10:51.530 --> 01:10:54.730 align:middle line:90% that this here is very small. 01:10:54.730 --> 01:10:57.110 align:middle line:90% But I now make this one here-- 01:10:57.110 --> 01:10:59.370 align:middle line:90% so if I take m-- 01:10:59.370 --> 01:11:03.770 align:middle line:84% when I choose my partition so that this number here-- 01:11:03.770 --> 01:11:06.610 align:middle line:84% so then I can just make sure that this number 01:11:06.610 --> 01:11:12.450 align:middle line:84% here, when I choose the partition, 01:11:12.450 --> 01:11:15.650 align:middle line:84% then I just said that I choose the partition so 01:11:15.650 --> 01:11:23.450 align:middle line:84% that this thing here minus this lower here for this capital N, 01:11:23.450 --> 01:11:27.050 align:middle line:84% that this here was smaller than epsilon over 3. 01:11:27.050 --> 01:11:29.690 align:middle line:90% And so this means again-- 01:11:29.690 --> 01:11:35.210 align:middle line:84% but now you see that this integral 01:11:35.210 --> 01:11:39.330 align:middle line:84% of this integral here, this is always squeezed 01:11:39.330 --> 01:11:40.750 align:middle line:90% between these two numbers. 01:11:40.750 --> 01:11:47.740 align:middle line:84% So this is always squeezed like this. 01:11:47.740 --> 01:11:56.640 align:middle line:90% 01:11:56.640 --> 01:12:04.850 align:middle line:84% And so now you see that this means that this integral is 01:12:04.850 --> 01:12:08.530 align:middle line:90% in this interval here. 01:12:08.530 --> 01:12:16.930 align:middle line:84% But in fact, of course, in this interval between these two 01:12:16.930 --> 01:12:17.430 align:middle line:90% numbers-- 01:12:17.430 --> 01:12:20.960 align:middle line:90% 01:12:20.960 --> 01:12:24.080 align:middle line:90% between these two numbers-- 01:12:24.080 --> 01:12:27.880 align:middle line:84% even if there wasn't a plus in that interval, 01:12:27.880 --> 01:12:31.520 align:middle line:84% that this integral here would also lie. 01:12:31.520 --> 01:12:37.040 align:middle line:84% So it lies even more comfortable in this larger interval, 01:12:37.040 --> 01:12:39.240 align:middle line:84% where you're adding epsilon over 3 01:12:39.240 --> 01:12:41.880 align:middle line:90% and subtracting epsilon over 3. 01:12:41.880 --> 01:12:46.560 align:middle line:84% So this integral lie between these two. 01:12:46.560 --> 01:12:49.350 align:middle line:90% The same is the case for this. 01:12:49.350 --> 01:12:53.720 align:middle line:90% 01:12:53.720 --> 01:12:56.960 align:middle line:84% And this interval, I can have length epsilon. 01:12:56.960 --> 01:12:59.280 align:middle line:84% So this means that this difference between these two 01:12:59.280 --> 01:13:01.790 align:middle line:84% numbers is also less than epsilon. 01:13:01.790 --> 01:13:09.720 align:middle line:90% 01:13:09.720 --> 01:13:14.080 align:middle line:84% And so this really proves that if it was sufficiently far out, 01:13:14.080 --> 01:13:16.920 align:middle line:84% then the difference between these two integrals 01:13:16.920 --> 01:13:18.800 align:middle line:90% was bounded by epsilon. 01:13:18.800 --> 01:13:22.800 align:middle line:84% So this proves that the integrals are converging. 01:13:22.800 --> 01:13:25.300 align:middle line:90% OK? 01:13:25.300 --> 01:13:28.080 align:middle line:84% Let's look at-- so how do we use this? 01:13:28.080 --> 01:13:34.940 align:middle line:84% So what we were interested in was using this fact 01:13:34.940 --> 01:13:36.980 align:middle line:90% in the case of a power series. 01:13:36.980 --> 01:13:41.660 align:middle line:84% So let me just look at an example for that. 01:13:41.660 --> 01:13:47.200 align:middle line:90% 01:13:47.200 --> 01:13:50.360 align:middle line:90% So suppose that En of x-- 01:13:50.360 --> 01:13:54.280 align:middle line:84% so this is the same as in the previous example, 01:13:54.280 --> 01:14:00.280 align:middle line:84% where I'm summing here from k equal to 0 to n xn over-- 01:14:00.280 --> 01:14:03.820 align:middle line:90% xk, sorry, over k factorial. 01:14:03.820 --> 01:14:08.440 align:middle line:84% So these are the functions xn and x-- 01:14:08.440 --> 01:14:10.480 align:middle line:90% sorry, En. 01:14:10.480 --> 01:14:14.230 align:middle line:84% And the function-- so this was the function En. 01:14:14.230 --> 01:14:18.110 align:middle line:84% The function E is just the exponential function, 01:14:18.110 --> 01:14:24.950 align:middle line:84% which I think about as the infinite series, power series, 01:14:24.950 --> 01:14:27.950 align:middle line:90% like this. 01:14:27.950 --> 01:14:29.530 align:middle line:90% And now I had-- 01:14:29.530 --> 01:14:34.590 align:middle line:84% so we already observed that by the Weierstrass. 01:14:34.590 --> 01:14:41.390 align:middle line:90% So by Weierstrass M-test. 01:14:41.390 --> 01:14:45.150 align:middle line:90% 01:14:45.150 --> 01:14:53.240 align:middle line:84% By Weierstrass M-test, the En converges to E on any interval. 01:14:53.240 --> 01:14:56.070 align:middle line:90% 01:14:56.070 --> 01:14:58.110 align:middle line:84% You have to restrict the interval. 01:14:58.110 --> 01:15:00.710 align:middle line:84% But if you're fixing an interval like this, 01:15:00.710 --> 01:15:05.030 align:middle line:90% then it's converging uniformly. 01:15:05.030 --> 01:15:07.810 align:middle line:84% So this is what Weierstrass M-test gave. 01:15:07.810 --> 01:15:11.174 align:middle line:90% 01:15:11.174 --> 01:15:16.760 align:middle line:84% And now these functions here, they are polynomials. 01:15:16.760 --> 01:15:18.120 align:middle line:90% The En are polynomials. 01:15:18.120 --> 01:15:20.260 align:middle line:90% So it's easy to integrate them. 01:15:20.260 --> 01:15:29.260 align:middle line:84% So you have that the integral here of En dx from, 01:15:29.260 --> 01:15:32.620 align:middle line:84% let's say, some interval from a to b, this thing 01:15:32.620 --> 01:15:38.280 align:middle line:84% here is now equal to the sum here. 01:15:38.280 --> 01:15:41.580 align:middle line:84% So now it's just a finite sum, so you can exchange that. 01:15:41.580 --> 01:15:48.120 align:middle line:84% And then you're integrating from a to b of xk over k factorial. 01:15:48.120 --> 01:15:52.380 align:middle line:90% 01:15:52.380 --> 01:15:56.900 align:middle line:84% And of course, that's very easy to do. 01:15:56.900 --> 01:15:59.620 align:middle line:90% So dx. 01:15:59.620 --> 01:16:04.700 align:middle line:84% And the theorem that we just proved is that this thing 01:16:04.700 --> 01:16:09.780 align:middle line:84% here, because the convergence is uniform, each of these 01:16:09.780 --> 01:16:11.020 align:middle line:90% is integrable. 01:16:11.020 --> 01:16:16.260 align:middle line:84% This means that the limit here is also integrable, 01:16:16.260 --> 01:16:22.900 align:middle line:84% and the integral of the exponential function 01:16:22.900 --> 01:16:25.900 align:middle line:90% is the limit of these integrals. 01:16:25.900 --> 01:16:31.860 align:middle line:84% And so the point here, again, is that if you 01:16:31.860 --> 01:16:35.340 align:middle line:84% have uniform convergence, then you can-- 01:16:35.340 --> 01:16:38.100 align:middle line:84% I mean, you're using the uniform convergence 01:16:38.100 --> 01:16:40.900 align:middle line:84% that we have from Weierstrass M-test. 01:16:40.900 --> 01:16:45.500 align:middle line:84% Then we can interchange this infinite sum 01:16:45.500 --> 01:16:47.920 align:middle line:90% with taking integrals. 01:16:47.920 --> 01:16:48.420 align:middle line:90% OK? 01:16:48.420 --> 01:16:51.420 align:middle line:90% 01:16:51.420 --> 01:16:53.500 align:middle line:90% OK. 01:16:53.500 --> 01:16:55.040 align:middle line:90% Let me just-- 01:16:55.040 --> 01:16:56.880 align:middle line:84% I had wanted to do one more thing, 01:16:56.880 --> 01:17:02.220 align:middle line:84% but I don't have time to do that quite. 01:17:02.220 --> 01:17:04.320 align:middle line:84% So I'll pick up on that next time. 01:17:04.320 --> 01:17:09.240 align:middle line:84% But let me just observe that when you're integrating here, 01:17:09.240 --> 01:17:12.730 align:middle line:84% when you're integrating this thing here, 01:17:12.730 --> 01:17:14.590 align:middle line:90% what is the integral of this? 01:17:14.590 --> 01:17:17.050 align:middle line:84% So when we just calculating this. 01:17:17.050 --> 01:17:22.410 align:middle line:84% So you're integrating x to the k over k factorial. 01:17:22.410 --> 01:17:26.730 align:middle line:84% When you're integrating a polynomial, you're looking for-- 01:17:26.730 --> 01:17:29.570 align:middle line:84% you're using the fundamental theorem of calculus. 01:17:29.570 --> 01:17:32.970 align:middle line:84% So you're trying to look for a function whose derivative is 01:17:32.970 --> 01:17:34.090 align:middle line:90% this. 01:17:34.090 --> 01:17:37.850 align:middle line:84% Such a function is often called the antiderivative. 01:17:37.850 --> 01:17:40.170 align:middle line:90% And so what is that function? 01:17:40.170 --> 01:17:43.610 align:middle line:84% So a function-- so if you're looking 01:17:43.610 --> 01:17:48.698 align:middle line:84% at x to the power k plus 1, you know 01:17:48.698 --> 01:17:50.490 align:middle line:84% that you have to-- because it's polynomial, 01:17:50.490 --> 01:17:55.850 align:middle line:84% you have to add 1 to the exponent. 01:17:55.850 --> 01:18:00.530 align:middle line:84% And then, when you take derivatives, this comes down. 01:18:00.530 --> 01:18:06.270 align:middle line:84% So you have to look at k plus 1 factorial. 01:18:06.270 --> 01:18:10.240 align:middle line:84% Because when you take the derivative, then k plus 1 01:18:10.240 --> 01:18:10.760 align:middle line:90% comes down. 01:18:10.760 --> 01:18:15.600 align:middle line:84% And it kills the first one, and the rest is this. 01:18:15.600 --> 01:18:19.240 align:middle line:84% And so when you evaluate this integral, 01:18:19.240 --> 01:18:22.920 align:middle line:84% then this here is the antiderivative. 01:18:22.920 --> 01:18:30.460 align:middle line:84% And you have to insert the boundary values like this. 01:18:30.460 --> 01:18:31.880 align:middle line:90% So that would be the integral. 01:18:31.880 --> 01:18:34.440 align:middle line:90% 01:18:34.440 --> 01:18:36.960 align:middle line:90% So I'll just stop here. 01:18:36.960 --> 01:18:41.960 align:middle line:84% And next time, I'll go through the one theorem I didn't get to, 01:18:41.960 --> 01:18:45.940 align:middle line:84% which is how to interchange the differentiation. 01:18:45.940 --> 01:18:48.480 align:middle line:90% 01:18:48.480 --> 01:18:51.840 align:middle line:84% Well, if you take a sequence of functions 01:18:51.840 --> 01:18:53.960 align:middle line:84% that converge uniformly and you want 01:18:53.960 --> 01:18:57.840 align:middle line:84% to take the derivative of these, how 01:18:57.840 --> 01:19:00.960 align:middle line:84% do you relate the derivative of the sequence 01:19:00.960 --> 01:19:03.672 align:middle line:90% to the derivative of the limit? 01:19:03.672 --> 01:19:05.000 align:middle line:90% But it's in the notes. 01:19:05.000 --> 01:19:09.320 align:middle line:84% I already posted the notes, but I'll go through that next time. 01:19:09.320 --> 01:19:11.620 align:middle line:90% Any questions? 01:19:11.620 --> 01:19:12.120 align:middle line:90% Yeah? 01:19:12.120 --> 01:19:16.925 align:middle line:84% AUDIENCE: I had a question about the upper-number sums. 01:19:16.925 --> 01:19:17.800 align:middle line:90% TOBIAS COLDING: Yeah. 01:19:17.800 --> 01:19:19.230 align:middle line:90% AUDIENCE: I think there was a-- 01:19:19.230 --> 01:19:21.040 align:middle line:84% [INAUDIBLE] I've seen lot of difference 01:19:21.040 --> 01:19:23.460 align:middle line:84% between epsilon and numbers less than or equal to. 01:19:23.460 --> 01:19:23.880 align:middle line:90% TOBIAS COLDING: Yeah. 01:19:23.880 --> 01:19:24.520 align:middle line:90% AUDIENCE: Is that OK? 01:19:24.520 --> 01:19:25.978 align:middle line:84% TOBIAS COLDING: That's OK because-- 01:19:25.978 --> 01:19:30.760 align:middle line:90% 01:19:30.760 --> 01:19:34.460 align:middle line:84% I'm not 100% sure I needed epsilon less than 01:19:34.460 --> 01:19:35.640 align:middle line:90% or equal to epsilon. 01:19:35.640 --> 01:19:38.400 align:middle line:84% It doesn't really matter because you just pick that. 01:19:38.400 --> 01:19:46.440 align:middle line:84% I had chosen it with the epsilon over 3 and epsilon over 3 times 01:19:46.440 --> 01:19:47.800 align:middle line:90% b minus a. 01:19:47.800 --> 01:19:50.100 align:middle line:84% That was just to have it worked out 01:19:50.100 --> 01:19:51.700 align:middle line:90% so it was epsilon in the end. 01:19:51.700 --> 01:19:54.280 align:middle line:84% But you could, of course, have chosen it even smaller. 01:19:54.280 --> 01:19:56.840 align:middle line:84% Then you would have had strict inequality. 01:19:56.840 --> 01:19:59.880 align:middle line:84% It's possible it was actually strict inequality. 01:19:59.880 --> 01:20:00.800 align:middle line:90% Anyway. 01:20:00.800 --> 01:20:02.050 align:middle line:90% OK 01:20:02.050 --> 01:20:07.000 align:middle line:90%