WEBVTT 00:00:00.000 --> 00:00:01.968 align:middle line:90% [SQUEAKING] 00:00:01.968 --> 00:00:04.428 align:middle line:90% [RUSTLING] 00:00:04.428 --> 00:00:06.888 align:middle line:90% [CLICKING] 00:00:06.888 --> 00:00:10.608 align:middle line:90% 00:00:10.608 --> 00:00:11.400 align:middle line:90% TOBIAS COLDING: OK. 00:00:11.400 --> 00:00:17.200 align:middle line:84% So today, we want to talk a little bit more about sequences 00:00:17.200 --> 00:00:25.080 align:middle line:84% and how sequences connect to a function being continuous. 00:00:25.080 --> 00:00:27.080 align:middle line:84% And then the second thing we will talk about 00:00:27.080 --> 00:00:29.800 align:middle line:90% is metric spaces. 00:00:29.800 --> 00:00:38.640 align:middle line:84% So let me just remind you first that if you take a subset here, 00:00:38.640 --> 00:00:45.400 align:middle line:84% say, of R and you have a function A into R, 00:00:45.400 --> 00:01:09.810 align:middle line:84% then f is said to be continuous if for all x0 in A, 00:01:09.810 --> 00:01:23.130 align:middle line:84% the function is continuous at x0. 00:01:23.130 --> 00:01:49.690 align:middle line:84% And the function is continuous at x0 00:01:49.690 --> 00:01:56.250 align:middle line:84% if for all epsilon greater than 0, 00:01:56.250 --> 00:02:11.930 align:middle line:84% there exists delta greater than 0 such that if the two points-- 00:02:11.930 --> 00:02:19.830 align:middle line:84% so x0 and this other point, x, they deviate at most by delta. 00:02:19.830 --> 00:02:23.840 align:middle line:84% Then their images deviate at most by epsilon. 00:02:23.840 --> 00:02:27.950 align:middle line:90% 00:02:27.950 --> 00:02:34.390 align:middle line:84% And in general, this delta here depends not only on x0 but also 00:02:34.390 --> 00:02:36.450 align:middle line:90% on epsilon. 00:02:36.450 --> 00:02:40.270 align:middle line:84% So delta would in general be expected 00:02:40.270 --> 00:02:42.600 align:middle line:84% to depend on both of these things. 00:02:42.600 --> 00:02:45.210 align:middle line:90% 00:02:45.210 --> 00:02:47.750 align:middle line:84% So we talked about this a number of times. 00:02:47.750 --> 00:02:51.950 align:middle line:84% And we will also generalize this later in the lecture. 00:02:51.950 --> 00:02:54.950 align:middle line:84% So this is what it means to be continuous, that it needs 00:02:54.950 --> 00:02:57.100 align:middle line:90% to be continuous at all points. 00:02:57.100 --> 00:03:03.360 align:middle line:84% And this is what it means to be continuous at a point. 00:03:03.360 --> 00:03:06.400 align:middle line:84% Now, two things we talked about before 00:03:06.400 --> 00:03:20.130 align:middle line:84% is the extreme value theorem and the intermediate value theorem. 00:03:20.130 --> 00:03:29.200 align:middle line:90% 00:03:29.200 --> 00:03:32.600 align:middle line:84% And I want to think about these two theorems 00:03:32.600 --> 00:03:34.140 align:middle line:90% in terms of sequences. 00:03:34.140 --> 00:03:37.520 align:middle line:90% 00:03:37.520 --> 00:03:42.240 align:middle line:84% And that's often a really useful way of thinking about it. 00:03:42.240 --> 00:03:46.140 align:middle line:84% So the extreme value theorem is that we have a function-- 00:03:46.140 --> 00:03:52.160 align:middle line:84% so the extreme value theorem, abbreviated like this, 00:03:52.160 --> 00:03:56.700 align:middle line:84% that's the theorem that says that if you take a function 00:03:56.700 --> 00:04:00.060 align:middle line:90% on some interval from a to b-- 00:04:00.060 --> 00:04:04.980 align:middle line:84% a and b are both included into R. 00:04:04.980 --> 00:04:07.040 align:middle line:90% And this function is continuous. 00:04:07.040 --> 00:04:12.460 align:middle line:90% 00:04:12.460 --> 00:04:14.790 align:middle line:84% Then the claim is that then there exist-- 00:04:14.790 --> 00:04:18.380 align:middle line:90% 00:04:18.380 --> 00:04:28.300 align:middle line:84% so one way of formulating it is that then there exists xM-- 00:04:28.300 --> 00:04:32.400 align:middle line:84% capital M. This is something in that interval. 00:04:32.400 --> 00:04:37.860 align:middle line:84% So there exists some points that we call x capital M-- 00:04:37.860 --> 00:04:43.380 align:middle line:84% and x little m also in the interval. 00:04:43.380 --> 00:04:45.500 align:middle line:84% And the first one has the property 00:04:45.500 --> 00:04:56.150 align:middle line:84% that for all x in the interval, you 00:04:56.150 --> 00:05:04.310 align:middle line:84% have that f of x is less than or equal to the value at xM. 00:05:04.310 --> 00:05:05.830 align:middle line:90% So you have this. 00:05:05.830 --> 00:05:10.030 align:middle line:84% And you also have that f of x, that this here 00:05:10.030 --> 00:05:16.270 align:middle line:84% is bigger or equal to the value at this other point. 00:05:16.270 --> 00:05:21.310 align:middle line:84% So this means in particular that the maximum and the minimum 00:05:21.310 --> 00:05:23.310 align:middle line:90% is achieved. 00:05:23.310 --> 00:05:25.550 align:middle line:84% It also means that the function is actually 00:05:25.550 --> 00:05:27.750 align:middle line:90% bounded on that interval. 00:05:27.750 --> 00:05:29.660 align:middle line:84% So that's an extreme value theorem. 00:05:29.660 --> 00:05:32.630 align:middle line:90% 00:05:32.630 --> 00:05:34.270 align:middle line:84% And we talked about that before, but I 00:05:34.270 --> 00:05:36.690 align:middle line:84% want to think about it in terms of sequences. 00:05:36.690 --> 00:05:38.670 align:middle line:84% And the other thing I want to think 00:05:38.670 --> 00:05:45.990 align:middle line:84% about in terms of sequences is the intermediate value theorem. 00:05:45.990 --> 00:05:53.370 align:middle line:84% So the intermediate value theorem say that all values-- 00:05:53.370 --> 00:05:56.490 align:middle line:84% let me just formulate it loosely. 00:05:56.490 --> 00:06:03.610 align:middle line:90% All values between-- sorry. 00:06:03.610 --> 00:06:05.360 align:middle line:90% f here is as above. 00:06:05.360 --> 00:06:08.850 align:middle line:90% 00:06:08.850 --> 00:06:14.090 align:middle line:84% So you have, again, a function from the integral, 00:06:14.090 --> 00:06:19.430 align:middle line:84% like this, into R. And this is continuous. 00:06:19.430 --> 00:06:23.410 align:middle line:90% 00:06:23.410 --> 00:06:26.850 align:middle line:84% So you have this continuous function. 00:06:26.850 --> 00:06:30.530 align:middle line:84% And then the intermediate value theorem-- so loosely formulated 00:06:30.530 --> 00:06:46.860 align:middle line:84% is, say that all values between f of a and f of b are achieved. 00:06:46.860 --> 00:06:51.460 align:middle line:90% 00:06:51.460 --> 00:06:54.920 align:middle line:84% So this means that if you take a function-- 00:06:54.920 --> 00:06:56.980 align:middle line:90% so you have a function. 00:06:56.980 --> 00:07:01.460 align:middle line:84% This here is the value at the endpoint. 00:07:01.460 --> 00:07:04.420 align:middle line:84% Here, you have the value in the beginning. 00:07:04.420 --> 00:07:09.080 align:middle line:84% Then it say that everything in between these two-- 00:07:09.080 --> 00:07:14.420 align:middle line:84% these here-- is the image of something. 00:07:14.420 --> 00:07:18.820 align:middle line:84% That's what the intermediate value theorem say. 00:07:18.820 --> 00:07:19.320 align:middle line:90% OK. 00:07:19.320 --> 00:07:24.900 align:middle line:84% And I want to prove these two using sequences. 00:07:24.900 --> 00:07:31.540 align:middle line:84% And so in order to do that, we need the following lemma, 00:07:31.540 --> 00:07:34.220 align:middle line:90% which is often-- 00:07:34.220 --> 00:07:38.040 align:middle line:84% so, again, this here is like the definition of being continuous. 00:07:38.040 --> 00:07:43.820 align:middle line:84% But often, when you're proving that something is continuous, 00:07:43.820 --> 00:07:47.220 align:middle line:84% you would instead use this lemma here. 00:07:47.220 --> 00:07:50.700 align:middle line:84% So it's really an if and only if statement, 00:07:50.700 --> 00:07:54.460 align:middle line:84% but I will just prove the if statement. 00:07:54.460 --> 00:07:56.380 align:middle line:90% So if f is continuous-- 00:07:56.380 --> 00:08:03.840 align:middle line:90% 00:08:03.840 --> 00:08:07.480 align:middle line:84% so it's only the if statement I need in this case here. 00:08:07.480 --> 00:08:11.800 align:middle line:84% But in other cases, it's useful to have the if and only 00:08:11.800 --> 00:08:12.840 align:middle line:90% if statement. 00:08:12.840 --> 00:08:14.720 align:middle line:90% So if f is continuous-- 00:08:14.720 --> 00:08:25.660 align:middle line:90% 00:08:25.660 --> 00:08:29.120 align:middle line:84% let me just say where the function is defined. 00:08:29.120 --> 00:08:37.799 align:middle line:84% So in this case, it's just defined on some subset of r. 00:08:37.799 --> 00:08:44.680 align:middle line:84% If this here is continuous and xn 00:08:44.680 --> 00:08:51.690 align:middle line:84% is a sequence in a and xn converge to x and x 00:08:51.690 --> 00:09:01.730 align:middle line:84% is also in a, then the claim is that then the sequence here 00:09:01.730 --> 00:09:07.810 align:middle line:90% converge to this. 00:09:07.810 --> 00:09:13.530 align:middle line:84% And remember that if this here is the case, 00:09:13.530 --> 00:09:19.050 align:middle line:84% then we write this here as x is the limit of xn 00:09:19.050 --> 00:09:21.610 align:middle line:90% as n goes to infinity. 00:09:21.610 --> 00:09:24.210 align:middle line:84% This statement here can be written 00:09:24.210 --> 00:09:33.970 align:middle line:84% as that f of x is the limit as n goes to infinity of this thing 00:09:33.970 --> 00:09:35.930 align:middle line:90% here. 00:09:35.930 --> 00:09:38.730 align:middle line:84% And so if you combine these two things, 00:09:38.730 --> 00:09:42.130 align:middle line:84% if you fill in x is equal to that, 00:09:42.130 --> 00:09:44.610 align:middle line:84% then this here can also be written 00:09:44.610 --> 00:09:52.870 align:middle line:84% as that f of the limit of xn as n goes to infinity-- 00:09:52.870 --> 00:09:55.910 align:middle line:84% this is this by filling in this-- 00:09:55.910 --> 00:09:58.570 align:middle line:90% and then equal to the limit. 00:09:58.570 --> 00:10:04.230 align:middle line:84% So in a way, if you want to think about it 00:10:04.230 --> 00:10:06.710 align:middle line:84% like it's something about-- that if it's continuous, 00:10:06.710 --> 00:10:11.010 align:middle line:84% you can exchange the order of taking limits. 00:10:11.010 --> 00:10:13.530 align:middle line:90% OK. 00:10:13.530 --> 00:10:14.030 align:middle line:90% Yeah. 00:10:14.030 --> 00:10:16.170 align:middle line:84% But that's a more formal way, in a way. 00:10:16.170 --> 00:10:19.930 align:middle line:84% But I was just thinking about it like this. 00:10:19.930 --> 00:10:20.430 align:middle line:90% OK. 00:10:20.430 --> 00:10:23.770 align:middle line:90% So we want to prove this lemma. 00:10:23.770 --> 00:10:26.110 align:middle line:84% And so then we want to use this lemma 00:10:26.110 --> 00:10:27.540 align:middle line:90% to prove these two theorems. 00:10:27.540 --> 00:10:31.670 align:middle line:90% 00:10:31.670 --> 00:10:32.170 align:middle line:90% OK. 00:10:32.170 --> 00:10:35.400 align:middle line:90% So now let's prove this lemma. 00:10:35.400 --> 00:10:45.640 align:middle line:90% 00:10:45.640 --> 00:10:56.440 align:middle line:84% So we want to show that f of xn converge to f of x. 00:10:56.440 --> 00:10:58.780 align:middle line:84% So I'm first going to use that f. 00:10:58.780 --> 00:11:15.600 align:middle line:84% So use first that the f is continuous at this point, x. 00:11:15.600 --> 00:11:19.560 align:middle line:90% So this is some point in a. 00:11:19.560 --> 00:11:29.920 align:middle line:84% So this means that, therefore, given epsilon greater than 0, 00:11:29.920 --> 00:11:40.600 align:middle line:84% there exist delta greater than 0 such that. 00:11:40.600 --> 00:11:48.780 align:middle line:84% If x minus-- so any other element, y, 00:11:48.780 --> 00:11:52.780 align:middle line:84% if this thing here is less than delta, 00:11:52.780 --> 00:12:02.220 align:middle line:84% then their images deviate at most epsilon. 00:12:02.220 --> 00:12:04.580 align:middle line:84% This is just using the definition 00:12:04.580 --> 00:12:09.060 align:middle line:84% that x here is now thought about as being some fixed 00:12:09.060 --> 00:12:10.600 align:middle line:90% point, a fixed point. 00:12:10.600 --> 00:12:13.840 align:middle line:84% And then y is like any other point. 00:12:13.840 --> 00:12:14.340 align:middle line:90% OK. 00:12:14.340 --> 00:12:17.300 align:middle line:90% So we have that. 00:12:17.300 --> 00:12:29.460 align:middle line:84% But then, second, we use that the sequence xn converges to x. 00:12:29.460 --> 00:12:33.300 align:middle line:84% So this means that now thinking about this delta 00:12:33.300 --> 00:12:36.820 align:middle line:84% here that this epsilon is spitting out-- 00:12:36.820 --> 00:12:49.850 align:middle line:84% so this means that for delta, there exist capital N such that 00:12:49.850 --> 00:12:56.230 align:middle line:84% if little n is bigger than the capital N, 00:12:56.230 --> 00:13:03.718 align:middle line:84% then actually, xn minus x is smaller than delta. 00:13:03.718 --> 00:13:07.030 align:middle line:90% 00:13:07.030 --> 00:13:09.430 align:middle line:84% But you see that-- so now combining this-- 00:13:09.430 --> 00:13:17.270 align:middle line:90% 00:13:17.270 --> 00:13:19.510 align:middle line:90% so maybe I'll go over here. 00:13:19.510 --> 00:13:34.910 align:middle line:84% So combining this, then we get that if n here 00:13:34.910 --> 00:13:41.010 align:middle line:84% is bigger than capital N, then xn minus x 00:13:41.010 --> 00:13:43.690 align:middle line:90% is smaller than delta. 00:13:43.690 --> 00:13:48.030 align:middle line:84% But this means that now if they are smaller than delta, 00:13:48.030 --> 00:13:51.410 align:middle line:84% then the first part, say, that their images-- 00:13:51.410 --> 00:13:59.530 align:middle line:84% so this means that f of xn minus f of x is smaller than epsilon. 00:13:59.530 --> 00:14:01.130 align:middle line:84% And that's exactly what we wanted 00:14:01.130 --> 00:14:06.450 align:middle line:84% to prove to show that if you are sufficiently far out, then 00:14:06.450 --> 00:14:09.490 align:middle line:84% these two things are very close together. 00:14:09.490 --> 00:14:13.330 align:middle line:90% So that was quite easy. 00:14:13.330 --> 00:14:15.450 align:middle line:84% Now I want to use these-- but this 00:14:15.450 --> 00:14:21.630 align:middle line:84% is like you use this sort of thing for continuity. 00:14:21.630 --> 00:14:24.370 align:middle line:90% This is a standard thing to use. 00:14:24.370 --> 00:14:26.190 align:middle line:90% It's used implicitly. 00:14:26.190 --> 00:14:29.730 align:middle line:84% And again, this one has a converse. 00:14:29.730 --> 00:14:31.540 align:middle line:84% And the converse is also very useful. 00:14:31.540 --> 00:14:34.170 align:middle line:90% 00:14:34.170 --> 00:14:39.100 align:middle line:84% So now let's come back to the extreme value 00:14:39.100 --> 00:14:41.370 align:middle line:84% theorem and the intermediate value theorem. 00:14:41.370 --> 00:14:47.780 align:middle line:90% 00:14:47.780 --> 00:14:50.140 align:middle line:90% So now let's try to prove-- 00:14:50.140 --> 00:15:09.540 align:middle line:84% so using the lemma, we'll show the extreme value 00:15:09.540 --> 00:15:16.080 align:middle line:84% theorem and the intermediate value theorem. 00:15:16.080 --> 00:15:20.980 align:middle line:84% And so now first, to prove the extreme value theorem-- 00:15:20.980 --> 00:15:26.070 align:middle line:84% so proof of the extreme value theorem. 00:15:26.070 --> 00:15:29.540 align:middle line:90% 00:15:29.540 --> 00:15:33.300 align:middle line:84% I'm first going to prove that the image-- 00:15:33.300 --> 00:15:47.850 align:middle line:84% so first show that the image of this interval here is bounded. 00:15:47.850 --> 00:15:51.520 align:middle line:90% 00:15:51.520 --> 00:15:54.400 align:middle line:84% So this is what I first want to show. 00:15:54.400 --> 00:15:56.360 align:middle line:90% And so I suppose not-- 00:15:56.360 --> 00:15:57.310 align:middle line:90% so suppose not. 00:15:57.310 --> 00:16:02.520 align:middle line:90% 00:16:02.520 --> 00:16:15.160 align:middle line:84% This means that for each n which is an integer, 00:16:15.160 --> 00:16:17.360 align:middle line:84% let's say that it's not bounded from above. 00:16:17.360 --> 00:16:20.700 align:middle line:84% To show that it's not bounded from below is similar. 00:16:20.700 --> 00:16:25.360 align:middle line:84% So I'm just going to prove that it's bounded from above. 00:16:25.360 --> 00:16:31.960 align:middle line:84% But to prove that it is bounded from below 00:16:31.960 --> 00:16:35.810 align:middle line:84% is similar with obvious modifications. 00:16:35.810 --> 00:16:38.970 align:middle line:84% So suppose it's not bounded from above. 00:16:38.970 --> 00:16:43.210 align:middle line:84% Then for each integer, there must 00:16:43.210 --> 00:16:57.890 align:middle line:84% exist x in this interval such that f of x 00:16:57.890 --> 00:16:59.630 align:middle line:90% is bigger or equal to n. 00:16:59.630 --> 00:17:02.250 align:middle line:84% Because otherwise, n would have been an upper bound. 00:17:02.250 --> 00:17:11.170 align:middle line:84% And so I just pick such a n-- so pick. 00:17:11.170 --> 00:17:18.310 align:middle line:84% such n, x-- and let's call it xn. 00:17:18.310 --> 00:17:21.530 align:middle line:90% So for each n, you now have xn. 00:17:21.530 --> 00:17:27.569 align:middle line:84% And it has a property that f of xn is bigger or equal to n. 00:17:27.569 --> 00:17:32.230 align:middle line:90% 00:17:32.230 --> 00:17:36.690 align:middle line:84% Now you have that this gives you a sequence. 00:17:36.690 --> 00:17:40.390 align:middle line:84% So you have that xn here is a sequence. 00:17:40.390 --> 00:17:47.030 align:middle line:84% It's a sequence in-- sorry, it's in this interval here. 00:17:47.030 --> 00:17:50.790 align:middle line:84% But this means that it's a bounded sequence-- 00:17:50.790 --> 00:18:17.670 align:middle line:84% so a bounded sequence so that by the Bolzano-Weierstrass theorem, 00:18:17.670 --> 00:18:30.390 align:middle line:84% it has a convergent subsequence xnk. 00:18:30.390 --> 00:18:33.120 align:middle line:90% 00:18:33.120 --> 00:18:35.460 align:middle line:90% So you have that xnk-- 00:18:35.460 --> 00:18:41.040 align:middle line:84% so this is this subsequence, and it converges to some x. 00:18:41.040 --> 00:18:44.340 align:middle line:90% 00:18:44.340 --> 00:18:48.640 align:middle line:84% But now this means that by the lemma over there, 00:18:48.640 --> 00:18:52.680 align:middle line:84% you have that this sequence then, 00:18:52.680 --> 00:18:56.760 align:middle line:84% because the function was continuous, converge to this. 00:18:56.760 --> 00:18:59.880 align:middle line:84% But on the other hand, this here, 00:18:59.880 --> 00:19:09.800 align:middle line:90% xnk, is a subsequence of xn. 00:19:09.800 --> 00:19:12.300 align:middle line:84% But this means that this number here-- 00:19:12.300 --> 00:19:16.820 align:middle line:84% remember that a subsequence just means that this here-- 00:19:16.820 --> 00:19:18.760 align:middle line:84% it means in particular that this here 00:19:18.760 --> 00:19:21.240 align:middle line:90% would be further out than n. 00:19:21.240 --> 00:19:26.280 align:middle line:84% So you have that f of xnk is supposed 00:19:26.280 --> 00:19:28.640 align:middle line:90% to be bigger or equal to nk. 00:19:28.640 --> 00:19:32.260 align:middle line:84% But this is supposed to be bigger than n. 00:19:32.260 --> 00:19:37.780 align:middle line:84% So these guys here, these here goes to-- 00:19:37.780 --> 00:19:42.180 align:middle line:84% you know these here are unbounded. 00:19:42.180 --> 00:19:44.860 align:middle line:84% But on the other hand, we're saying that this sequence 00:19:44.860 --> 00:19:46.660 align:middle line:90% converges to this sequence. 00:19:46.660 --> 00:19:49.700 align:middle line:84% So any convergent sequence is bounded. 00:19:49.700 --> 00:19:51.140 align:middle line:90% So that's a contradiction. 00:19:51.140 --> 00:19:56.386 align:middle line:90% 00:19:56.386 --> 00:19:56.886 align:middle line:90% OK. 00:19:56.886 --> 00:20:01.780 align:middle line:84% So this is like proving first using the lemma, 00:20:01.780 --> 00:20:08.340 align:middle line:84% we have now that the image here is bounded from above. 00:20:08.340 --> 00:20:12.820 align:middle line:84% The almost exact same proof with obvious modification 00:20:12.820 --> 00:20:15.340 align:middle line:84% give that it's bounded from below. 00:20:15.340 --> 00:20:18.140 align:middle line:84% So now you have that the image of this 00:20:18.140 --> 00:20:20.690 align:middle line:84% is actually bounded from both above and below. 00:20:20.690 --> 00:20:28.990 align:middle line:90% 00:20:28.990 --> 00:20:33.530 align:middle line:84% So now this means that if you're looking at the image here-- 00:20:33.530 --> 00:20:39.310 align:middle line:90% 00:20:39.310 --> 00:20:41.170 align:middle line:90% this here is a bounded subset. 00:20:41.170 --> 00:20:43.200 align:middle line:90% So this is a bounded subset. 00:20:43.200 --> 00:20:48.670 align:middle line:90% 00:20:48.670 --> 00:20:52.560 align:middle line:84% And so you can look at the sup of the set. 00:20:52.560 --> 00:21:00.230 align:middle line:90% 00:21:00.230 --> 00:21:04.470 align:middle line:84% You can look at the sup of this set. 00:21:04.470 --> 00:21:06.630 align:middle line:90% And I'm going to take the sup-- 00:21:06.630 --> 00:21:11.630 align:middle line:84% I'm going to call this capital M. 00:21:11.630 --> 00:21:20.040 align:middle line:84% And now for each little n-- the n is an integer. 00:21:20.040 --> 00:21:22.910 align:middle line:90% 00:21:22.910 --> 00:21:29.520 align:middle line:90% Then I can look at M minus 1/n. 00:21:29.520 --> 00:21:33.040 align:middle line:84% So this is obviously strictly smaller than capital M. 00:21:33.040 --> 00:21:35.780 align:middle line:90% So it's not an upper bound. 00:21:35.780 --> 00:21:39.800 align:middle line:84% So this means that for little n, this here is not 00:21:39.800 --> 00:21:42.240 align:middle line:90% an upper bound for this set. 00:21:42.240 --> 00:21:45.040 align:middle line:84% So this means that there must exist 00:21:45.040 --> 00:21:57.612 align:middle line:84% an x so that you have this for some x in this interval here. 00:21:57.612 --> 00:22:01.000 align:middle line:84% Because this here is not an upper bound, 00:22:01.000 --> 00:22:05.080 align:middle line:84% which means that there must be something larger than that. 00:22:05.080 --> 00:22:07.800 align:middle line:90% 00:22:07.800 --> 00:22:12.280 align:middle line:84% So for each little n, you just pick such x. 00:22:12.280 --> 00:22:15.280 align:middle line:90% And we call this xn. 00:22:15.280 --> 00:22:18.880 align:middle line:90% So xn is in the interval. 00:22:18.880 --> 00:22:26.020 align:middle line:84% And we have that m minus 1 over little n is strictly less than f 00:22:26.020 --> 00:22:27.660 align:middle line:90% of xn. 00:22:27.660 --> 00:22:31.900 align:middle line:84% Remember that this thing here is, of course obviously less 00:22:31.900 --> 00:22:34.040 align:middle line:90% than the sup over all images. 00:22:34.040 --> 00:22:35.200 align:middle line:90% So you even have that. 00:22:35.200 --> 00:22:38.020 align:middle line:90% 00:22:38.020 --> 00:22:40.220 align:middle line:90% OK. 00:22:40.220 --> 00:22:43.330 align:middle line:84% But now you see that now you have a sequence here. 00:22:43.330 --> 00:22:46.860 align:middle line:90% 00:22:46.860 --> 00:22:50.540 align:middle line:90% So now you have a sequence. 00:22:50.540 --> 00:22:55.260 align:middle line:90% And so you have a sequence, xn. 00:22:55.260 --> 00:23:14.340 align:middle line:84% And so by the Bolzano-Weierstrass theorem, 00:23:14.340 --> 00:23:19.040 align:middle line:84% this sequence here has a convergent subsequence. 00:23:19.040 --> 00:23:25.070 align:middle line:90% 00:23:25.070 --> 00:23:27.630 align:middle line:84% This is because this is sequence, 00:23:27.630 --> 00:23:30.730 align:middle line:84% the xn lie in this bounded interval. 00:23:30.730 --> 00:23:32.810 align:middle line:90% So it's a bounded sequence. 00:23:32.810 --> 00:23:35.350 align:middle line:84% So it has a convergent subsequence. 00:23:35.350 --> 00:23:41.110 align:middle line:84% And this convergent subsequence, I call xn k, like that. 00:23:41.110 --> 00:23:46.270 align:middle line:84% So this is converging to something. 00:23:46.270 --> 00:23:51.070 align:middle line:84% And that something, I'm going to call xM. 00:23:51.070 --> 00:23:55.160 align:middle line:84% And I then will prove that this has the property that I wanted. 00:23:55.160 --> 00:23:57.758 align:middle line:90% 00:23:57.758 --> 00:23:59.470 align:middle line:90% So I have that. 00:23:59.470 --> 00:24:03.510 align:middle line:84% And so because f was continuous, then I 00:24:03.510 --> 00:24:10.170 align:middle line:84% have that the images converge to the image of this. 00:24:10.170 --> 00:24:21.010 align:middle line:90% This is by continuity of f-- 00:24:21.010 --> 00:24:24.370 align:middle line:84% and by the continuity of f and the little lemma 00:24:24.370 --> 00:24:27.450 align:middle line:90% we proved before. 00:24:27.450 --> 00:24:29.370 align:middle line:90% So you have that. 00:24:29.370 --> 00:24:35.490 align:middle line:84% But now you see that this here tells us-- 00:24:35.490 --> 00:24:37.250 align:middle line:90% we have this inequality here. 00:24:37.250 --> 00:24:43.210 align:middle line:84% But this is just saying that as n goes to infinity, 00:24:43.210 --> 00:24:49.810 align:middle line:84% this sequence here converge to M. So we also have 00:24:49.810 --> 00:24:54.930 align:middle line:84% that this sequence here converges to m. 00:24:54.930 --> 00:24:57.370 align:middle line:84% This is just because it's squeezed. 00:24:57.370 --> 00:24:59.670 align:middle line:84% It squeezed here in this interval. 00:24:59.670 --> 00:25:03.560 align:middle line:84% And this here, this lower bound is going to the upper bound. 00:25:03.560 --> 00:25:06.595 align:middle line:90% 00:25:06.595 --> 00:25:10.530 align:middle line:84% So if you have that this here is converging, 00:25:10.530 --> 00:25:14.410 align:middle line:84% then any subsequence would also converge to the same thing. 00:25:14.410 --> 00:25:19.450 align:middle line:84% So this implies that for this here, 00:25:19.450 --> 00:25:22.060 align:middle line:90% that is a subsequence of these. 00:25:22.060 --> 00:25:25.900 align:middle line:84% But this is then converging to the same thing. 00:25:25.900 --> 00:25:30.420 align:middle line:84% So now you see that we have that this is converging to that, 00:25:30.420 --> 00:25:32.130 align:middle line:84% but it's also converging to this. 00:25:32.130 --> 00:25:34.660 align:middle line:90% 00:25:34.660 --> 00:25:45.660 align:middle line:84% We have, by uniqueness of limits, then 00:25:45.660 --> 00:25:54.036 align:middle line:84% we have that f of xM is equal to this. 00:25:54.036 --> 00:25:57.340 align:middle line:84% And capital M, again, was the sup. 00:25:57.340 --> 00:25:59.760 align:middle line:84% So this means that the sup is actually achieved. 00:25:59.760 --> 00:26:02.800 align:middle line:84% You found a point where the sup was achieved. 00:26:02.800 --> 00:26:03.300 align:middle line:90% OK. 00:26:03.300 --> 00:26:06.420 align:middle line:84% So that proves the extreme value theorem. 00:26:06.420 --> 00:26:09.360 align:middle line:84% By one of the two parts of the extreme value theorem, 00:26:09.360 --> 00:26:11.340 align:middle line:84% we proved that the sup was achieved. 00:26:11.340 --> 00:26:13.420 align:middle line:84% And the proof that the inverse achieved 00:26:13.420 --> 00:26:18.840 align:middle line:84% is almost identical, with some obvious changes. 00:26:18.840 --> 00:26:22.920 align:middle line:84% Now for the intermediate value theorem-- 00:26:22.920 --> 00:26:25.280 align:middle line:90% so let's try to prove that. 00:26:25.280 --> 00:26:27.630 align:middle line:84% So how do we prove the intermediate value theorem? 00:26:27.630 --> 00:26:36.800 align:middle line:90% 00:26:36.800 --> 00:26:41.450 align:middle line:84% And so proof of intermediate value theorem-- 00:26:41.450 --> 00:26:47.160 align:middle line:90% 00:26:47.160 --> 00:26:52.480 align:middle line:84% and I'm going to prove, but it's not really-- 00:26:52.480 --> 00:26:56.380 align:middle line:84% the general case follows easily from this. 00:26:56.380 --> 00:26:58.260 align:middle line:84% So I'm going to make an assumption. 00:26:58.260 --> 00:27:06.560 align:middle line:90% 00:27:06.560 --> 00:27:11.440 align:middle line:84% I'm going to assume that the value at a is strictly 00:27:11.440 --> 00:27:16.260 align:middle line:84% less than 0 and the value of b is strictly bigger than 0. 00:27:16.260 --> 00:27:21.530 align:middle line:84% And then I want to show that 0 is achieved 00:27:21.530 --> 00:27:24.830 align:middle line:90% as the image of something. 00:27:24.830 --> 00:27:27.630 align:middle line:84% You can say that sounds like a special case. 00:27:27.630 --> 00:27:29.410 align:middle line:90% It is a special case. 00:27:29.410 --> 00:27:32.370 align:middle line:84% But once you can prove this, it's 00:27:32.370 --> 00:27:35.830 align:middle line:90% easy to get the general case. 00:27:35.830 --> 00:27:39.330 align:middle line:84% You can either get it directly from this 00:27:39.330 --> 00:27:41.970 align:middle line:90% by modifying the function. 00:27:41.970 --> 00:27:46.030 align:middle line:84% Or once you see how the proof of this goes, 00:27:46.030 --> 00:27:49.010 align:middle line:84% then it's easy to prove it, to set up 00:27:49.010 --> 00:27:51.790 align:middle line:84% what are the different possible cases 00:27:51.790 --> 00:27:58.970 align:middle line:84% and then prove using the same sort of argument as we do here. 00:27:58.970 --> 00:28:01.330 align:middle line:90% OK. 00:28:01.330 --> 00:28:07.522 align:middle line:84% So this is what I'm going to show want to show, 00:28:07.522 --> 00:28:21.950 align:middle line:84% that there exist x between a and b such that f of x 00:28:21.950 --> 00:28:24.670 align:middle line:90% is equal to 0. 00:28:24.670 --> 00:28:26.070 align:middle line:90% So this is what we want to show. 00:28:26.070 --> 00:28:30.550 align:middle line:90% 00:28:30.550 --> 00:28:33.030 align:middle line:84% And again, of course, the function f 00:28:33.030 --> 00:28:35.410 align:middle line:90% is obviously continuous. 00:28:35.410 --> 00:28:49.850 align:middle line:90% 00:28:49.850 --> 00:28:50.350 align:middle line:90% OK. 00:28:50.350 --> 00:28:56.550 align:middle line:90% So now let me define into a set. 00:28:56.550 --> 00:28:58.510 align:middle line:90% It's actually an interval. 00:28:58.510 --> 00:29:09.590 align:middle line:84% So I define E here to be the sets of x between-- 00:29:09.590 --> 00:29:12.750 align:middle line:90% 00:29:12.750 --> 00:29:13.940 align:middle line:90% let's call this here. 00:29:13.940 --> 00:29:18.020 align:middle line:84% So I want to show that there exists some x. 00:29:18.020 --> 00:29:22.200 align:middle line:84% Let's call it x0, just to have something to refer to. 00:29:22.200 --> 00:29:27.440 align:middle line:84% I call it x0 because the value's going to be 0. 00:29:27.440 --> 00:29:37.880 align:middle line:84% So I define a set, E. And this is the x in the interval. 00:29:37.880 --> 00:29:44.640 align:middle line:84% But I'm now requiring that f of y 00:29:44.640 --> 00:29:56.980 align:middle line:84% is less than or equal to 0 for all y which is between a and x. 00:29:56.980 --> 00:30:01.325 align:middle line:90% 00:30:01.325 --> 00:30:04.880 align:middle line:90% Now, E here is nonempty. 00:30:04.880 --> 00:30:06.440 align:middle line:90% E is nonempty. 00:30:06.440 --> 00:30:09.360 align:middle line:90% E here is not the empty set. 00:30:09.360 --> 00:30:11.040 align:middle line:90% Why? 00:30:11.040 --> 00:30:14.500 align:middle line:90% Because actually, a here-- 00:30:14.500 --> 00:30:23.180 align:middle line:84% this is since a here is in E. Because if you take a, 00:30:23.180 --> 00:30:26.180 align:middle line:84% then the value is actually strictly less than 0. 00:30:26.180 --> 00:30:29.280 align:middle line:90% 00:30:29.280 --> 00:30:33.740 align:middle line:84% And everything that lies in here but lies 00:30:33.740 --> 00:30:38.200 align:middle line:84% to the left of a-- well, that's only a right. 00:30:38.200 --> 00:30:40.220 align:middle line:84% So this is a kind of trivial statement. 00:30:40.220 --> 00:30:44.260 align:middle line:90% So this is not 0. 00:30:44.260 --> 00:30:46.770 align:middle line:84% So now you can take the sup of this. 00:30:46.770 --> 00:30:49.500 align:middle line:90% 00:30:49.500 --> 00:30:52.940 align:middle line:84% So maybe I should make one more remark-- 00:30:52.940 --> 00:30:59.220 align:middle line:84% that, of course, not only is it not the empty set, 00:30:59.220 --> 00:31:01.960 align:middle line:90% but E here is bounded. 00:31:01.960 --> 00:31:08.940 align:middle line:84% So the sup of E is less than or equal to b. 00:31:08.940 --> 00:31:13.210 align:middle line:84% Because all of the elements in E are in this set. 00:31:13.210 --> 00:31:17.510 align:middle line:84% So they're trivially bounded by b. 00:31:17.510 --> 00:31:19.870 align:middle line:90% So I now can look at the sup. 00:31:19.870 --> 00:31:24.070 align:middle line:84% So the sup here is a nice number. 00:31:24.070 --> 00:31:26.000 align:middle line:90% And I can look at the sup. 00:31:26.000 --> 00:31:31.250 align:middle line:90% 00:31:31.250 --> 00:31:32.610 align:middle line:90% So the claim here is-- 00:31:32.610 --> 00:31:41.990 align:middle line:90% 00:31:41.990 --> 00:31:45.950 align:middle line:84% or maybe let me just first say set 00:31:45.950 --> 00:31:51.310 align:middle line:84% x not equal to that sup of E. And then 00:31:51.310 --> 00:31:58.550 align:middle line:84% the claim is that f of x0 is actually 0. 00:31:58.550 --> 00:32:08.970 align:middle line:90% 00:32:08.970 --> 00:32:18.870 align:middle line:84% And so now you see that if you take-- 00:32:18.870 --> 00:32:22.550 align:middle line:90% 00:32:22.550 --> 00:32:23.050 align:middle line:90% right. 00:32:23.050 --> 00:32:26.240 align:middle line:90% So now look at the sup. 00:32:26.240 --> 00:32:29.770 align:middle line:90% 00:32:29.770 --> 00:32:49.570 align:middle line:84% So for each n, say, integer, if you're looking at x0 minus 1/n, 00:32:49.570 --> 00:32:52.850 align:middle line:90% this is not the sup of e. 00:32:52.850 --> 00:32:55.270 align:middle line:84% So this means that there exists an element. 00:32:55.270 --> 00:32:59.830 align:middle line:90% 00:32:59.830 --> 00:33:05.930 align:middle line:84% For each n, there exists an element, which are called xn. 00:33:05.930 --> 00:33:09.500 align:middle line:90% And this xn is larger than this. 00:33:09.500 --> 00:33:27.860 align:middle line:84% And the xn is in E. But you see that for it 00:33:27.860 --> 00:33:33.220 align:middle line:84% to be an E, then, in fact, this element, 00:33:33.220 --> 00:33:36.500 align:middle line:84% the value has to be less than or equal to 0. 00:33:36.500 --> 00:33:44.980 align:middle line:84% So f of xn has to be less than or equal to 0 since it's 00:33:44.980 --> 00:33:49.300 align:middle line:84% in E. So since it's in E, then we have that. 00:33:49.300 --> 00:33:55.300 align:middle line:90% 00:33:55.300 --> 00:34:02.480 align:middle line:84% But now, of course, the xn's, the xn's, they converge. 00:34:02.480 --> 00:34:05.240 align:middle line:84% It's bigger than this, but they are less 00:34:05.240 --> 00:34:09.440 align:middle line:90% than the sup of all such guys. 00:34:09.440 --> 00:34:12.100 align:middle line:90% So this means that the xn-- 00:34:12.100 --> 00:34:18.120 align:middle line:90% so the xn converges to x0. 00:34:18.120 --> 00:34:24.210 align:middle line:84% And so this means that f of xn converges to f of x0. 00:34:24.210 --> 00:34:28.932 align:middle line:90% 00:34:28.932 --> 00:34:33.199 align:middle line:84% Now this means that this here is-- 00:34:33.199 --> 00:34:35.739 align:middle line:84% and these here are all non-negative numbers. 00:34:35.739 --> 00:34:38.746 align:middle line:84% This means that this number here must be non-negative. 00:34:38.746 --> 00:34:42.920 align:middle line:90% 00:34:42.920 --> 00:34:47.320 align:middle line:84% If x is 0, then that's where we want it. 00:34:47.320 --> 00:35:00.360 align:middle line:84% So we're done if f of x0 is actually 0. 00:35:00.360 --> 00:35:09.000 align:middle line:84% So assume that f of x0 is then strictly less than 0. 00:35:09.000 --> 00:35:13.050 align:middle line:90% 00:35:13.050 --> 00:35:19.730 align:middle line:84% But if the value here is strictly less than 0, 00:35:19.730 --> 00:35:23.190 align:middle line:84% if this is the case, then there exists-- 00:35:23.190 --> 00:35:25.590 align:middle line:90% then if you take-- 00:35:25.590 --> 00:35:32.690 align:middle line:90% so here, you have 0. 00:35:32.690 --> 00:35:35.250 align:middle line:90% Here, you have f of x0. 00:35:35.250 --> 00:35:38.090 align:middle line:90% 00:35:38.090 --> 00:35:40.870 align:middle line:84% You can then take a little interval around it. 00:35:40.870 --> 00:35:45.590 align:middle line:90% 00:35:45.590 --> 00:36:04.110 align:middle line:84% So take epsilon bigger than 0 such that this thing here-- 00:36:04.110 --> 00:36:10.530 align:middle line:84% such that f of x0 plus epsilon, this is still negative. 00:36:10.530 --> 00:36:18.950 align:middle line:90% 00:36:18.950 --> 00:36:24.550 align:middle line:84% So I'm going to look at-- so here, I have f of x0. 00:36:24.550 --> 00:36:28.030 align:middle line:90% This is actually negative. 00:36:28.030 --> 00:36:30.670 align:middle line:84% And I'm taking an interval around it. 00:36:30.670 --> 00:36:32.490 align:middle line:90% So this is like an interval. 00:36:32.490 --> 00:36:36.270 align:middle line:84% Here, we have f of x0 plus epsilon. 00:36:36.270 --> 00:36:40.830 align:middle line:84% And here, we have f of x0 minus epsilon. 00:36:40.830 --> 00:36:44.950 align:middle line:84% So I'm looking at this little interval here around f of x0. 00:36:44.950 --> 00:36:47.510 align:middle line:90% 00:36:47.510 --> 00:36:52.590 align:middle line:90% And here, I have x0. 00:36:52.590 --> 00:36:56.410 align:middle line:84% But now because the function is continuous, 00:36:56.410 --> 00:36:58.890 align:middle line:84% I know that there exists an interval. 00:36:58.890 --> 00:37:01.120 align:middle line:90% So there exists some delta. 00:37:01.120 --> 00:37:08.920 align:middle line:84% There exists delta bigger than 0 so 00:37:08.920 --> 00:37:11.380 align:middle line:84% that if I'm in this little interval-- 00:37:11.380 --> 00:37:17.800 align:middle line:84% this is the interval from x0 minus delta, x0 plus delta. 00:37:17.800 --> 00:37:20.360 align:middle line:84% If I'm looking at this little interval, 00:37:20.360 --> 00:37:26.720 align:middle line:84% then anything here is taken into this thing here. 00:37:26.720 --> 00:37:32.840 align:middle line:84% So in other words, I have that if x minus x0 00:37:32.840 --> 00:37:41.240 align:middle line:84% is less than delta, then f of dx minus f of x0 00:37:41.240 --> 00:37:44.640 align:middle line:90% is less than epsilon. 00:37:44.640 --> 00:37:47.960 align:middle line:84% But everything that is in this interval here 00:37:47.960 --> 00:37:49.660 align:middle line:90% is still strictly negative. 00:37:49.660 --> 00:37:51.340 align:middle line:90% So this means that in fact-- 00:37:51.340 --> 00:37:56.220 align:middle line:90% 00:37:56.220 --> 00:38:00.100 align:middle line:84% in particular, this implies such that f of x 00:38:00.100 --> 00:38:01.400 align:middle line:90% is actually negative. 00:38:01.400 --> 00:38:07.240 align:middle line:90% 00:38:07.240 --> 00:38:11.490 align:middle line:84% So now I want to get a contradiction because I know-- 00:38:11.490 --> 00:38:14.060 align:middle line:90% 00:38:14.060 --> 00:38:15.880 align:middle line:90% so now I have this picture here. 00:38:15.880 --> 00:38:23.540 align:middle line:90% 00:38:23.540 --> 00:38:26.220 align:middle line:90% I have my x0 here. 00:38:26.220 --> 00:38:37.160 align:middle line:84% The x0 was supposed to be the sup of the set A. 00:38:37.160 --> 00:38:42.900 align:middle line:84% So I know that, actually, on a little interval around this x0, 00:38:42.900 --> 00:38:45.660 align:middle line:90% everything is-- 00:38:45.660 --> 00:38:49.900 align:middle line:84% if I take anything in a little interval around this x0, 00:38:49.900 --> 00:38:54.180 align:middle line:84% then the values of f are actually negative. 00:38:54.180 --> 00:38:56.860 align:middle line:90% 00:38:56.860 --> 00:39:04.070 align:middle line:84% But you see that this x0 was limits or points where 00:39:04.070 --> 00:39:08.550 align:middle line:84% everything to the left was not positive. 00:39:08.550 --> 00:39:13.310 align:middle line:84% So it means that everything to the left was not positive. 00:39:13.310 --> 00:39:20.150 align:middle line:84% But then even a little bit further was still not positive. 00:39:20.150 --> 00:39:22.390 align:middle line:90% But then it wasn't like the sup. 00:39:22.390 --> 00:39:26.350 align:middle line:84% Because anything here would be larger, 00:39:26.350 --> 00:39:28.310 align:middle line:84% and it still would have the property 00:39:28.310 --> 00:39:34.130 align:middle line:84% that everything to the left under f was non-negative. 00:39:34.130 --> 00:39:39.386 align:middle line:90% 00:39:39.386 --> 00:39:43.030 align:middle line:84% So that's the proof of the extreme value theorem 00:39:43.030 --> 00:39:44.870 align:middle line:84% and the intermediate value theorem 00:39:44.870 --> 00:39:50.790 align:middle line:84% using sequences and using that little lemma from before. 00:39:50.790 --> 00:39:51.930 align:middle line:90% OK. 00:39:51.930 --> 00:39:54.150 align:middle line:84% So now the next thing I want to talk about 00:39:54.150 --> 00:39:57.550 align:middle line:84% is another really important concept. 00:39:57.550 --> 00:40:00.970 align:middle line:84% And that, to some extent, is a generalization 00:40:00.970 --> 00:40:04.560 align:middle line:84% of things we know from the real numbers. 00:40:04.560 --> 00:40:08.130 align:middle line:90% 00:40:08.130 --> 00:40:10.760 align:middle line:84% And so this is this concept of a metric space. 00:40:10.760 --> 00:40:17.010 align:middle line:90% 00:40:17.010 --> 00:40:19.430 align:middle line:84% And so let me just give you a definition, 00:40:19.430 --> 00:40:23.050 align:middle line:84% and then I'll give you a bunch of examples. 00:40:23.050 --> 00:40:29.450 align:middle line:84% So a metric space is a set, X. So this is the set. 00:40:29.450 --> 00:40:36.250 align:middle line:84% And then it's a map for each pair of points. 00:40:36.250 --> 00:40:39.250 align:middle line:84% So let me explain what I mean by this. 00:40:39.250 --> 00:40:51.330 align:middle line:84% So whenever you have x in X and another element in this space, 00:40:51.330 --> 00:40:54.980 align:middle line:84% then when you have any such pair, 00:40:54.980 --> 00:41:00.960 align:middle line:84% then d x comma y, this gives you a real number. 00:41:00.960 --> 00:41:04.420 align:middle line:90% 00:41:04.420 --> 00:41:07.700 align:middle line:84% And so this here is what you usually call 00:41:07.700 --> 00:41:19.020 align:middle line:90% the distance between x and y. 00:41:19.020 --> 00:41:21.580 align:middle line:84% That's the distance between x and y. 00:41:21.580 --> 00:41:24.340 align:middle line:84% Now, what properties should this have? 00:41:24.340 --> 00:41:32.380 align:middle line:84% It's not just any map from this space across itself into R. 00:41:32.380 --> 00:41:34.740 align:middle line:90% It has three properties. 00:41:34.740 --> 00:41:43.700 align:middle line:84% The first is that the distance between pair of points-- 00:41:43.700 --> 00:41:46.380 align:middle line:84% and then there are generalization of this. 00:41:46.380 --> 00:41:47.920 align:middle line:84% But anyway, so I'll talk about it. 00:41:47.920 --> 00:41:48.920 align:middle line:90% That's generalization. 00:41:48.920 --> 00:41:52.460 align:middle line:84% Like, if you're working with general relativity, 00:41:52.460 --> 00:41:55.730 align:middle line:84% then you would actually not quite work with this definition. 00:41:55.730 --> 00:41:58.990 align:middle line:84% You would work with some generalization of this. 00:41:58.990 --> 00:42:02.770 align:middle line:84% But so what properties should this distance have? 00:42:02.770 --> 00:42:05.430 align:middle line:90% It should have three properties. 00:42:05.430 --> 00:42:10.430 align:middle line:84% The first is that for any x and y, 00:42:10.430 --> 00:42:12.910 align:middle line:84% the distance between any two points 00:42:12.910 --> 00:42:16.310 align:middle line:84% should always give you a non-negative number. 00:42:16.310 --> 00:42:18.870 align:middle line:84% So this here should be non-negative. 00:42:18.870 --> 00:42:25.230 align:middle line:84% And if the distance between two points is 0, 00:42:25.230 --> 00:42:31.750 align:middle line:84% then those two points should be the same. 00:42:31.750 --> 00:42:36.030 align:middle line:84% So in fact, the distance here is equal to 0 00:42:36.030 --> 00:42:39.150 align:middle line:90% if and only if x is equal to y. 00:42:39.150 --> 00:42:41.950 align:middle line:90% 00:42:41.950 --> 00:42:45.550 align:middle line:90% So that's the first property. 00:42:45.550 --> 00:42:49.230 align:middle line:90% And the second property is-- 00:42:49.230 --> 00:42:52.010 align:middle line:84% just their distances are non-negative, if you will. 00:42:52.010 --> 00:42:55.210 align:middle line:84% The second property is that if you're looking at, 00:42:55.210 --> 00:42:57.770 align:middle line:84% what is the distance from x to y? 00:42:57.770 --> 00:43:00.370 align:middle line:84% The distance from x to y should be the same 00:43:00.370 --> 00:43:02.350 align:middle line:90% as the distance from y to x. 00:43:02.350 --> 00:43:06.210 align:middle line:90% 00:43:06.210 --> 00:43:07.190 align:middle line:90% So this is this. 00:43:07.190 --> 00:43:10.690 align:middle line:84% And this is sometimes referred to that it's symmetric. 00:43:10.690 --> 00:43:12.730 align:middle line:84% It doesn't matter whether you first-- 00:43:12.730 --> 00:43:18.690 align:middle line:84% you begin at x and go to y or you begin at y and go to x. 00:43:18.690 --> 00:43:21.010 align:middle line:84% And then the third thing, which is 00:43:21.010 --> 00:43:24.030 align:middle line:84% kind of like the nontrivial part of it 00:43:24.030 --> 00:43:26.330 align:middle line:84% and the thing that makes it really useful, 00:43:26.330 --> 00:43:29.330 align:middle line:84% which is called the triangle inequality-- 00:43:29.330 --> 00:43:34.650 align:middle line:84% that if you have three points, x, y, and z, in this space 00:43:34.650 --> 00:43:42.530 align:middle line:84% and you're looking at the distance from x to z-- 00:43:42.530 --> 00:43:46.010 align:middle line:84% so if you imagine that you have three points-- 00:43:46.010 --> 00:43:47.930 align:middle line:90% x, y. 00:43:47.930 --> 00:43:50.500 align:middle line:84% And then you have another point, z. 00:43:50.500 --> 00:43:51.860 align:middle line:90% Sorry, no. 00:43:51.860 --> 00:43:53.100 align:middle line:90% That's not how I [INAUDIBLE]. 00:43:53.100 --> 00:43:57.700 align:middle line:84% So you have three points, x, y, and z. 00:43:57.700 --> 00:44:00.620 align:middle line:84% But you think about that you're looking at the distance 00:44:00.620 --> 00:44:03.180 align:middle line:90% from x to z. 00:44:03.180 --> 00:44:06.680 align:middle line:84% Well, instead of going directly from x to z, 00:44:06.680 --> 00:44:10.180 align:middle line:84% if you went over y, then this distance 00:44:10.180 --> 00:44:13.380 align:middle line:84% here should be longer than if you went directly. 00:44:13.380 --> 00:44:15.240 align:middle line:84% So this thing here should be less than 00:44:15.240 --> 00:44:21.420 align:middle line:84% or equal to the distance from x to y, plus the distance from y 00:44:21.420 --> 00:44:22.740 align:middle line:90% to z. 00:44:22.740 --> 00:44:27.540 align:middle line:84% And this here is the nontrivial part of it. 00:44:27.540 --> 00:44:29.880 align:middle line:84% And this is called the triangle inequality. 00:44:29.880 --> 00:44:40.060 align:middle line:90% 00:44:40.060 --> 00:44:42.513 align:middle line:84% Now, what are some examples of this? 00:44:42.513 --> 00:44:43.930 align:middle line:84% Let's look at a bunch of examples. 00:44:43.930 --> 00:44:55.880 align:middle line:90% 00:44:55.880 --> 00:45:03.640 align:middle line:84% The first example is just where x is R. 00:45:03.640 --> 00:45:07.960 align:middle line:84% And so you have two points in x in R. 00:45:07.960 --> 00:45:15.360 align:middle line:84% And now the distance between them is simply x minus 1. 00:45:15.360 --> 00:45:17.880 align:middle line:90% This is obviously non-negative. 00:45:17.880 --> 00:45:21.480 align:middle line:84% If it's 0, that means that x is equal to y. 00:45:21.480 --> 00:45:24.160 align:middle line:90% It's obviously also symmetric. 00:45:24.160 --> 00:45:26.600 align:middle line:84% And the triangle inequality is this inequality 00:45:26.600 --> 00:45:28.040 align:middle line:90% that we're using all the time. 00:45:28.040 --> 00:45:32.280 align:middle line:90% 00:45:32.280 --> 00:45:39.520 align:middle line:84% Another example is if you're looking at where x here 00:45:39.520 --> 00:45:41.800 align:middle line:90% is R squared, say. 00:45:41.800 --> 00:45:46.410 align:middle line:84% And then you're thinking about x and y. 00:45:46.410 --> 00:45:47.885 align:middle line:84% You may think about a mesh vector 00:45:47.885 --> 00:45:49.510 align:middle line:84% or whatever you want to think about it. 00:45:49.510 --> 00:45:55.410 align:middle line:84% So this means that x here has coordinates x1 and x2. 00:45:55.410 --> 00:45:59.580 align:middle line:90% And y has coordinates y1 and y2. 00:45:59.580 --> 00:46:03.130 align:middle line:90% 00:46:03.130 --> 00:46:09.360 align:middle line:84% And now the distance between these two points 00:46:09.360 --> 00:46:17.930 align:middle line:84% is just the square root of x1 minus y1 squared 00:46:17.930 --> 00:46:25.330 align:middle line:90% plus x2 minus y2 squared. 00:46:25.330 --> 00:46:26.990 align:middle line:90% That's the distance. 00:46:26.990 --> 00:46:31.040 align:middle line:84% And this here is often referred to as the Euclidean distance. 00:46:31.040 --> 00:46:33.850 align:middle line:90% 00:46:33.850 --> 00:46:37.770 align:middle line:90% So this distance that you know. 00:46:37.770 --> 00:46:41.370 align:middle line:84% The distance is given by Pythagoras' theorem. 00:46:41.370 --> 00:46:46.430 align:middle line:84% And so it's clear the first two property, 00:46:46.430 --> 00:46:48.110 align:middle line:90% those are always clear. 00:46:48.110 --> 00:46:50.390 align:middle line:84% And the triangle inequality, that's 00:46:50.390 --> 00:46:53.710 align:middle line:84% the one that sometimes is tricky to prove. 00:46:53.710 --> 00:46:54.210 align:middle line:90% OK. 00:46:54.210 --> 00:46:57.310 align:middle line:84% So that's another example of a distance. 00:46:57.310 --> 00:47:00.590 align:middle line:84% I want to give you a few more examples. 00:47:00.590 --> 00:47:06.430 align:middle line:84% And so another example is like where you do this in R3. 00:47:06.430 --> 00:47:10.730 align:middle line:84% And so you have, instead of points having two coordinates, 00:47:10.730 --> 00:47:12.630 align:middle line:90% they now have three coordinates. 00:47:12.630 --> 00:47:14.590 align:middle line:84% And you do the sort of same of thing. 00:47:14.590 --> 00:47:18.590 align:middle line:84% But now there would be a third term under the square root sign, 00:47:18.590 --> 00:47:21.630 align:middle line:84% would be the square of the difference 00:47:21.630 --> 00:47:23.410 align:middle line:90% between the third coordinates. 00:47:23.410 --> 00:47:26.216 align:middle line:90% 00:47:26.216 --> 00:47:27.870 align:middle line:90% OK. 00:47:27.870 --> 00:47:32.900 align:middle line:84% Then let me give you a few more examples. 00:47:32.900 --> 00:47:36.390 align:middle line:90% 00:47:36.390 --> 00:47:41.150 align:middle line:84% There's one that is, to some extent, simpler than this one 00:47:41.150 --> 00:47:43.240 align:middle line:90% here. 00:47:43.240 --> 00:47:45.820 align:middle line:90% And it's sometimes useful. 00:47:45.820 --> 00:47:55.080 align:middle line:84% But it's not as useful as the Euclidean distances. 00:47:55.080 --> 00:47:58.820 align:middle line:84% And again, these sort of things are generalized. 00:47:58.820 --> 00:48:01.640 align:middle line:84% They're an example of something much more general. 00:48:01.640 --> 00:48:04.520 align:middle line:84% Like, this here is what's called a Hilbert space, 00:48:04.520 --> 00:48:06.160 align:middle line:90% example of a Hilbert space. 00:48:06.160 --> 00:48:09.560 align:middle line:84% The next is something which is an example of something 00:48:09.560 --> 00:48:10.900 align:middle line:90% called a Banach space. 00:48:10.900 --> 00:48:12.640 align:middle line:84% But so the next example I will look at 00:48:12.640 --> 00:48:16.050 align:middle line:84% is what is sometimes called the box distance. 00:48:16.050 --> 00:48:20.640 align:middle line:90% 00:48:20.640 --> 00:48:27.420 align:middle line:84% And so if space in this case is R2-- 00:48:27.420 --> 00:48:30.320 align:middle line:84% you could do it in R3 also if you want. 00:48:30.320 --> 00:48:36.700 align:middle line:84% And you take two points here, x and y in R2. 00:48:36.700 --> 00:48:43.700 align:middle line:84% And so this here, underline x, has coordinates x1 and x2. 00:48:43.700 --> 00:48:49.180 align:middle line:84% And now the distance between these two points 00:48:49.180 --> 00:48:54.880 align:middle line:84% would be-- you didn't do this fancy thing with square root, 00:48:54.880 --> 00:48:59.340 align:middle line:84% but you did the more trivial thing, where you're just adding. 00:48:59.340 --> 00:49:03.340 align:middle line:84% So you're taking two points here in R2. 00:49:03.340 --> 00:49:06.060 align:middle line:90% This is maybe x1 comma x2. 00:49:06.060 --> 00:49:11.740 align:middle line:90% This is y1 So y2. 00:49:11.740 --> 00:49:15.260 align:middle line:84% The Euclidean distance is simply this distance here. 00:49:15.260 --> 00:49:20.160 align:middle line:84% The box distance is where you take this distance here 00:49:20.160 --> 00:49:22.860 align:middle line:84% and you're adding this distance here. 00:49:22.860 --> 00:49:29.340 align:middle line:84% So this is this here plus x2 minus y2. 00:49:29.340 --> 00:49:31.500 align:middle line:90% That's a box distance. 00:49:31.500 --> 00:49:33.640 align:middle line:84% And obviously, this is a non-negative. 00:49:33.640 --> 00:49:36.380 align:middle line:84% It gives you-- for any pair, it's non-negative. 00:49:36.380 --> 00:49:38.200 align:middle line:90% It's obviously symmetric. 00:49:38.200 --> 00:49:41.750 align:middle line:84% It doesn't matter the ordering of these two. 00:49:41.750 --> 00:49:44.230 align:middle line:84% Its just switching the order between absolute value sign 00:49:44.230 --> 00:49:46.390 align:middle line:90% anyway, so it doesn't matter. 00:49:46.390 --> 00:49:49.950 align:middle line:84% And it's also clear that it's 0 if and only 00:49:49.950 --> 00:49:54.590 align:middle line:84% if these two points-- it means that the coordinates are 00:49:54.590 --> 00:49:57.163 align:middle line:90% the same. 00:49:57.163 --> 00:50:00.670 align:middle line:84% And then, in this case, the triangle inequality 00:50:00.670 --> 00:50:03.170 align:middle line:90% is not very hard to establish. 00:50:03.170 --> 00:50:05.630 align:middle line:84% Proving the triangle even just for the Euclidean 00:50:05.630 --> 00:50:07.050 align:middle line:90% is a nontrivial thing. 00:50:07.050 --> 00:50:10.550 align:middle line:90% 00:50:10.550 --> 00:50:12.710 align:middle line:90% So that's another example. 00:50:12.710 --> 00:50:14.990 align:middle line:84% And I want to give you a few more examples. 00:50:14.990 --> 00:50:19.790 align:middle line:84% Here's a sort of interesting example. 00:50:19.790 --> 00:50:24.950 align:middle line:90% 00:50:24.950 --> 00:50:28.750 align:middle line:90% Suppose you take a interval-- 00:50:28.750 --> 00:50:32.030 align:middle line:84% let's just make the interval from 0 to 1. 00:50:32.030 --> 00:50:36.110 align:middle line:84% So suppose you're looking at the interval from 0 to 1. 00:50:36.110 --> 00:50:42.170 align:middle line:84% And now looking at-- so this here, this is now your space, x. 00:50:42.170 --> 00:50:52.970 align:middle line:84% And so by that, I mean that x here consists of-- 00:50:52.970 --> 00:50:57.110 align:middle line:90% so an element in x-- 00:50:57.110 --> 00:50:58.290 align:middle line:90% let me write. 00:50:58.290 --> 00:50:58.920 align:middle line:90% An element. 00:50:58.920 --> 00:51:01.770 align:middle line:90% 00:51:01.770 --> 00:51:17.130 align:middle line:84% In x is a continuous function on the interval from 0 to 1. 00:51:17.130 --> 00:51:20.690 align:middle line:84% So this means that if you take two elements-- 00:51:20.690 --> 00:51:25.170 align:middle line:84% so if you take one continuous function on the interval 00:51:25.170 --> 00:51:29.170 align:middle line:84% from 0 to 1 and another continuous function, 00:51:29.170 --> 00:51:31.290 align:middle line:84% what should be the distance between two 00:51:31.290 --> 00:51:32.910 align:middle line:90% continuous functions? 00:51:32.910 --> 00:51:35.650 align:middle line:90% 00:51:35.650 --> 00:51:40.300 align:middle line:84% Well, the distance here-- so this here is defined to be 00:51:40.300 --> 00:51:43.540 align:middle line:90% the maximum of-- 00:51:43.540 --> 00:51:50.260 align:middle line:84% so if you're looking f minus g, If f is continuous, 00:51:50.260 --> 00:51:52.580 align:middle line:84% g is continuous, this is continuous. 00:51:52.580 --> 00:51:56.460 align:middle line:90% 00:51:56.460 --> 00:51:58.820 align:middle line:84% And now this is a continuous function 00:51:58.820 --> 00:52:01.540 align:middle line:90% defined on this interval. 00:52:01.540 --> 00:52:08.060 align:middle line:84% So the extreme value theorem say that the maximum in-- so 00:52:08.060 --> 00:52:09.860 align:middle line:90% this here is continuous. 00:52:09.860 --> 00:52:13.420 align:middle line:84% And in fact, even this here is continuous. 00:52:13.420 --> 00:52:18.660 align:middle line:84% So the maximum of this here is achieved. 00:52:18.660 --> 00:52:24.480 align:middle line:84% So this is the maximum, where x is in 0 to 1. 00:52:24.480 --> 00:52:26.840 align:middle line:90% So this is achieved. 00:52:26.840 --> 00:52:30.180 align:middle line:84% And this is clearly, again, symmetric. 00:52:30.180 --> 00:52:32.060 align:middle line:84% Because of the absolute value sign, 00:52:32.060 --> 00:52:37.000 align:middle line:84% it's clearly symmetric, in terms of f and g. 00:52:37.000 --> 00:52:39.920 align:middle line:84% It's also clear that it's non-negative. 00:52:39.920 --> 00:52:43.040 align:middle line:84% And it's also clear that it's 0 if 00:52:43.040 --> 00:52:46.680 align:middle line:84% and only if the two functions are the same. 00:52:46.680 --> 00:52:48.720 align:middle line:84% And in this case, it's quite easy 00:52:48.720 --> 00:52:56.760 align:middle line:90% to prove that this is a metric. 00:52:56.760 --> 00:53:01.500 align:middle line:84% OK, so maybe I'll just leave that. 00:53:01.500 --> 00:53:02.000 align:middle line:90% OK. 00:53:02.000 --> 00:53:06.860 align:middle line:84% Now I want to give you a couple of more examples. 00:53:06.860 --> 00:53:09.545 align:middle line:84% And then I'll start with example and talk about-- 00:53:09.545 --> 00:53:11.420 align:middle line:84% I want to give you a couple of more examples. 00:53:11.420 --> 00:53:16.020 align:middle line:84% And the two next examples, one is-- 00:53:16.020 --> 00:53:20.587 align:middle line:90% 00:53:20.587 --> 00:53:22.920 align:middle line:84% and the reason why I want to give you two other examples 00:53:22.920 --> 00:53:24.440 align:middle line:90% is that-- 00:53:24.440 --> 00:53:28.120 align:middle line:90% 00:53:28.120 --> 00:53:31.600 align:middle line:84% so you know a lot of things about distances already. 00:53:31.600 --> 00:53:34.130 align:middle line:84% And this gives you excellent intuition 00:53:34.130 --> 00:53:36.850 align:middle line:90% to proving many things. 00:53:36.850 --> 00:53:40.210 align:middle line:84% But you should just keep in mind that a metric space 00:53:40.210 --> 00:53:42.650 align:middle line:90% could be very wild. 00:53:42.650 --> 00:53:46.690 align:middle line:84% So I'll give you an example of some strange and directly 00:53:46.690 --> 00:53:48.930 align:middle line:90% wild metric spaces. 00:53:48.930 --> 00:53:51.490 align:middle line:84% So the first one is kind of strange. 00:53:51.490 --> 00:53:57.130 align:middle line:84% And then I'll give you another example, which is wild. 00:53:57.130 --> 00:54:00.100 align:middle line:84% So the strange one, so the strange example-- 00:54:00.100 --> 00:54:03.170 align:middle line:90% 00:54:03.170 --> 00:54:08.230 align:middle line:84% and so this is like if you're looking at the integers. 00:54:08.230 --> 00:54:09.920 align:middle line:84% So the space is now the integers. 00:54:09.920 --> 00:54:13.330 align:middle line:90% 00:54:13.330 --> 00:54:16.450 align:middle line:84% And so you want to look at, what is the distance 00:54:16.450 --> 00:54:20.423 align:middle line:90% between two integers? 00:54:20.423 --> 00:54:22.770 align:middle line:84% Now, of course, your intuition would 00:54:22.770 --> 00:54:26.890 align:middle line:84% say that if one integer is really large 00:54:26.890 --> 00:54:28.510 align:middle line:84% and the other one is not very large, 00:54:28.510 --> 00:54:31.770 align:middle line:84% you would think that the distance was large. 00:54:31.770 --> 00:54:34.070 align:middle line:90% But it doesn't have to be large. 00:54:34.070 --> 00:54:40.510 align:middle line:84% So I define this distance to be 1/n minus 1/m. 00:54:40.510 --> 00:54:43.150 align:middle line:90% 00:54:43.150 --> 00:54:47.630 align:middle line:84% This here gives you so the absolute value of the difference 00:54:47.630 --> 00:54:50.870 align:middle line:90% of 1 over each of them. 00:54:50.870 --> 00:54:53.830 align:middle line:90% This here gives you a distance. 00:54:53.830 --> 00:55:00.590 align:middle line:84% And in particular, you see that if both n and m are 00:55:00.590 --> 00:55:04.790 align:middle line:84% really large, then in this metric, 00:55:04.790 --> 00:55:06.390 align:middle line:90% they're very close together. 00:55:06.390 --> 00:55:10.310 align:middle line:84% It looks like the distance is very small. 00:55:10.310 --> 00:55:16.110 align:middle line:84% Now, instead of actually proving that this thing here is-- 00:55:16.110 --> 00:55:18.150 align:middle line:90% so you could try to prove-- 00:55:18.150 --> 00:55:22.110 align:middle line:84% so as usual, to prove that this is 00:55:22.110 --> 00:55:24.930 align:middle line:84% a metric, the first and second property, 00:55:24.930 --> 00:55:28.030 align:middle line:84% the property that is non-negative, that's obvious. 00:55:28.030 --> 00:55:31.360 align:middle line:84% It's also obvious that it's 0 if and only 00:55:31.360 --> 00:55:34.120 align:middle line:90% if n and m are the same. 00:55:34.120 --> 00:55:37.120 align:middle line:84% It's also obvious that it's symmetric. 00:55:37.120 --> 00:55:39.700 align:middle line:84% The triangle inequality might be not quite as obvious. 00:55:39.700 --> 00:55:41.600 align:middle line:90% It's not very hard in this case. 00:55:41.600 --> 00:55:46.080 align:middle line:84% But it actually is a triviality because you could think 00:55:46.080 --> 00:55:50.240 align:middle line:90% about this one here as a map. 00:55:50.240 --> 00:55:58.400 align:middle line:84% Suppose you take a map from the integers into the real numbers. 00:55:58.400 --> 00:56:00.840 align:middle line:84% And the map is just that you take-- 00:56:00.840 --> 00:56:06.320 align:middle line:84% the image of an integer is just 1 over it. 00:56:06.320 --> 00:56:11.600 align:middle line:84% If you think about the usual distance over here-- 00:56:11.600 --> 00:56:13.380 align:middle line:90% and so the usual distance. 00:56:13.380 --> 00:56:17.560 align:middle line:84% By that, I mean that the distance between x and y 00:56:17.560 --> 00:56:19.840 align:middle line:90% is just this. 00:56:19.840 --> 00:56:24.840 align:middle line:84% If you think about this, but now thinking about that things 00:56:24.840 --> 00:56:27.360 align:middle line:90% here-- 00:56:27.360 --> 00:56:30.820 align:middle line:84% and so you think about the usual distance. 00:56:30.820 --> 00:56:36.900 align:middle line:84% But you're now restricted to not just all real numbers, 00:56:36.900 --> 00:56:40.540 align:middle line:84% but just things that are 1 over integers. 00:56:40.540 --> 00:56:45.740 align:middle line:84% On that space, it gives you a metric. 00:56:45.740 --> 00:56:48.100 align:middle line:84% But then you can just think about 00:56:48.100 --> 00:56:51.540 align:middle line:84% that this gives you another way of labeling things. 00:56:51.540 --> 00:56:55.360 align:middle line:84% And so it's kind of a triviality that it is a metric space. 00:56:55.360 --> 00:56:59.060 align:middle line:84% But distances looks a little counterintuitive. 00:56:59.060 --> 00:57:04.100 align:middle line:84% Because if you have two very large numbers, then actually, 00:57:04.100 --> 00:57:06.520 align:middle line:84% the distance between them is small, 00:57:06.520 --> 00:57:10.860 align:middle line:84% even though those two large numbers could be very far apart. 00:57:10.860 --> 00:57:14.820 align:middle line:84% So the last example I want to give you, that's 00:57:14.820 --> 00:57:16.590 align:middle line:90% a example of a wild metric. 00:57:16.590 --> 00:57:23.860 align:middle line:90% 00:57:23.860 --> 00:57:26.880 align:middle line:90% This is one wild. 00:57:26.880 --> 00:57:31.450 align:middle line:84% And so, essentially, all of these occur naturally. 00:57:31.450 --> 00:57:36.550 align:middle line:90% 00:57:36.550 --> 00:57:39.190 align:middle line:84% And so this example is what's called the French railway 00:57:39.190 --> 00:57:45.630 align:middle line:84% metric, which is maybe not a super flattering name 00:57:45.630 --> 00:57:46.850 align:middle line:90% but probably accurate. 00:57:46.850 --> 00:57:50.650 align:middle line:84% So the thing is that if you travel in France-- 00:57:50.650 --> 00:57:54.550 align:middle line:84% so if you imagine that this is France here. 00:57:54.550 --> 00:57:57.030 align:middle line:90% Here, you have Paris. 00:57:57.030 --> 00:58:01.970 align:middle line:84% And if you want to go to the French Riviera, 00:58:01.970 --> 00:58:04.030 align:middle line:90% which is maybe down here-- 00:58:04.030 --> 00:58:07.030 align:middle line:84% so two cities on the French Riviera 00:58:07.030 --> 00:58:13.590 align:middle line:84% that are incredibly close together is Cannes and Nice. 00:58:13.590 --> 00:58:14.610 align:middle line:90% This is Cannes. 00:58:14.610 --> 00:58:16.150 align:middle line:90% And here, you have Nice. 00:58:16.150 --> 00:58:18.950 align:middle line:84% They're very, very close together. 00:58:18.950 --> 00:58:22.630 align:middle line:84% But if you were going to go with a French railroad, 00:58:22.630 --> 00:58:26.150 align:middle line:84% then you would have to go to Paris first. 00:58:26.150 --> 00:58:29.330 align:middle line:84% So this here would take you maybe 20 minutes by train 00:58:29.330 --> 00:58:30.930 align:middle line:90% if there was a train. 00:58:30.930 --> 00:58:32.950 align:middle line:84% But you have to go to Paris first. 00:58:32.950 --> 00:58:35.510 align:middle line:90% And that's really far. 00:58:35.510 --> 00:58:38.330 align:middle line:84% The good thing is that the French have this TGV, which 00:58:38.330 --> 00:58:41.730 align:middle line:84% is this incredible fast train, but it still 00:58:41.730 --> 00:58:44.730 align:middle line:90% would take many hours. 00:58:44.730 --> 00:58:48.450 align:middle line:84% And so that's the idea of the French railway network, 00:58:48.450 --> 00:58:53.810 align:middle line:90% that the space here is R2. 00:58:53.810 --> 00:58:57.890 align:middle line:84% And if you take two points, x and y, 00:58:57.890 --> 00:59:08.210 align:middle line:84% in R2, then what is the distance between this pair? 00:59:08.210 --> 00:59:10.700 align:middle line:84% So the distance between the pair is-- 00:59:10.700 --> 00:59:19.290 align:middle line:90% 00:59:19.290 --> 00:59:25.196 align:middle line:84% the distance between x and y is-- 00:59:25.196 --> 00:59:33.860 align:middle line:90% 00:59:33.860 --> 00:59:41.380 align:middle line:84% if x and y lie on the same-- so imagine that this here is Paris. 00:59:41.380 --> 00:59:45.300 align:middle line:84% And so just think about Paris as being the origin. 00:59:45.300 --> 00:59:49.100 align:middle line:84% So if x and y lie on the same line-- 00:59:49.100 --> 00:59:52.660 align:middle line:84% so lying on the same line just means that x here 00:59:52.660 --> 00:59:54.980 align:middle line:90% is equal to some constant. 00:59:54.980 --> 01:00:01.340 align:middle line:84% If this here is y, then x is like a constant times y. 01:00:01.340 --> 01:00:06.300 align:middle line:84% Or if one of them was 0, then you could have that y 01:00:06.300 --> 01:00:09.580 align:middle line:90% is equal to a constant times x. 01:00:09.580 --> 01:00:16.680 align:middle line:84% So this just means that these x and y lie on the same line. 01:00:16.680 --> 01:00:18.840 align:middle line:84% So if they lie on the same line-- 01:00:18.840 --> 01:00:25.920 align:middle line:84% so if x here is equal to c times y or y 01:00:25.920 --> 01:00:37.880 align:middle line:84% is equal to c times x for some real number, if that's the case, 01:00:37.880 --> 01:00:42.580 align:middle line:84% then it's just that the distance between them, 01:00:42.580 --> 01:00:50.160 align:middle line:84% it's just like this usual distance here between them. 01:00:50.160 --> 01:00:52.660 align:middle line:84% But if they don't lie on the same line-- 01:00:52.660 --> 01:00:58.000 align:middle line:84% they may lie on two lines that are very, very close together-- 01:00:58.000 --> 01:01:06.920 align:middle line:84% then the distance here would be the sum of these two otherwise. 01:01:06.920 --> 01:01:10.320 align:middle line:90% 01:01:10.320 --> 01:01:10.820 align:middle line:90% OK. 01:01:10.820 --> 01:01:14.120 align:middle line:84% So that's the French railway metric 01:01:14.120 --> 01:01:18.440 align:middle line:84% if I didn't have some sort of strange properties, somewhat 01:01:18.440 --> 01:01:22.050 align:middle line:90% counterintuitive properties. 01:01:22.050 --> 01:01:25.250 align:middle line:84% The next thing I want to talk about 01:01:25.250 --> 01:01:30.070 align:middle line:84% is convergence in a metric space. 01:01:30.070 --> 01:01:33.730 align:middle line:84% So a lot of things we have done generalizes 01:01:33.730 --> 01:01:37.770 align:middle line:90% to general metric spaces. 01:01:37.770 --> 01:01:41.960 align:middle line:84% So first, let's talk about, what is a sequence in a metric space? 01:01:41.960 --> 01:01:46.810 align:middle line:90% 01:01:46.810 --> 01:01:50.570 align:middle line:84% So now you have a metric space here. 01:01:50.570 --> 01:01:51.960 align:middle line:90% So this is a metric space. 01:01:51.960 --> 01:01:56.810 align:middle line:90% 01:01:56.810 --> 01:02:00.850 align:middle line:90% And a sequence xn is a sequence. 01:02:00.850 --> 01:02:03.890 align:middle line:84% So maybe I should say it like before, 01:02:03.890 --> 01:02:11.690 align:middle line:84% that if you take a map from the integers into x, 01:02:11.690 --> 01:02:13.020 align:middle line:90% this is a sequence. 01:02:13.020 --> 01:02:18.210 align:middle line:90% 01:02:18.210 --> 01:02:23.420 align:middle line:84% And as before, the image of an integer, you 01:02:23.420 --> 01:02:26.020 align:middle line:84% would usually denote it like this. 01:02:26.020 --> 01:02:31.340 align:middle line:84% So this is the same definition that we use for the real. 01:02:31.340 --> 01:02:32.360 align:middle line:90% So this is a sequence. 01:02:32.360 --> 01:02:39.460 align:middle line:90% 01:02:39.460 --> 01:02:42.220 align:middle line:84% A subsequence of a given sequence-- 01:02:42.220 --> 01:02:48.700 align:middle line:84% well, then you have another map, g, from the integers 01:02:48.700 --> 01:02:50.940 align:middle line:84% into the integers and is supposed 01:02:50.940 --> 01:02:52.480 align:middle line:90% to be strictly increasing. 01:02:52.480 --> 01:02:59.740 align:middle line:90% 01:02:59.740 --> 01:03:07.540 align:middle line:84% And then you're looking f of g of n. 01:03:07.540 --> 01:03:08.800 align:middle line:90% And you think about this. 01:03:08.800 --> 01:03:13.140 align:middle line:84% You will typically write it like this, f of g of k, say. 01:03:13.140 --> 01:03:16.620 align:middle line:84% And you would typically write this as nk right. 01:03:16.620 --> 01:03:18.172 align:middle line:90% This here is a subsequences. 01:03:18.172 --> 01:03:21.840 align:middle line:90% 01:03:21.840 --> 01:03:25.360 align:middle line:84% It's just exactly like the way we define it 01:03:25.360 --> 01:03:26.980 align:middle line:90% for the real numbers. 01:03:26.980 --> 01:03:29.600 align:middle line:90% 01:03:29.600 --> 01:03:39.300 align:middle line:84% Now, that a sequence is converging in a metric space, 01:03:39.300 --> 01:03:41.400 align:middle line:90% that makes also sense. 01:03:41.400 --> 01:03:48.960 align:middle line:84% So suppose that you now have a general metric space, 01:03:48.960 --> 01:03:57.380 align:middle line:90% and xn is a sequence in X. 01:03:57.380 --> 01:04:07.240 align:middle line:84% Then we say that xn converges to x. 01:04:07.240 --> 01:04:17.410 align:middle line:84% So x here is another element in X 01:04:17.410 --> 01:04:25.170 align:middle line:84% if for all epsilon greater than 0, 01:04:25.170 --> 01:04:36.730 align:middle line:84% there exist a capital N so that if little n is bigger 01:04:36.730 --> 01:04:46.970 align:middle line:84% or equal to capital N, then the distance between xn and x 01:04:46.970 --> 01:04:48.450 align:middle line:90% is smaller than epsilon. 01:04:48.450 --> 01:04:50.110 align:middle line:84% This is what it means to converge. 01:04:50.110 --> 01:04:56.330 align:middle line:90% 01:04:56.330 --> 01:05:02.890 align:middle line:84% Now, similarly, the concept of being a Cauchy sequence, 01:05:02.890 --> 01:05:06.370 align:middle line:84% this also extends to metric spaces. 01:05:06.370 --> 01:05:24.138 align:middle line:84% So a Cauchy sequence in a metric space is the following. 01:05:24.138 --> 01:05:35.910 align:middle line:90% 01:05:35.910 --> 01:05:38.820 align:middle line:84% So a Cauchy sequence-- let's call it xn. 01:05:38.820 --> 01:05:41.670 align:middle line:90% 01:05:41.670 --> 01:05:49.350 align:middle line:84% It's the following, that for all epsilon greater than 0, 01:05:49.350 --> 01:06:05.550 align:middle line:84% there exist capital N such that if little m and little n 01:06:05.550 --> 01:06:15.240 align:middle line:84% are bigger than this capital N, then the distance between them 01:06:15.240 --> 01:06:16.800 align:middle line:90% is smaller than epsilon. 01:06:16.800 --> 01:06:18.930 align:middle line:84% This is what it means to be a Cauchy sequence. 01:06:18.930 --> 01:06:21.480 align:middle line:90% 01:06:21.480 --> 01:06:26.080 align:middle line:84% Now we have a theorem that is like the same theorem. 01:06:26.080 --> 01:06:27.660 align:middle line:84% The proof is like the same proof, 01:06:27.660 --> 01:06:29.020 align:middle line:90% but we'll just go over it. 01:06:29.020 --> 01:06:31.600 align:middle line:90% 01:06:31.600 --> 01:06:33.900 align:middle line:90% So we have a theorem. 01:06:33.900 --> 01:06:37.412 align:middle line:90% Let's say that if you have a-- 01:06:37.412 --> 01:06:39.060 align:middle line:90% so theorem is the following. 01:06:39.060 --> 01:06:44.280 align:middle line:90% 01:06:44.280 --> 01:06:48.370 align:middle line:84% So, again, X comma d is a metric space. 01:06:48.370 --> 01:06:57.880 align:middle line:90% 01:06:57.880 --> 01:07:15.620 align:middle line:84% And if xn is a convergent sequence in X, 01:07:15.620 --> 01:07:25.530 align:middle line:90% then xn also a Cauchy sequence. 01:07:25.530 --> 01:07:32.970 align:middle line:90% 01:07:32.970 --> 01:07:35.580 align:middle line:90% Now, how do we prove this? 01:07:35.580 --> 01:07:39.500 align:middle line:84% So prove it really is kind of the same as we did before. 01:07:39.500 --> 01:07:43.760 align:middle line:84% So prove-- so we have that xn converges. 01:07:43.760 --> 01:07:47.620 align:middle line:90% Let's say that the limit is x. 01:07:47.620 --> 01:07:53.780 align:middle line:84% So this means that given epsilon greater than 0, 01:07:53.780 --> 01:08:05.660 align:middle line:84% there exist capital N such that if little n is 01:08:05.660 --> 01:08:11.030 align:middle line:84% bigger or equal to capital N, then the distance here-- 01:08:11.030 --> 01:08:16.069 align:middle line:84% and I'm using now, in the definition of convergence, 01:08:16.069 --> 01:08:19.310 align:middle line:90% not epsilon but epsilon over 2. 01:08:19.310 --> 01:08:21.630 align:middle line:90% So I have that. 01:08:21.630 --> 01:08:31.870 align:middle line:84% But this means that therefore, if little m and little 01:08:31.870 --> 01:08:35.189 align:middle line:84% n, if they are both bigger than capital N 01:08:35.189 --> 01:08:39.590 align:middle line:84% and you're now looking at the distance from xm 01:08:39.590 --> 01:08:46.189 align:middle line:84% to xn, well, by the triangle inequality, this thing here, 01:08:46.189 --> 01:08:47.229 align:middle line:90% it's worse. 01:08:47.229 --> 01:08:50.960 align:middle line:84% Or it's certainly not better if you go over this other point. 01:08:50.960 --> 01:08:56.590 align:middle line:90% 01:08:56.590 --> 01:08:59.510 align:middle line:84% But you see, this one here and this one, each of them 01:08:59.510 --> 01:09:01.330 align:middle line:90% is smaller than epsilon over 2. 01:09:01.330 --> 01:09:08.490 align:middle line:90% 01:09:08.490 --> 01:09:11.649 align:middle line:84% So the difference here is bounded by epsilon, 01:09:11.649 --> 01:09:13.569 align:middle line:90% as it should be. 01:09:13.569 --> 01:09:17.050 align:middle line:84% So this proves that if you take a convergent sequence, then 01:09:17.050 --> 01:09:18.590 align:middle line:90% it is a Cauchy sequence. 01:09:18.590 --> 01:09:22.569 align:middle line:84% Remember that Cauchy convergence theorem was saying that 01:09:22.569 --> 01:09:28.569 align:middle line:84% in R, if you have a Cauchy sequence, 01:09:28.569 --> 01:09:30.450 align:middle line:90% then it was also convergent. 01:09:30.450 --> 01:09:32.170 align:middle line:90% But that actually fails. 01:09:32.170 --> 01:09:33.970 align:middle line:90% This is not true. 01:09:33.970 --> 01:09:37.290 align:middle line:84% And it's some important property of a metric space 01:09:37.290 --> 01:09:42.290 align:middle line:84% if it has the Cauchy convergence property. 01:09:42.290 --> 01:09:46.210 align:middle line:84% So let me give you a very simple example. 01:09:46.210 --> 01:09:51.370 align:middle line:84% So suppose that your metric space-- 01:09:51.370 --> 01:09:58.650 align:middle line:84% so the space here, let's just take the interval from 0 to 1. 01:09:58.650 --> 01:10:03.290 align:middle line:84% And what is the distance between two elements? 01:10:03.290 --> 01:10:09.380 align:middle line:84% So x and y are in this integral from 0 to 1. 01:10:09.380 --> 01:10:11.060 align:middle line:90% What is the distance? 01:10:11.060 --> 01:10:12.570 align:middle line:90% I just take the usual distance. 01:10:12.570 --> 01:10:15.220 align:middle line:90% 01:10:15.220 --> 01:10:23.540 align:middle line:84% But now if I'm looking at this sequence here, xn, which is 1/n, 01:10:23.540 --> 01:10:25.710 align:middle line:84% this thing here is a Cauchy sequence. 01:10:25.710 --> 01:10:32.940 align:middle line:90% 01:10:32.940 --> 01:10:34.310 align:middle line:90% It is a Cauchy sequence. 01:10:34.310 --> 01:10:37.460 align:middle line:90% 01:10:37.460 --> 01:10:43.020 align:middle line:84% Because when n and n are really large, then both of them 01:10:43.020 --> 01:10:46.540 align:middle line:90% is close to this endpoint. 01:10:46.540 --> 01:10:49.940 align:middle line:84% But they don't converge because-- 01:10:49.940 --> 01:10:53.780 align:middle line:84% in the larger space, R, they converge to 0. 01:10:53.780 --> 01:10:55.900 align:middle line:90% But 0 is not part of this space. 01:10:55.900 --> 01:10:58.110 align:middle line:84% So they don't converge to anything in this space. 01:10:58.110 --> 01:11:00.900 align:middle line:90% 01:11:00.900 --> 01:11:02.660 align:middle line:90% So this is an example-- 01:11:02.660 --> 01:11:21.640 align:middle line:84% so it's a Cauchy sequence, but xn does not converge in x 0. 01:11:21.640 --> 01:11:25.720 align:middle line:84% This is because 0 is not part of this interval. 01:11:25.720 --> 01:11:27.638 align:middle line:84% And 0 would have to be the limit-- yeah? 01:11:27.638 --> 01:11:29.680 align:middle line:84% STUDENT: Is that the only type of counterexample? 01:11:29.680 --> 01:11:30.760 align:middle line:90% TOBIAS COLDING: No, no. 01:11:30.760 --> 01:11:34.200 align:middle line:84% But this does provide an important example 01:11:34.200 --> 01:11:35.560 align:middle line:90% of counterexamples. 01:11:35.560 --> 01:11:37.620 align:middle line:84% Because then it becomes like a question. 01:11:37.620 --> 01:11:41.120 align:middle line:84% Can you make the space Cauchy complete? 01:11:41.120 --> 01:11:43.100 align:middle line:90% And sometimes you can. 01:11:43.100 --> 01:11:44.280 align:middle line:90% Sometimes you can't. 01:11:44.280 --> 01:11:46.810 align:middle line:90% But that's a good question. 01:11:46.810 --> 01:11:53.780 align:middle line:84% But it seems utterly trivial example, but it's not. 01:11:53.780 --> 01:11:56.880 align:middle line:84% It does show you this important part, 01:11:56.880 --> 01:12:03.970 align:middle line:84% that it's often useful to have the Cauchy convergence property. 01:12:03.970 --> 01:12:08.180 align:middle line:84% And so can you enlarge the space to get that property? 01:12:08.180 --> 01:12:12.370 align:middle line:90% 01:12:12.370 --> 01:12:14.810 align:middle line:90% OK. 01:12:14.810 --> 01:12:20.930 align:middle line:84% Let's talk about what it means for a function 01:12:20.930 --> 01:12:22.700 align:middle line:90% to be continuous also. 01:12:22.700 --> 01:12:26.570 align:middle line:90% 01:12:26.570 --> 01:12:30.920 align:middle line:90% So now I want to think about-- 01:12:30.920 --> 01:12:34.170 align:middle line:90% 01:12:34.170 --> 01:12:40.370 align:middle line:84% so I want to think about continuity 01:12:40.370 --> 01:12:41.700 align:middle line:90% for general metric spaces. 01:12:41.700 --> 01:12:45.370 align:middle line:90% 01:12:45.370 --> 01:12:48.490 align:middle line:84% And one can do this even more generally. 01:12:48.490 --> 01:12:54.450 align:middle line:84% But let me settle on something in between something 01:12:54.450 --> 01:12:58.210 align:middle line:84% more general and what we already have. 01:12:58.210 --> 01:13:03.430 align:middle line:90% So I have a metric space here. 01:13:03.430 --> 01:13:09.150 align:middle line:84% And I'm thinking about that I have a map from this space 01:13:09.150 --> 01:13:16.950 align:middle line:84% into R. And then I want to answer, what does it 01:13:16.950 --> 01:13:20.110 align:middle line:84% mean for this function here to be continuous? 01:13:20.110 --> 01:13:33.870 align:middle line:84% So f is continuous at some point x0 in this space 01:13:33.870 --> 01:13:41.190 align:middle line:84% if for all epsilon greater than 0, 01:13:41.190 --> 01:13:48.870 align:middle line:84% there exist delta greater than 0 so 01:13:48.870 --> 01:13:54.110 align:middle line:84% that if the distance between x and this x0 01:13:54.110 --> 01:14:06.640 align:middle line:84% is smaller than delta, then the images is 01:14:06.640 --> 01:14:08.900 align:middle line:90% epsilon close together. 01:14:08.900 --> 01:14:15.040 align:middle line:84% So you see this is a kind of obvious generalization of how 01:14:15.040 --> 01:14:21.280 align:middle line:84% we define continuity at a point for when x was part Of r. 01:14:21.280 --> 01:14:23.720 align:middle line:84% And so, of course, the function is-- so this 01:14:23.720 --> 01:14:25.640 align:middle line:90% is a continuous in a point. 01:14:25.640 --> 01:14:38.280 align:middle line:84% And so f is said to be continuous 01:14:38.280 --> 01:14:46.350 align:middle line:84% if it is continuous at all points. 01:14:46.350 --> 01:14:52.910 align:middle line:90% 01:14:52.910 --> 01:14:55.980 align:middle line:84% So this is what it means for a function to be continuous. 01:14:55.980 --> 01:15:00.660 align:middle line:90% 01:15:00.660 --> 01:15:01.880 align:middle line:90% OK. 01:15:01.880 --> 01:15:07.440 align:middle line:84% Now let me just give you an example of something like this. 01:15:07.440 --> 01:15:10.460 align:middle line:90% 01:15:10.460 --> 01:15:15.960 align:middle line:84% So remember, before, we looked at the continuous functions. 01:15:15.960 --> 01:15:20.100 align:middle line:84% So we looked at where the space was the continuous function 01:15:20.100 --> 01:15:22.740 align:middle line:90% on the interval from 0 to 1. 01:15:22.740 --> 01:15:28.180 align:middle line:84% And we said that the distance between two continuous functions 01:15:28.180 --> 01:15:31.740 align:middle line:84% defined on the interval, this thing here, 01:15:31.740 --> 01:15:48.140 align:middle line:84% is the max of the differences here, like that. 01:15:48.140 --> 01:15:52.780 align:middle line:84% And that's quite easy to see, but it's a good exercise to do. 01:15:52.780 --> 01:15:55.030 align:middle line:84% It's quite easy to see that this is a metric. 01:15:55.030 --> 01:16:00.410 align:middle line:90% 01:16:00.410 --> 01:16:02.710 align:middle line:90% So this is a metric space. 01:16:02.710 --> 01:16:04.850 align:middle line:90% So now I can think about-- 01:16:04.850 --> 01:16:08.870 align:middle line:84% so now here's an example of a function. 01:16:08.870 --> 01:16:13.550 align:middle line:84% This is a function that is like evaluating. 01:16:13.550 --> 01:16:16.270 align:middle line:90% 01:16:16.270 --> 01:16:19.030 align:middle line:84% So you see this getting a little bit abstract because now I'm 01:16:19.030 --> 01:16:22.150 align:middle line:90% thinking about a function on-- 01:16:22.150 --> 01:16:26.630 align:middle line:84% so I now looking at the function-- 01:16:26.630 --> 01:16:31.430 align:middle line:84% so from x into R. But in this case 01:16:31.430 --> 01:16:35.510 align:middle line:84% x here is the space of continuous function 01:16:35.510 --> 01:16:37.530 align:middle line:90% on the interval. 01:16:37.530 --> 01:16:38.030 align:middle line:90% OK. 01:16:38.030 --> 01:16:41.130 align:middle line:90% 01:16:41.130 --> 01:16:45.150 align:middle line:84% So my function is now supposed to be defined 01:16:45.150 --> 01:16:48.070 align:middle line:90% on a continuous function. 01:16:48.070 --> 01:16:52.030 align:middle line:84% And I just take it to be equal to the value 01:16:52.030 --> 01:16:57.970 align:middle line:84% of this continuous function at the initial point 01:16:57.970 --> 01:16:59.514 align:middle line:90% in the interval. 01:16:59.514 --> 01:17:00.014 align:middle line:90% OK. 01:17:00.014 --> 01:17:03.402 align:middle line:90% 01:17:03.402 --> 01:17:07.130 align:middle line:84% And I claim that this function here-- 01:17:07.130 --> 01:17:14.430 align:middle line:84% so the claim is that this capital F is continuous. 01:17:14.430 --> 01:17:19.810 align:middle line:90% 01:17:19.810 --> 01:17:26.010 align:middle line:84% So again, it's continuous in the sense 01:17:26.010 --> 01:17:28.730 align:middle line:84% that you have this general metric space. 01:17:28.730 --> 01:17:42.210 align:middle line:84% And this here-- just so that it lines up with the example. 01:17:42.210 --> 01:17:45.970 align:middle line:84% So we have this general metric space. 01:17:45.970 --> 01:17:48.650 align:middle line:84% And we have now a map from this general metric space 01:17:48.650 --> 01:17:53.170 align:middle line:84% into R. This is definitely a map from this general metric space 01:17:53.170 --> 01:18:02.304 align:middle line:84% into R. And I claim that this is continuous in this sense here. 01:18:02.304 --> 01:18:05.800 align:middle line:84% And that's also quite easy to prove. 01:18:05.800 --> 01:18:09.160 align:middle line:90% 01:18:09.160 --> 01:18:12.940 align:middle line:84% So let me just make one observation, that if I'm 01:18:12.940 --> 01:18:14.960 align:middle line:90% looking at this thing here-- 01:18:14.960 --> 01:18:20.460 align:middle line:84% so this is the image under this map in one continuous function. 01:18:20.460 --> 01:18:26.540 align:middle line:84% And then I subtract the image in another continuous function. 01:18:26.540 --> 01:18:30.260 align:middle line:84% And now I'm looking at this difference here. 01:18:30.260 --> 01:18:35.100 align:middle line:84% But this thing here is nothing but f of 0. 01:18:35.100 --> 01:18:37.310 align:middle line:90% And this is nothing but g of 0. 01:18:37.310 --> 01:18:40.380 align:middle line:90% 01:18:40.380 --> 01:18:43.420 align:middle line:84% But you see that this is obviously less 01:18:43.420 --> 01:18:50.540 align:middle line:84% than or equal to the maximum of all x in this interval of f 01:18:50.540 --> 01:18:56.600 align:middle line:90% of x minus g of x, like that. 01:18:56.600 --> 01:19:03.400 align:middle line:84% But that is the distance between these two elements 01:19:03.400 --> 01:19:05.680 align:middle line:90% of this general metric space. 01:19:05.680 --> 01:19:11.920 align:middle line:84% And so you see that this here is an example that if-- 01:19:11.920 --> 01:19:14.760 align:middle line:84% so given epsilon, I'm going to pick 01:19:14.760 --> 01:19:17.280 align:middle line:90% delta equal to that epsilon. 01:19:17.280 --> 01:19:19.680 align:middle line:84% Because if these guys here are now 01:19:19.680 --> 01:19:22.280 align:middle line:84% epsilon close by this inequality, 01:19:22.280 --> 01:19:25.973 align:middle line:84% their images is also epsilon close. 01:19:25.973 --> 01:19:30.040 align:middle line:84% So this map is clearly continuous. 01:19:30.040 --> 01:19:30.732 align:middle line:90% OK. 01:19:30.732 --> 01:19:33.160 align:middle line:90% So I'll stop here. 01:19:33.160 --> 01:19:37.600 align:middle line:84% So there's a little lemma in the notes that we didn't go through. 01:19:37.600 --> 01:19:40.800 align:middle line:84% And that lemma is just the lemma that I had over here, 01:19:40.800 --> 01:19:43.720 align:middle line:84% that if I had a continuous function 01:19:43.720 --> 01:19:47.680 align:middle line:84% and I have a sequence that converges to a limit, 01:19:47.680 --> 01:19:50.680 align:middle line:84% then I could exchange the order of limits. 01:19:50.680 --> 01:19:53.760 align:middle line:84% And the same proof works in a general metric space. 01:19:53.760 --> 01:19:56.810 align:middle line:90% 01:19:56.810 --> 01:19:58.050 align:middle line:90% OK, great. 01:19:58.050 --> 01:19:58.730 align:middle line:90% Yeah? 01:19:58.730 --> 01:20:00.570 align:middle line:84% STUDENT: What kind of properties do you 01:20:00.570 --> 01:20:05.023 align:middle line:84% need to generalize [INAUDIBLE] space? 01:20:05.023 --> 01:20:07.245 align:middle line:84% Is it some kind of closure under addition? 01:20:07.245 --> 01:20:08.370 align:middle line:90% TOBIAS COLDING: Yeah, yeah. 01:20:08.370 --> 01:20:08.870 align:middle line:90% Right. 01:20:08.870 --> 01:20:10.590 align:middle line:90% That's right. 01:20:10.590 --> 01:20:14.910 align:middle line:84% So addition is something about a field. 01:20:14.910 --> 01:20:21.870 align:middle line:84% And so for these spaces, most of the things I looked at here, 01:20:21.870 --> 01:20:23.370 align:middle line:90% some of it does. 01:20:23.370 --> 01:20:26.310 align:middle line:84% Like, the continuous function has some structure. 01:20:26.310 --> 01:20:29.530 align:middle line:84% The space of continuous function has some structure. 01:20:29.530 --> 01:20:31.450 align:middle line:84% It's not exactly a field, but it has 01:20:31.450 --> 01:20:33.830 align:middle line:84% some of those properties, some of these algebra. 01:20:33.830 --> 01:20:37.890 align:middle line:84% It's like an important example of something 01:20:37.890 --> 01:20:40.790 align:middle line:90% called a sea star of algebra. 01:20:40.790 --> 01:20:45.090 align:middle line:84% So it have this algebraic structure. 01:20:45.090 --> 01:20:47.450 align:middle line:84% But it's not the algebraic structure 01:20:47.450 --> 01:20:50.550 align:middle line:84% you need in order to be completeness. 01:20:50.550 --> 01:20:51.610 align:middle line:90% It's not that. 01:20:51.610 --> 01:20:53.810 align:middle line:90% Yeah. 01:20:53.810 --> 01:20:59.490 align:middle line:84% And yeah, I think it takes us a little bit far afield. 01:20:59.490 --> 01:21:01.670 align:middle line:90% But it's the same-- 01:21:01.670 --> 01:21:04.430 align:middle line:84% you see this sort of property also in-- 01:21:04.430 --> 01:21:07.750 align:middle line:90% 01:21:07.750 --> 01:21:10.370 align:middle line:84% like, metric spaces, extremely useful. 01:21:10.370 --> 01:21:12.670 align:middle line:90% You see it all over the place. 01:21:12.670 --> 01:21:15.030 align:middle line:84% One of many places where you see it 01:21:15.030 --> 01:21:17.390 align:middle line:90% is also in general relativity. 01:21:17.390 --> 01:21:20.470 align:middle line:84% And in general relativity, the space you're looking at 01:21:20.470 --> 01:21:23.270 align:middle line:90% are typically not complete. 01:21:23.270 --> 01:21:27.950 align:middle line:84% So there's something about that, that you 01:21:27.950 --> 01:21:30.490 align:middle line:90% reach something in finite time. 01:21:30.490 --> 01:21:31.290 align:middle line:90% You reach it. 01:21:31.290 --> 01:21:32.050 align:middle line:90% You're going. 01:21:32.050 --> 01:21:33.790 align:middle line:84% And this is like the same thing, that you 01:21:33.790 --> 01:21:36.590 align:middle line:90% reach the end of things. 01:21:36.590 --> 01:21:40.910 align:middle line:84% Anyway, but it's not the algebraic property 01:21:40.910 --> 01:21:43.740 align:middle line:90% that is used for this. 01:21:43.740 --> 01:21:49.000 align:middle line:90%