WEBVTT 00:00:00.000 --> 00:00:02.450 align:middle line:90% [SQUEAKING] 00:00:02.450 --> 00:00:04.410 align:middle line:90% [RUSTLING] 00:00:04.410 --> 00:00:07.350 align:middle line:90% [CLICKING] 00:00:07.350 --> 00:00:09.520 align:middle line:90% 00:00:09.520 --> 00:00:12.800 align:middle line:84% TOBIAS COLDING: Just begin with-- 00:00:12.800 --> 00:00:31.800 align:middle line:84% so if we have a metric space here, then a cover of x 00:00:31.800 --> 00:00:51.080 align:middle line:84% is a collection of subsets u alpha so that the x is contained 00:00:51.080 --> 00:00:55.120 align:middle line:90% in the union of u alpha. 00:00:55.120 --> 00:01:03.470 align:middle line:84% So that's a cover of X and likewise, 00:01:03.470 --> 00:01:11.070 align:middle line:84% if a here is a subset of x, then u alpha 00:01:11.070 --> 00:01:23.630 align:middle line:84% is a cover of a if a here is in the union of the u alpha. 00:01:23.630 --> 00:01:29.950 align:middle line:84% So that's what it means to cover the whole set or just a subset. 00:01:29.950 --> 00:01:35.990 align:middle line:84% Then there are some covers that are particularly important, 00:01:35.990 --> 00:01:38.570 align:middle line:90% and those are the open covers. 00:01:38.570 --> 00:01:55.840 align:middle line:84% So a open cover is a cover that consists of open subsets. 00:01:55.840 --> 00:02:02.110 align:middle line:90% 00:02:02.110 --> 00:02:07.190 align:middle line:84% And then finally, we have that, so if again, 00:02:07.190 --> 00:02:13.230 align:middle line:84% x,d is a metric space, a here is a subset of x, 00:02:13.230 --> 00:02:39.630 align:middle line:84% then we say that a is compact if every open cover of a has 00:02:39.630 --> 00:02:51.350 align:middle line:90% a finite subcover, OK. 00:02:51.350 --> 00:02:56.110 align:middle line:84% And one thing we proved last time, was that if you have-- 00:02:56.110 --> 00:02:59.380 align:middle line:90% 00:02:59.380 --> 00:03:13.620 align:middle line:84% a so we proved last time, that if you have a here is compact, 00:03:13.620 --> 00:03:26.330 align:middle line:90% then a is bounded and closed. 00:03:26.330 --> 00:03:34.460 align:middle line:90% 00:03:34.460 --> 00:03:37.460 align:middle line:84% Right, so it's a is a closed subset, meaning 00:03:37.460 --> 00:03:44.380 align:middle line:84% that the complement is open, and a itself 00:03:44.380 --> 00:03:49.900 align:middle line:84% is contained in some large ball centered at some point in x. 00:03:49.900 --> 00:03:55.780 align:middle line:84% Now, the converse is not the case always. 00:03:55.780 --> 00:04:19.180 align:middle line:84% So In Rn the usual metric, Euclidean metric, 00:04:19.180 --> 00:04:32.300 align:middle line:84% the converse is the case but in general not-- 00:04:32.300 --> 00:04:33.230 align:middle line:90% but not in general. 00:04:33.230 --> 00:04:42.400 align:middle line:90% 00:04:42.400 --> 00:04:50.220 align:middle line:84% And so we looked at one example on this last time. 00:04:50.220 --> 00:04:52.930 align:middle line:84% So we looked at the following example. 00:04:52.930 --> 00:04:55.450 align:middle line:84% But it was a rather trivial example 00:04:55.450 --> 00:04:58.250 align:middle line:90% that if you take x here-- 00:04:58.250 --> 00:05:05.770 align:middle line:84% if you take the space x to be, say, the interval from 0 to 1, 00:05:05.770 --> 00:05:10.530 align:middle line:84% and the metric here, this here is the usual metric, 00:05:10.530 --> 00:05:15.530 align:middle line:84% so the distance between two elements 00:05:15.530 --> 00:05:20.930 align:middle line:84% here is just the norm of the difference. 00:05:20.930 --> 00:05:27.650 align:middle line:84% And then well, x is the whole set, so with this here, 00:05:27.650 --> 00:05:28.830 align:middle line:90% x is the whole set. 00:05:28.830 --> 00:05:32.530 align:middle line:84% So x here, if this here is the metric space, 00:05:32.530 --> 00:05:38.650 align:middle line:84% then x is, by definition, closed because the whole set is closed. 00:05:38.650 --> 00:05:41.450 align:middle line:84% It's not closed as a subset of r, 00:05:41.450 --> 00:05:48.410 align:middle line:84% but if the metric space just is x, then it is closed, 00:05:48.410 --> 00:05:52.490 align:middle line:90% and it's bounded-- 00:05:52.490 --> 00:05:55.570 align:middle line:90% and bounded. 00:05:55.570 --> 00:05:58.890 align:middle line:84% In particular, this interval here 00:05:58.890 --> 00:06:05.610 align:middle line:84% is contained in the ball of radius one half around one half. 00:06:05.610 --> 00:06:08.930 align:middle line:84% But it's actually equal to that ball, if you will. 00:06:08.930 --> 00:06:15.790 align:middle line:84% So it's definitely bounded, but it's not compact, right? 00:06:15.790 --> 00:06:16.840 align:middle line:90% So it's not compact. 00:06:16.840 --> 00:06:22.050 align:middle line:90% 00:06:22.050 --> 00:06:31.610 align:middle line:84% Because you can write x here as the union of 1 over n to 1. 00:06:31.610 --> 00:06:38.215 align:middle line:84% And now you take the union over all positive integers. 00:06:38.215 --> 00:06:41.050 align:middle line:84% But if you only take finitely many of them, 00:06:41.050 --> 00:06:45.490 align:middle line:84% then you don't get all the way down to zero, right? 00:06:45.490 --> 00:06:47.390 align:middle line:90% So it's not a union. 00:06:47.390 --> 00:06:49.440 align:middle line:90% It's union of all of these. 00:06:49.440 --> 00:06:53.800 align:middle line:90% Each of these r is open, But. 00:06:53.800 --> 00:07:00.200 align:middle line:84% There's no finite union of those sets that cover x. 00:07:00.200 --> 00:07:07.640 align:middle line:84% A more interesting example is that-- suppose you take-- 00:07:07.640 --> 00:07:12.520 align:middle line:84% suppose you take-- so here's another example that we 00:07:12.520 --> 00:07:17.680 align:middle line:84% didn't get to last time, that if you take the continuous function 00:07:17.680 --> 00:07:23.380 align:middle line:84% on the unit interval, and so if you take-- 00:07:23.380 --> 00:07:30.640 align:middle line:84% and now we use this metric that we have looked at before, 00:07:30.640 --> 00:07:36.400 align:middle line:84% where the distance between two functions 00:07:36.400 --> 00:07:47.640 align:middle line:84% is the max over x in this interval here of the norm of f 00:07:47.640 --> 00:07:50.006 align:middle line:90% of x minus g of x. 00:07:50.006 --> 00:07:53.080 align:middle line:90% 00:07:53.080 --> 00:07:57.900 align:middle line:84% We've already seen that this here is a nice metric. 00:07:57.900 --> 00:08:01.320 align:middle line:90% And if I now look at-- 00:08:01.320 --> 00:08:06.720 align:middle line:84% in this metric, I look at the unit ball, 00:08:06.720 --> 00:08:12.040 align:middle line:84% so I'm looking at the closed unit ball, which 00:08:12.040 --> 00:08:19.800 align:middle line:84% means that I'm looking at the set of functions here, 00:08:19.800 --> 00:08:26.240 align:middle line:84% so that the distance of f to the function that 00:08:26.240 --> 00:08:31.800 align:middle line:84% is constant equal to zero, this distance here, 00:08:31.800 --> 00:08:35.419 align:middle line:90% is less or equal to one. 00:08:35.419 --> 00:08:38.039 align:middle line:90% 00:08:38.039 --> 00:08:41.440 align:middle line:84% So I'm looking here at the subset 00:08:41.440 --> 00:08:47.590 align:middle line:84% of the continuous function on this interval, 00:08:47.590 --> 00:08:52.590 align:middle line:84% so that the distance to the zero function is at most one. 00:08:52.590 --> 00:08:54.810 align:middle line:90% So let me just write this out. 00:08:54.810 --> 00:08:58.110 align:middle line:84% So this means that this is the function 00:08:58.110 --> 00:09:04.190 align:middle line:84% f in this continuous function here, 00:09:04.190 --> 00:09:11.630 align:middle line:84% so that the max of f of x minus 0, 00:09:11.630 --> 00:09:19.190 align:middle line:84% so I don't really have to write 0, is less than or equal to 1. 00:09:19.190 --> 00:09:23.030 align:middle line:90% 00:09:23.030 --> 00:09:28.790 align:middle line:84% Now, this set again, this set here, is what we denote by the-- 00:09:28.790 --> 00:09:32.550 align:middle line:84% this is the closed unit ball because you have less than 00:09:32.550 --> 00:09:33.930 align:middle line:90% or equal to here. 00:09:33.930 --> 00:09:39.870 align:middle line:84% And the closed unit ball, the open unit ball we have written, 00:09:39.870 --> 00:09:44.750 align:middle line:84% this is now the function that is identically equal to 0. 00:09:44.750 --> 00:09:47.750 align:middle line:84% The open unit ball, we wrote like this. 00:09:47.750 --> 00:09:51.710 align:middle line:84% This is if you have strict inequality. 00:09:51.710 --> 00:09:56.030 align:middle line:84% And to denote that it's less or equal to, 00:09:56.030 --> 00:09:57.890 align:middle line:90% I'm just putting a bar here. 00:09:57.890 --> 00:10:01.950 align:middle line:90% 00:10:01.950 --> 00:10:04.310 align:middle line:84% And so we already know that this thing 00:10:04.310 --> 00:10:06.830 align:middle line:84% here, if you take a metric space, 00:10:06.830 --> 00:10:09.970 align:middle line:84% and you're looking at a closed unit ball like this, 00:10:09.970 --> 00:10:13.210 align:middle line:84% we already proved that this here is closed. 00:10:13.210 --> 00:10:16.590 align:middle line:84% Because we proved directly, using the triangle inequality, 00:10:16.590 --> 00:10:18.340 align:middle line:90% that the complement is open. 00:10:18.340 --> 00:10:20.980 align:middle line:90% 00:10:20.980 --> 00:10:24.650 align:middle line:90% OK, so this is a closed subset. 00:10:24.650 --> 00:10:28.000 align:middle line:84% It's obviously bounded because it's actually even just a ball, 00:10:28.000 --> 00:10:28.500 align:middle line:90% right? 00:10:28.500 --> 00:10:33.990 align:middle line:90% 00:10:33.990 --> 00:10:35.310 align:middle line:90% But I claim that-- 00:10:35.310 --> 00:10:37.170 align:middle line:90% so it's closed. 00:10:37.170 --> 00:10:42.220 align:middle line:84% It's bounded, but it's not compact. 00:10:42.220 --> 00:10:49.660 align:middle line:84% And here's a interesting example of a sequence 00:10:49.660 --> 00:10:51.680 align:middle line:90% of elements in this ball. 00:10:51.680 --> 00:11:01.540 align:middle line:84% So I will now construct for each n, which is a natural number, 00:11:01.540 --> 00:11:10.700 align:middle line:84% we construct a function fn in this 00:11:10.700 --> 00:11:14.300 align:middle line:84% that is a continuous function on this unit interval. 00:11:14.300 --> 00:11:18.490 align:middle line:84% And the function f will be given as follows. 00:11:18.490 --> 00:11:23.820 align:middle line:90% 00:11:23.820 --> 00:11:33.670 align:middle line:90% It will be one from when-- 00:11:33.670 --> 00:11:37.340 align:middle line:90% 00:11:37.340 --> 00:11:37.880 align:middle line:90% let's see. 00:11:37.880 --> 00:11:48.190 align:middle line:84% Let me make it 0/ I guess it doesn't matter. 00:11:48.190 --> 00:11:56.530 align:middle line:84% I'll make it one when x here is between 1 over n plus 1. 00:11:56.530 --> 00:12:01.170 align:middle line:84% No, no, I would make it equal to 1 00:12:01.170 --> 00:12:06.850 align:middle line:84% when x here is bigger or equal to 0 and less than or equal to 1 00:12:06.850 --> 00:12:08.190 align:middle line:90% over n plus 1. 00:12:08.190 --> 00:12:10.970 align:middle line:90% 00:12:10.970 --> 00:12:21.010 align:middle line:84% Then I'll make it when x here is bigger than or equal to 1 over n 00:12:21.010 --> 00:12:28.650 align:middle line:84% but less or equal to 1, then I'll set it equal to 0. 00:12:28.650 --> 00:12:34.250 align:middle line:84% And then you see now, I wanted to do so-- 00:12:34.250 --> 00:12:36.310 align:middle line:90% so let's look at it. 00:12:36.310 --> 00:12:54.930 align:middle line:84% So this function here is if you draw this function-- 00:12:54.930 --> 00:12:58.710 align:middle line:90% so I have here 1. 00:12:58.710 --> 00:13:00.770 align:middle line:90% Here I have 0. 00:13:00.770 --> 00:13:08.190 align:middle line:84% And this function is 1 up to 1 over n plus 1. 00:13:08.190 --> 00:13:11.850 align:middle line:84% Maybe I'll draw it a little bit further out. 00:13:11.850 --> 00:13:19.170 align:middle line:84% Let me actually make the picture bigger, so like that. 00:13:19.170 --> 00:13:22.170 align:middle line:90% Here I have 1. 00:13:22.170 --> 00:13:27.810 align:middle line:90% Here I have 1 over n plus 1. 00:13:27.810 --> 00:13:32.090 align:middle line:90% And here I have 1 over n. 00:13:32.090 --> 00:13:35.120 align:middle line:84% And the function is supposed to be 1 here. 00:13:35.120 --> 00:13:38.520 align:middle line:90% 00:13:38.520 --> 00:13:48.240 align:middle line:84% And then it's supposed to die off between here and here. 00:13:48.240 --> 00:13:51.760 align:middle line:84% And I choose it to die off linearly. 00:13:51.760 --> 00:13:54.160 align:middle line:90% And then it's 0 here. 00:13:54.160 --> 00:13:57.520 align:middle line:84% So if it's going down linearly, this 00:13:57.520 --> 00:14:00.840 align:middle line:84% means that I have my-- what is the difference 00:14:00.840 --> 00:14:02.920 align:middle line:90% between these two? 00:14:02.920 --> 00:14:07.220 align:middle line:84% It's 1 over n minus 1 over n plus 1. 00:14:07.220 --> 00:14:09.320 align:middle line:90% This is this distance here. 00:14:09.320 --> 00:14:13.640 align:middle line:84% And so this here, putting it on a common denominator n plus 1 00:14:13.640 --> 00:14:16.000 align:middle line:90% over n times n plus 1. 00:14:16.000 --> 00:14:18.280 align:middle line:90% This is what you get from this. 00:14:18.280 --> 00:14:22.240 align:middle line:84% And if you put it on the common denominator for the second one, 00:14:22.240 --> 00:14:24.160 align:middle line:90% this is what you get here. 00:14:24.160 --> 00:14:30.240 align:middle line:84% And so this is just 1 over n like this. 00:14:30.240 --> 00:14:34.200 align:middle line:84% And so this is this distance here. 00:14:34.200 --> 00:14:39.820 align:middle line:84% And so if I define the function to be-- 00:14:39.820 --> 00:14:46.260 align:middle line:90% so I'm setting it equal to this. 00:14:46.260 --> 00:14:49.520 align:middle line:84% And now in between when x here is 00:14:49.520 --> 00:14:55.120 align:middle line:84% bigger or equal to 1 over n plus 1 and less or equal to 1 over n, 00:14:55.120 --> 00:15:11.640 align:middle line:84% I will set it equal to 1 minus n times n plus 1 times-- 00:15:11.640 --> 00:15:13.850 align:middle line:90% sorry, let me make more space. 00:15:13.850 --> 00:15:19.680 align:middle line:90% 00:15:19.680 --> 00:15:22.600 align:middle line:90% Yeah, let me make more space. 00:15:22.600 --> 00:15:27.180 align:middle line:84% So here, this here was when 0 is less or equal to x, less 00:15:27.180 --> 00:15:30.280 align:middle line:90% or equal to 1 over n plus 1. 00:15:30.280 --> 00:15:38.110 align:middle line:84% This is when x here is between these two numbers. 00:15:38.110 --> 00:15:42.470 align:middle line:84% And this here is when x is larger than 1 over n 00:15:42.470 --> 00:15:46.810 align:middle line:84% And it's x is always in the interval between 0 and 1. 00:15:46.810 --> 00:15:59.550 align:middle line:84% And here, I'm choosing it's x minus and then 1 over n plus 1. 00:15:59.550 --> 00:16:06.870 align:middle line:84% So you see that if you take x to be 1 over n plus 1, 00:16:06.870 --> 00:16:08.830 align:middle line:90% then here contributes 0. 00:16:08.830 --> 00:16:11.750 align:middle line:84% So at this left point, this here is equal 00:16:11.750 --> 00:16:15.670 align:middle line:84% to-- it's the function is equal to 1, as it should be. 00:16:15.670 --> 00:16:19.590 align:middle line:84% And when x here is at the other endpoint, 00:16:19.590 --> 00:16:23.350 align:middle line:84% then this here becomes exactly 1 over this, 00:16:23.350 --> 00:16:24.990 align:middle line:90% So this whole thing becomes 0. 00:16:24.990 --> 00:16:28.230 align:middle line:90% 00:16:28.230 --> 00:16:36.370 align:middle line:84% So this function here I drew, is if you write it down precisely, 00:16:36.370 --> 00:16:37.650 align:middle line:90% this is what the function is. 00:16:37.650 --> 00:16:46.340 align:middle line:90% 00:16:46.340 --> 00:16:57.230 align:middle line:84% And now if I'm looking at the difference between two 00:16:57.230 --> 00:17:01.550 align:middle line:84% such functions, so for the next one-- 00:17:01.550 --> 00:17:04.030 align:middle line:90% you see the next one here? 00:17:04.030 --> 00:17:06.050 align:middle line:90% The next one here is-- 00:17:06.050 --> 00:17:10.430 align:middle line:90% so this here is fn. 00:17:10.430 --> 00:17:12.130 align:middle line:90% This is how fn looks like. 00:17:12.130 --> 00:17:18.069 align:middle line:84% If I now look at fn plus 1, this start off being it's 00:17:18.069 --> 00:17:21.390 align:middle line:84% 0 to the right of this thing here. 00:17:21.390 --> 00:17:28.950 align:middle line:84% And then it's 1 once you get to n plus 2. 00:17:28.950 --> 00:17:34.540 align:middle line:84% So the next function looks like this. 00:17:34.540 --> 00:17:36.440 align:middle line:84% I don't know if there's any color chalk, 00:17:36.440 --> 00:17:42.960 align:middle line:84% but this thing here, that's the next function. 00:17:42.960 --> 00:17:48.540 align:middle line:84% So the first function is this function here. 00:17:48.540 --> 00:17:50.540 align:middle line:84% That's a graph of the first function. 00:17:50.540 --> 00:17:53.420 align:middle line:90% The next one is this one here. 00:17:53.420 --> 00:17:56.300 align:middle line:90% 00:17:56.300 --> 00:18:02.500 align:middle line:84% And so if you're now looking at the distance between fn and fn 00:18:02.500 --> 00:18:09.020 align:middle line:84% plus 1, the distance here was the sup 00:18:09.020 --> 00:18:11.440 align:middle line:90% of the difference of the two. 00:18:11.440 --> 00:18:18.980 align:middle line:84% This here was just a max of f of-- 00:18:18.980 --> 00:18:19.720 align:middle line:90% like this. 00:18:19.720 --> 00:18:22.900 align:middle line:90% 00:18:22.900 --> 00:18:25.740 align:middle line:84% It was just a max of this, right? 00:18:25.740 --> 00:18:29.620 align:middle line:84% And now I know that these two numbers is always 00:18:29.620 --> 00:18:31.620 align:middle line:90% between 0 and 1. 00:18:31.620 --> 00:18:36.640 align:middle line:90% So this max here is at most 1. 00:18:36.640 --> 00:18:40.540 align:middle line:84% But if I now evaluate it, if I'm now looking at-- 00:18:40.540 --> 00:18:42.500 align:middle line:84% if I now just evaluate it, if I'm 00:18:42.500 --> 00:18:48.460 align:middle line:84% looking at fn, and now I just evaluate it in this point 00:18:48.460 --> 00:18:53.580 align:middle line:84% here, in this point 1 over n plus 1, 00:18:53.580 --> 00:18:59.820 align:middle line:84% so if I'm looking at this thing here fn plus 1, 00:18:59.820 --> 00:19:02.580 align:middle line:90% 1 over n plus 1-- 00:19:02.580 --> 00:19:07.820 align:middle line:84% you see this value here, this first function here, 00:19:07.820 --> 00:19:10.340 align:middle line:90% and at this point, it's 1. 00:19:10.340 --> 00:19:12.500 align:middle line:90% This here is 1. 00:19:12.500 --> 00:19:15.940 align:middle line:84% But for the next one, you see at that point it's 0. 00:19:15.940 --> 00:19:19.220 align:middle line:90% 00:19:19.220 --> 00:19:26.210 align:middle line:84% So this means that this max here, is for all n and n plus 1, 00:19:26.210 --> 00:19:31.330 align:middle line:90% is actually equal to 1. 00:19:31.330 --> 00:19:44.580 align:middle line:84% And so now if you look at the balls-- 00:19:44.580 --> 00:19:52.530 align:middle line:90% 00:19:52.530 --> 00:19:55.970 align:middle line:84% so now, in particular, if I'm looking 00:19:55.970 --> 00:20:01.250 align:middle line:84% at the balls of radius one half, suppose 00:20:01.250 --> 00:20:05.150 align:middle line:84% I'm looking at the ball of radius one half around-- 00:20:05.150 --> 00:20:10.850 align:middle line:84% so this is an open ball, so around fn. 00:20:10.850 --> 00:20:12.590 align:middle line:84% Because this here is an open ball, 00:20:12.590 --> 00:20:15.050 align:middle line:84% so it means that's strictly less than one half, 00:20:15.050 --> 00:20:20.370 align:middle line:84% so this means that fn plus 1 is not in that ball. 00:20:20.370 --> 00:20:23.210 align:middle line:90% 00:20:23.210 --> 00:20:24.150 align:middle line:90% So it's not in. 00:20:24.150 --> 00:20:26.730 align:middle line:90% So it's not in that ball. 00:20:26.730 --> 00:20:32.410 align:middle line:84% And so now it's easy to see that all of these balls 00:20:32.410 --> 00:20:36.690 align:middle line:90% here are disjoint. 00:20:36.690 --> 00:20:40.930 align:middle line:84% So all of these balls here will be disjoint. 00:20:40.930 --> 00:20:42.570 align:middle line:90% They're all disjoint. 00:20:42.570 --> 00:20:49.730 align:middle line:84% And so if you're looking at if you can write this initial ball, 00:20:49.730 --> 00:20:52.290 align:middle line:84% this ball here, this was this close ball 00:20:52.290 --> 00:20:55.130 align:middle line:84% that we started with in the beginning, 00:20:55.130 --> 00:21:00.170 align:middle line:90% you can write it as a union. 00:21:00.170 --> 00:21:05.010 align:middle line:84% You can write it, of course, as a union of half balls 00:21:05.010 --> 00:21:14.210 align:middle line:84% where you take the union over all centers, 00:21:14.210 --> 00:21:17.570 align:middle line:90% all possible centers like this. 00:21:17.570 --> 00:21:21.250 align:middle line:84% Because it's clearly cover this set 00:21:21.250 --> 00:21:23.920 align:middle line:84% because it contains all the centers. 00:21:23.920 --> 00:21:27.800 align:middle line:84% But that's already the whole set. 00:21:27.800 --> 00:21:32.040 align:middle line:84% But using this here, you can see that there is not-- 00:21:32.040 --> 00:21:36.080 align:middle line:84% that you can't have finitely many of these half 00:21:36.080 --> 00:21:39.360 align:middle line:90% balls that cover the whole set. 00:21:39.360 --> 00:21:57.440 align:middle line:84% So using the functions fn, it follows easily 00:21:57.440 --> 00:22:13.070 align:middle line:84% that finitely many of these balls does not cover. 00:22:13.070 --> 00:22:19.960 align:middle line:90% 00:22:19.960 --> 00:22:22.670 align:middle line:90% And let me just cover this here. 00:22:22.670 --> 00:22:25.430 align:middle line:90% Why is this the case? 00:22:25.430 --> 00:22:30.710 align:middle line:84% Actually, sometimes you put the bar all the way across. 00:22:30.710 --> 00:22:33.230 align:middle line:90% But I haven't done that. 00:22:33.230 --> 00:22:41.240 align:middle line:84% OK, suppose for a second, that you have-- 00:22:41.240 --> 00:22:46.710 align:middle line:90% 00:22:46.710 --> 00:22:54.170 align:middle line:84% suppose that you have a ball here of radius one 00:22:54.170 --> 00:22:56.430 align:middle line:90% half around some point. 00:22:56.430 --> 00:23:00.630 align:middle line:84% And suppose that you have a function, 00:23:00.630 --> 00:23:04.950 align:middle line:84% g-- two functions g and h that are in that ball, 00:23:04.950 --> 00:23:07.590 align:middle line:84% just using that it's a metric space. 00:23:07.590 --> 00:23:13.710 align:middle line:84% So now we have that this thing here, g, is strictly 00:23:13.710 --> 00:23:19.430 align:middle line:84% less than one half, and the distance from f to h 00:23:19.430 --> 00:23:22.470 align:middle line:90% is strictly less than one half. 00:23:22.470 --> 00:23:24.890 align:middle line:84% So now using the triangle inequality, 00:23:24.890 --> 00:23:30.630 align:middle line:84% we conclude that the distance between this pair here, g and h, 00:23:30.630 --> 00:23:34.570 align:middle line:84% is less than or equal to the distance where you go over f. 00:23:34.570 --> 00:23:38.150 align:middle line:90% 00:23:38.150 --> 00:23:41.510 align:middle line:84% And so this has strictly less than one half 00:23:41.510 --> 00:23:45.430 align:middle line:84% plus one half, so this is strictly less than 1. 00:23:45.430 --> 00:23:48.310 align:middle line:84% So you see from this, in particular, 00:23:48.310 --> 00:23:53.550 align:middle line:84% that if you take such a ball here, 00:23:53.550 --> 00:24:09.110 align:middle line:84% then you can't-- so it's impossible to have that fn, 00:24:09.110 --> 00:24:12.350 align:middle line:90% n fn plus 1. 00:24:12.350 --> 00:24:14.490 align:middle line:90% It's impossible to have that. 00:24:14.490 --> 00:24:17.940 align:middle line:90% They both are in that ball. 00:24:17.940 --> 00:24:20.940 align:middle line:84% Because if they were both in this ball, then 00:24:20.940 --> 00:24:23.360 align:middle line:84% the distance would be strictly less than 1. 00:24:23.360 --> 00:24:26.160 align:middle line:84% And we've already seen that distance is 1. 00:24:26.160 --> 00:24:32.340 align:middle line:84% And this holds, not just for fn and the next one, 00:24:32.340 --> 00:24:35.260 align:middle line:84% but it holds for fn-- the same argument gives 00:24:35.260 --> 00:24:39.500 align:middle line:84% that from fn and any n larger than n, 00:24:39.500 --> 00:24:42.260 align:middle line:90% this here would be impossible. 00:24:42.260 --> 00:24:44.980 align:middle line:84% And so from this, it follows that you can't have finitely 00:24:44.980 --> 00:24:51.900 align:middle line:90% many that cover the ball, Yeah? 00:24:51.900 --> 00:24:55.820 align:middle line:84% AUDIENCE: How do we know that the union of all 00:24:55.820 --> 00:24:59.100 align:middle line:90% of those functions and the one? 00:24:59.100 --> 00:25:01.500 align:middle line:84% TOBIAS COLDING: Yeah, so this is just-- 00:25:01.500 --> 00:25:03.260 align:middle line:84% and this is important because we're going 00:25:03.260 --> 00:25:06.460 align:middle line:90% to use this in a minute anyway. 00:25:06.460 --> 00:25:11.500 align:middle line:84% So again, if you take a set here-- 00:25:11.500 --> 00:25:16.540 align:middle line:84% so if you take a metric space, and suppose you take a set, 00:25:16.540 --> 00:25:23.780 align:middle line:84% a subset of x, then you can always find a cover, even 00:25:23.780 --> 00:25:25.780 align:middle line:90% an open cover. 00:25:25.780 --> 00:25:27.580 align:middle line:84% And in particular, in a minute, we 00:25:27.580 --> 00:25:31.420 align:middle line:84% will look at-- so if you take a here, so one 00:25:31.420 --> 00:25:35.340 align:middle line:84% example of an open cover of a, is that you're 00:25:35.340 --> 00:25:40.180 align:middle line:90% looking at the family of balls. 00:25:40.180 --> 00:25:43.800 align:middle line:84% You just pick some radius r as positive, 00:25:43.800 --> 00:25:46.180 align:middle line:90% And then you pick centers. 00:25:46.180 --> 00:25:48.880 align:middle line:90% 00:25:48.880 --> 00:25:50.040 align:middle line:90% But the centers-- sorry. 00:25:50.040 --> 00:25:57.660 align:middle line:84% You take the centers, x here is now anything in a. 00:25:57.660 --> 00:26:00.900 align:middle line:90% R is fixed, It's positive. 00:26:00.900 --> 00:26:04.060 align:middle line:90% X is anything in a. 00:26:04.060 --> 00:26:07.020 align:middle line:84% Well, the union of these guys, the union 00:26:07.020 --> 00:26:09.720 align:middle line:90% certainly contains the centers. 00:26:09.720 --> 00:26:14.250 align:middle line:90% So this here already cover a. 00:26:14.250 --> 00:26:16.690 align:middle line:90% And we will use this. 00:26:16.690 --> 00:26:18.910 align:middle line:84% This is like a standard way to get a cover. 00:26:18.910 --> 00:26:21.410 align:middle line:84% And then if it's compact, then finally, many of those 00:26:21.410 --> 00:26:23.530 align:middle line:90% should cover. 00:26:23.530 --> 00:26:25.840 align:middle line:90% So in fact, let's get to that. 00:26:25.840 --> 00:26:28.650 align:middle line:90% 00:26:28.650 --> 00:26:33.130 align:middle line:84% So the next thing I want to prove, 00:26:33.130 --> 00:26:44.960 align:middle line:84% is this Bolzano-Weierstrass version for a metric space. 00:26:44.960 --> 00:26:48.170 align:middle line:90% 00:26:48.170 --> 00:26:53.010 align:middle line:84% So you prove a Bolzano-Weierstrass theorem 00:26:53.010 --> 00:26:54.030 align:middle line:90% for metric spaces. 00:26:54.030 --> 00:26:56.930 align:middle line:84% Remember that the Bolzano-Weierstrass does not 00:26:56.930 --> 00:26:58.440 align:middle line:90% hold for every metric space-- 00:26:58.440 --> 00:27:14.090 align:middle line:90% 00:27:14.090 --> 00:27:18.250 align:middle line:90% for a metric space. 00:27:18.250 --> 00:27:22.370 align:middle line:84% So now, the Bolzano-Weierstrass does not hold for every metric 00:27:22.370 --> 00:27:23.890 align:middle line:90% space, but it holds-- 00:27:23.890 --> 00:27:30.900 align:middle line:84% so when this space here is compact. 00:27:30.900 --> 00:27:40.710 align:middle line:90% 00:27:40.710 --> 00:27:42.450 align:middle line:90% So that's what I want to prove. 00:27:42.450 --> 00:27:44.530 align:middle line:84% And that's one of the motivations 00:27:44.530 --> 00:27:50.130 align:middle line:84% here for the definition of being compact. 00:27:50.130 --> 00:27:54.570 align:middle line:90% And so let me just remind you. 00:27:54.570 --> 00:28:01.530 align:middle line:84% So on Rn, on r, say, what is the [? Balzani? ?] 00:28:01.530 --> 00:28:10.200 align:middle line:84% So on r the Bolzano-Weierstrass theorem, what did that say? 00:28:10.200 --> 00:28:15.440 align:middle line:84% So this is that any bounded sequence 00:28:15.440 --> 00:28:28.400 align:middle line:90% has a convergent subsequence. 00:28:28.400 --> 00:28:33.140 align:middle line:90% 00:28:33.140 --> 00:28:35.380 align:middle line:84% That was the Bolzano-Weierstrass theorem. 00:28:35.380 --> 00:28:38.560 align:middle line:90% 00:28:38.560 --> 00:28:42.720 align:middle line:84% And so what we want to show here, 00:28:42.720 --> 00:28:47.450 align:middle line:84% is the following theorem for a general metric space. 00:28:47.450 --> 00:28:57.760 align:middle line:90% 00:28:57.760 --> 00:29:05.240 align:middle line:84% So we want to prove the following theorem that if x,d is 00:29:05.240 --> 00:29:16.040 align:middle line:90% a metric space, a here is-- 00:29:16.040 --> 00:29:17.490 align:middle line:90% subset is compact. 00:29:17.490 --> 00:29:22.520 align:middle line:90% 00:29:22.520 --> 00:29:48.400 align:middle line:84% Then any sequence in a has a convergent subsequence. 00:29:48.400 --> 00:29:57.320 align:middle line:90% 00:29:57.320 --> 00:30:00.760 align:middle line:84% Now, let me try to divide it into two-- 00:30:00.760 --> 00:30:04.410 align:middle line:84% let me try to divide it into some subpieces. 00:30:04.410 --> 00:30:09.150 align:middle line:90% 00:30:09.150 --> 00:30:17.030 align:middle line:90% So I begin with a lemma. 00:30:17.030 --> 00:30:26.430 align:middle line:84% So I want to begin with a lemma that is the following. 00:30:26.430 --> 00:30:35.360 align:middle line:90% So I have a my metric space. 00:30:35.360 --> 00:30:39.150 align:middle line:90% 00:30:39.150 --> 00:30:44.900 align:middle line:84% I have a here is the subset that is compact. 00:30:44.900 --> 00:30:50.350 align:middle line:90% 00:30:50.350 --> 00:30:55.250 align:middle line:84% And then I have as a family of nested subsets, 00:30:55.250 --> 00:30:59.420 align:middle line:84% so let me just talk about nested subsets for a second. 00:30:59.420 --> 00:31:05.550 align:middle line:90% 00:31:05.550 --> 00:31:08.350 align:middle line:90% So I'll do this on this board. 00:31:08.350 --> 00:31:10.980 align:middle line:84% So what do I mean by nested subset? 00:31:10.980 --> 00:31:19.070 align:middle line:90% 00:31:19.070 --> 00:31:28.110 align:middle line:84% So again, I have some set, and then I have some subsets, c. 00:31:28.110 --> 00:31:34.470 align:middle line:84% It's a family, but it's now a family that 00:31:34.470 --> 00:31:38.990 align:middle line:90% is indexed by the integers. 00:31:38.990 --> 00:31:44.310 align:middle line:90% So Ci here, is subset. 00:31:44.310 --> 00:31:49.770 align:middle line:84% So this is i equal to 1, 2, and so on. 00:31:49.770 --> 00:31:53.890 align:middle line:90% 00:31:53.890 --> 00:31:58.070 align:middle line:84% In this case here, I am thinking about 00:31:58.070 --> 00:32:02.500 align:middle line:84% that the next subset is smaller than the previous, 00:32:02.500 --> 00:32:07.260 align:middle line:84% smaller or at most equal to the previous. 00:32:07.260 --> 00:32:09.500 align:middle line:90% So I have this. 00:32:09.500 --> 00:32:11.860 align:middle line:84% I could also have looked at where 00:32:11.860 --> 00:32:13.840 align:middle line:84% they were kind of increasing the subsets, 00:32:13.840 --> 00:32:18.030 align:middle line:84% but this here is what I have in mind. 00:32:18.030 --> 00:32:21.100 align:middle line:90% 00:32:21.100 --> 00:32:27.800 align:middle line:84% So the sets as I become larger, the set becomes smaller. 00:32:27.800 --> 00:32:30.500 align:middle line:90% So now back to the lemma. 00:32:30.500 --> 00:32:32.380 align:middle line:90% I have this metric space. 00:32:32.380 --> 00:32:34.340 align:middle line:90% I have a compact subset. 00:32:34.340 --> 00:32:42.860 align:middle line:84% And then I have a sequence here of Ci's is a sequence-- is 00:32:42.860 --> 00:32:53.780 align:middle line:90% a nested sequence of subsets. 00:32:53.780 --> 00:32:57.500 align:middle line:84% So again, this means that the next element is 00:32:57.500 --> 00:33:00.970 align:middle line:90% contained in the previous one. 00:33:00.970 --> 00:33:04.310 align:middle line:84% And this nested sequence of subsets, 00:33:04.310 --> 00:33:11.650 align:middle line:84% it's actually a nested sequence of closed subsets, 00:33:11.650 --> 00:33:16.690 align:middle line:90% closed subsets of a. 00:33:16.690 --> 00:33:24.810 align:middle line:84% So they're closed in x, and they're contained in a. 00:33:24.810 --> 00:33:29.850 align:middle line:90% And then I am assuming that-- 00:33:29.850 --> 00:33:42.570 align:middle line:90% and assume also that the Ci's-- 00:33:42.570 --> 00:33:46.010 align:middle line:84% that none of the Ci's are the empty sets. 00:33:46.010 --> 00:33:52.410 align:middle line:84% So I don't want any of them to be trivial. 00:33:52.410 --> 00:33:53.790 align:middle line:90% Then the claim is-- 00:33:53.790 --> 00:33:59.610 align:middle line:84% so the claim is that if that's the case, 00:33:59.610 --> 00:34:05.570 align:middle line:84% then actually, if you take the intersection of all of these, 00:34:05.570 --> 00:34:08.070 align:middle line:90% this here is non-empty. 00:34:08.070 --> 00:34:13.690 align:middle line:84% So there is something that is in all of the Ci's. 00:34:13.690 --> 00:34:16.889 align:middle line:90% That's what I want to prove. 00:34:16.889 --> 00:34:21.230 align:middle line:84% And in fact, this is quite straightforward. 00:34:21.230 --> 00:34:25.330 align:middle line:90% 00:34:25.330 --> 00:34:29.650 align:middle line:90% And because since the-- 00:34:29.650 --> 00:34:42.730 align:middle line:84% so the proof here, since Ci is closed, 00:34:42.730 --> 00:34:48.429 align:middle line:84% this means that the complement here is open. 00:34:48.429 --> 00:34:54.290 align:middle line:90% 00:34:54.290 --> 00:34:57.520 align:middle line:84% And now for the complements here-- 00:34:57.520 --> 00:35:00.560 align:middle line:84% so the complement, let's call it-- 00:35:00.560 --> 00:35:05.040 align:middle line:84% the compliments, let's call it Bi. 00:35:05.040 --> 00:35:06.820 align:middle line:90% So Bi is open again. 00:35:06.820 --> 00:35:14.600 align:middle line:84% And Bi plus 1, they actually have this property 00:35:14.600 --> 00:35:18.840 align:middle line:84% that they are because the previous ones was nested 00:35:18.840 --> 00:35:21.040 align:middle line:90% where they became smaller. 00:35:21.040 --> 00:35:23.540 align:middle line:84% Then when you're looking at the complements, 00:35:23.540 --> 00:35:24.540 align:middle line:90% then they become larger. 00:35:24.540 --> 00:35:30.800 align:middle line:90% 00:35:30.800 --> 00:35:35.120 align:middle line:84% So since this is closed, then we have this. 00:35:35.120 --> 00:35:40.400 align:middle line:84% If the intersection of these Ci, if there 00:35:40.400 --> 00:35:44.720 align:middle line:84% was nothing in the intersection of all of them, 00:35:44.720 --> 00:35:49.000 align:middle line:84% if this here was the empty set, but then it 00:35:49.000 --> 00:35:56.920 align:middle line:84% means that the complement of these guys would then cover a. 00:35:56.920 --> 00:35:59.880 align:middle line:84% It would actually bind the cover because there's nothing-- 00:35:59.880 --> 00:36:02.280 align:middle line:90% the union of the complement. 00:36:02.280 --> 00:36:04.460 align:middle line:84% So the union of the complement here. 00:36:04.460 --> 00:36:07.240 align:middle line:90% 00:36:07.240 --> 00:36:14.080 align:middle line:84% They would actually cover x, but it would in particular cover a. 00:36:14.080 --> 00:36:17.240 align:middle line:90% So this here called bi. 00:36:17.240 --> 00:36:20.280 align:middle line:90% So you have that. 00:36:20.280 --> 00:36:27.400 align:middle line:84% So this mean-- and again, this here, this is the same as bi. 00:36:27.400 --> 00:36:30.250 align:middle line:90% So now since a is compact-- 00:36:30.250 --> 00:36:37.080 align:middle line:90% 00:36:37.080 --> 00:36:49.240 align:middle line:84% since a is compact, this means that a here is contained 00:36:49.240 --> 00:36:53.200 align:middle line:90% in finitely many of these. 00:36:53.200 --> 00:36:59.730 align:middle line:84% Let's call them-- so you have finitely many like this 00:36:59.730 --> 00:37:01.450 align:middle line:90% of the B's. 00:37:01.450 --> 00:37:05.270 align:middle line:84% And I'm not sure whether it's the first, 00:37:05.270 --> 00:37:08.930 align:middle line:84% but there are finitely many of these that cover it. 00:37:08.930 --> 00:37:11.750 align:middle line:90% 00:37:11.750 --> 00:37:15.430 align:middle line:84% And so in particular by one of these 00:37:15.430 --> 00:37:18.470 align:middle line:84% is like by one of the indices here 00:37:18.470 --> 00:37:21.390 align:middle line:90% is the largest one of those. 00:37:21.390 --> 00:37:24.430 align:middle line:84% And so this means that it's actually contained 00:37:24.430 --> 00:37:26.310 align:middle line:90% in the largest of these. 00:37:26.310 --> 00:37:33.270 align:middle line:84% Let's say that the largest of these is Bn. 00:37:33.270 --> 00:37:36.950 align:middle line:90% So you have this for some n. 00:37:36.950 --> 00:37:40.990 align:middle line:84% But this here, remember that this here was 00:37:40.990 --> 00:37:46.950 align:middle line:90% the complement of Cn like this. 00:37:46.950 --> 00:37:53.390 align:middle line:84% So this means that a have nothing in Cn. 00:37:53.390 --> 00:37:54.710 align:middle line:90% But a was a-- 00:37:54.710 --> 00:38:00.350 align:middle line:90% Cn intersected with a. 00:38:00.350 --> 00:38:01.590 align:middle line:90% It's empty. 00:38:01.590 --> 00:38:04.390 align:middle line:90% But Cn was contained in a. 00:38:04.390 --> 00:38:13.310 align:middle line:84% So this means that since Cn was assumed to be contained in a, 00:38:13.310 --> 00:38:19.510 align:middle line:84% these were subsets of a, then this means that Cn-- 00:38:19.510 --> 00:38:24.190 align:middle line:84% so this means that this here then implies 00:38:24.190 --> 00:38:30.950 align:middle line:84% that this thing here, is this Cn. 00:38:30.950 --> 00:38:34.350 align:middle line:84% So you have that one of them was empty, 00:38:34.350 --> 00:38:37.620 align:middle line:84% but that we assume that none of the sets of Cn was empty. 00:38:37.620 --> 00:38:42.290 align:middle line:90% 00:38:42.290 --> 00:38:43.270 align:middle line:90% Was that clear? 00:38:43.270 --> 00:38:45.994 align:middle line:90% Sorry, Yeah? 00:38:45.994 --> 00:38:48.882 align:middle line:84% AUDIENCE: Could it be like a sequence of Cis? 00:38:48.882 --> 00:38:50.090 align:middle line:90% TOBIAS COLDING: That's right. 00:38:50.090 --> 00:38:54.400 align:middle line:84% So I'm trying to prove this theorem here. 00:38:54.400 --> 00:39:00.900 align:middle line:84% And it's not super complicated, but it's a little bit involved. 00:39:00.900 --> 00:39:02.660 align:middle line:90% And so I'm trying to-- 00:39:02.660 --> 00:39:04.780 align:middle line:84% of course, this is always a great idea, 00:39:04.780 --> 00:39:08.860 align:middle line:84% and it's kind of like an important thing in this class, 00:39:08.860 --> 00:39:14.500 align:middle line:84% and also if you take other more advanced math classes, 00:39:14.500 --> 00:39:18.780 align:middle line:84% is that when you're trying to prove something, 00:39:18.780 --> 00:39:21.040 align:middle line:84% there are things that are really easy to prove, 00:39:21.040 --> 00:39:24.100 align:middle line:84% but then there are things that are much more complicated. 00:39:24.100 --> 00:39:26.700 align:middle line:84% So one thing is that you try to prove it. 00:39:26.700 --> 00:39:30.680 align:middle line:84% Another thing is that you try to explain to other the proof. 00:39:30.680 --> 00:39:36.980 align:middle line:84% And so in either case, if you can divide it into substeps-- 00:39:36.980 --> 00:39:42.660 align:middle line:84% and so this here is a substep in the proof. 00:39:42.660 --> 00:39:47.720 align:middle line:84% And so in a minute, we will see why this here is relevant. 00:39:47.720 --> 00:39:50.100 align:middle line:84% And so this answers your question. 00:39:50.100 --> 00:39:52.280 align:middle line:90% I mean, how does this come up? 00:39:52.280 --> 00:39:55.540 align:middle line:84% And in a minute, you'll see how it comes up. 00:39:55.540 --> 00:39:59.067 align:middle line:84% So let me get back to that question in just a minute. 00:39:59.067 --> 00:40:01.400 align:middle line:84% AUDIENCE: Are you able to explain the last remark again? 00:40:01.400 --> 00:40:02.275 align:middle line:90% TOBIAS COLDING: Yeah. 00:40:02.275 --> 00:40:03.460 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:40:03.460 --> 00:40:10.780 align:middle line:84% TOBIAS COLDING: So again, the Bi's are open 00:40:10.780 --> 00:40:14.580 align:middle line:84% and using compactness because all of the Bi's 00:40:14.580 --> 00:40:18.940 align:middle line:84% cover a then finally minimize coverage. 00:40:18.940 --> 00:40:23.020 align:middle line:84% But the Bi's are nested in this way 00:40:23.020 --> 00:40:25.660 align:middle line:84% that they are increasing as subsets. 00:40:25.660 --> 00:40:28.220 align:middle line:84% So if I'm thinking about maybe this here 00:40:28.220 --> 00:40:31.140 align:middle line:84% is the one where the index is large, 00:40:31.140 --> 00:40:33.420 align:middle line:90% this then contain all the other. 00:40:33.420 --> 00:40:36.130 align:middle line:84% So it's actually equal to in that case. 00:40:36.130 --> 00:40:36.880 align:middle line:90% It would be equal. 00:40:36.880 --> 00:40:40.940 align:middle line:90% This really is equal to this. 00:40:40.940 --> 00:40:45.620 align:middle line:84% So this means that a is contained 00:40:45.620 --> 00:40:49.890 align:middle line:90% in the complement of Cn. 00:40:49.890 --> 00:40:57.650 align:middle line:84% So in other words, there is no element in a that are in Cn. 00:40:57.650 --> 00:40:58.850 align:middle line:90% That's what it means. 00:40:58.850 --> 00:41:04.370 align:middle line:84% So there is no element in a that is in Cn This thing here, 00:41:04.370 --> 00:41:06.250 align:middle line:90% is empty. 00:41:06.250 --> 00:41:10.290 align:middle line:84% But the Cn's, remember that the Cn's was 00:41:10.290 --> 00:41:14.690 align:middle line:90% assumed to be subsets of An. 00:41:14.690 --> 00:41:21.650 align:middle line:84% So this means that there is nothing in Cn that is not in a. 00:41:21.650 --> 00:41:26.930 align:middle line:84% So this is really just equal to Cn. 00:41:26.930 --> 00:41:29.822 align:middle line:84% AUDIENCE: So is that essentially just creating a contradiction? 00:41:29.822 --> 00:41:31.030 align:middle line:90% TOBIAS COLDING: That's right. 00:41:31.030 --> 00:41:32.890 align:middle line:84% And that's a contradiction because now 00:41:32.890 --> 00:41:35.655 align:middle line:84% let's show that this one here, this Cn is empty. 00:41:35.655 --> 00:41:36.530 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:41:36.530 --> 00:41:38.238 align:middle line:84% TOBIAS COLDING: But they were all assumed 00:41:38.238 --> 00:41:42.090 align:middle line:90% to be non-empty, yeah, OK. 00:41:42.090 --> 00:41:45.810 align:middle line:90% So this is like-- 00:41:45.810 --> 00:41:47.252 align:middle line:90% yeah, sorry, 00:41:47.252 --> 00:41:48.710 align:middle line:84% AUDIENCE: I have a question about-- 00:41:48.710 --> 00:41:50.550 align:middle line:84% when you think it's a contradiction, 00:41:50.550 --> 00:41:55.450 align:middle line:84% you said If it is the case that the intersection of the Ci's is, 00:41:55.450 --> 00:41:59.490 align:middle line:84% in fact, empty, then is contained in them. 00:41:59.490 --> 00:42:00.170 align:middle line:90% Why? 00:42:00.170 --> 00:42:07.610 align:middle line:84% TOBIAS COLDING: OK, so if the intersection of the Ci-- 00:42:07.610 --> 00:42:09.990 align:middle line:90% so here, sorry. 00:42:09.990 --> 00:42:11.650 align:middle line:90% So I want to prove this. 00:42:11.650 --> 00:42:15.170 align:middle line:84% So I'm assuming that this is the case. 00:42:15.170 --> 00:42:19.410 align:middle line:84% And I define the Bi's to be the complement of the Ci's. 00:42:19.410 --> 00:42:27.610 align:middle line:84% So if there is nothing in the intersection of all of the Ci's, 00:42:27.610 --> 00:42:31.290 align:middle line:84% then it means that the union of the complements 00:42:31.290 --> 00:42:33.120 align:middle line:90% must cover the whole set. 00:42:33.120 --> 00:42:36.310 align:middle line:90% 00:42:36.310 --> 00:42:39.350 align:middle line:90% So if you take anything-- 00:42:39.350 --> 00:42:41.850 align:middle line:84% even in x, if you take anything in x, 00:42:41.850 --> 00:42:44.800 align:middle line:90% it's not in all of the Ci's. 00:42:44.800 --> 00:42:48.840 align:middle line:84% So this means that there is some i so that it's 00:42:48.840 --> 00:42:50.800 align:middle line:90% in the complement of that Ci. 00:42:50.800 --> 00:42:55.486 align:middle line:90% But that's exactly this. 00:42:55.486 --> 00:42:56.028 align:middle line:90% AUDIENCE: OK. 00:42:56.028 --> 00:42:58.800 align:middle line:90% 00:42:58.800 --> 00:43:01.500 align:middle line:84% TOBIAS COLDING: So now you're asking why is this relevant? 00:43:01.500 --> 00:43:03.760 align:middle line:90% And that's a good question. 00:43:03.760 --> 00:43:07.180 align:middle line:84% And I will answer it in a minute. 00:43:07.180 --> 00:43:09.400 align:middle line:90% But I will kind of elaborate. 00:43:09.400 --> 00:43:12.480 align:middle line:84% I will give you an example where I have it, 00:43:12.480 --> 00:43:17.020 align:middle line:84% and then we will eventually see how to get to that example. 00:43:17.020 --> 00:43:23.700 align:middle line:84% So the next thing is that suppose I take a-- 00:43:23.700 --> 00:43:26.760 align:middle line:90% 00:43:26.760 --> 00:43:29.360 align:middle line:90% suppose I have the same setup. 00:43:29.360 --> 00:43:36.320 align:middle line:84% But now my sets are the closed sets. 00:43:36.320 --> 00:43:38.420 align:middle line:90% So I have this setup again. 00:43:38.420 --> 00:43:47.750 align:middle line:84% So this is like a corner Of the lemma that we just proved. 00:43:47.750 --> 00:43:50.442 align:middle line:90% So we have x, d. 00:43:50.442 --> 00:43:51.860 align:middle line:90% This is metric space. 00:43:51.860 --> 00:43:55.230 align:middle line:90% 00:43:55.230 --> 00:43:59.802 align:middle line:84% And we have a here is a subset of x. 00:43:59.802 --> 00:44:00.920 align:middle line:90% Ut is compact. 00:44:00.920 --> 00:44:03.590 align:middle line:90% 00:44:03.590 --> 00:44:10.110 align:middle line:84% And now my sets Ci's is going to be-- 00:44:10.110 --> 00:44:20.110 align:middle line:84% so Ci's here, these are going to be balls of radius Ri and center 00:44:20.110 --> 00:44:22.070 align:middle line:90% Xi. 00:44:22.070 --> 00:44:27.430 align:middle line:84% And there will be balls like this. 00:44:27.430 --> 00:44:32.070 align:middle line:84% And I'm assuming that they are nested. 00:44:32.070 --> 00:44:37.670 align:middle line:84% So in the same way, I'm assuming that this thing here, 00:44:37.670 --> 00:44:39.110 align:middle line:90% the centers change. 00:44:39.110 --> 00:44:43.590 align:middle line:84% The radii and the center change is depending on i. 00:44:43.590 --> 00:44:45.670 align:middle line:84% They are all these closed balls, where 00:44:45.670 --> 00:44:47.870 align:middle line:90% it's less or equal to the i. 00:44:47.870 --> 00:44:50.910 align:middle line:84% So I'm assuming that the next one is 00:44:50.910 --> 00:44:53.250 align:middle line:90% contained in the previous one. 00:44:53.250 --> 00:44:59.950 align:middle line:90% 00:44:59.950 --> 00:45:05.270 align:middle line:84% And then I'm making one more assumption 00:45:05.270 --> 00:45:09.100 align:middle line:90% is that the Ri goes to 0. 00:45:09.100 --> 00:45:11.816 align:middle line:90% 00:45:11.816 --> 00:45:13.950 align:middle line:90% So I have that. 00:45:13.950 --> 00:45:18.750 align:middle line:84% So just let's observe that these sets here, 00:45:18.750 --> 00:45:20.730 align:middle line:90% they are nested by assumption. 00:45:20.730 --> 00:45:21.790 align:middle line:90% They are closed. 00:45:21.790 --> 00:45:25.010 align:middle line:90% Because it's this closed ball. 00:45:25.010 --> 00:45:27.470 align:middle line:90% So they're closed subsets. 00:45:27.470 --> 00:45:32.790 align:middle line:84% Also observe that each of these balls contain the center. 00:45:32.790 --> 00:45:36.850 align:middle line:84% So this means that each of these balls is not empty. 00:45:36.850 --> 00:45:39.100 align:middle line:84% So it's automatic that they're non-empty. 00:45:39.100 --> 00:45:41.740 align:middle line:90% 00:45:41.740 --> 00:45:47.940 align:middle line:84% So now I claim that the intersection 00:45:47.940 --> 00:45:53.290 align:middle line:84% of all of these balls is just one point. 00:45:53.290 --> 00:45:57.660 align:middle line:90% 00:45:57.660 --> 00:46:12.730 align:middle line:84% So now the claim here that the intersection of these balls 00:46:12.730 --> 00:46:21.410 align:middle line:84% here, these close balls here, is a single point. 00:46:21.410 --> 00:46:32.100 align:middle line:90% 00:46:32.100 --> 00:46:32.820 align:middle line:90% Yeah? 00:46:32.820 --> 00:46:34.778 align:middle line:84% AUDIENCE: How do we know that each of the balls 00:46:34.778 --> 00:46:35.820 align:middle line:90% contain the center? 00:46:35.820 --> 00:46:39.310 align:middle line:84% TOBIAS COLDING: Because if you take a ball-- 00:46:39.310 --> 00:46:43.820 align:middle line:90% 00:46:43.820 --> 00:46:48.780 align:middle line:84% so if you take a ball here, in any metric space, 00:46:48.780 --> 00:46:52.580 align:middle line:84% this is the y in this space, in this 00:46:52.580 --> 00:46:57.460 align:middle line:84% set so that the distance from x to-- now 00:46:57.460 --> 00:47:01.020 align:middle line:84% it's take the open one but similarly with a closed, 00:47:01.020 --> 00:47:05.740 align:middle line:84% this thing here is less than-- sorry, this here is 1. 00:47:05.740 --> 00:47:08.460 align:middle line:84% AUDIENCE: It contains its own center but not one single point 00:47:08.460 --> 00:47:09.440 align:middle line:90% across all the balls. 00:47:09.440 --> 00:47:11.023 align:middle line:84% TOBIAS COLDING: Yeah, so that's right. 00:47:11.023 --> 00:47:12.120 align:middle line:90% So I'm just saying that-- 00:47:12.120 --> 00:47:17.540 align:middle line:84% so if you plug-in x for y, you get 0. 00:47:17.540 --> 00:47:23.580 align:middle line:84% So this here x definite is in here. 00:47:23.580 --> 00:47:27.780 align:middle line:84% And I'm just saying that each of these guys is non-empty. 00:47:27.780 --> 00:47:32.380 align:middle line:90% 00:47:32.380 --> 00:47:35.010 align:middle line:84% And then I want to claim that actually there 00:47:35.010 --> 00:47:38.970 align:middle line:84% is something in the intersection of all of these balls. 00:47:38.970 --> 00:47:41.910 align:middle line:84% And I want to claim that there's only one such thing. 00:47:41.910 --> 00:47:45.050 align:middle line:84% There's a unique thing that is in intersection 00:47:45.050 --> 00:47:47.610 align:middle line:90% of all these balls. 00:47:47.610 --> 00:47:59.610 align:middle line:84% So now first, I want to prove that this thing here 00:47:59.610 --> 00:48:01.050 align:middle line:90% is non-empty. 00:48:01.050 --> 00:48:07.490 align:middle line:84% And this should have been the closed balls like this. 00:48:07.490 --> 00:48:20.160 align:middle line:84% But you see this just follows from the lemma. 00:48:20.160 --> 00:48:23.250 align:middle line:90% 00:48:23.250 --> 00:48:45.230 align:middle line:84% Because each B-- because each Bri xi is closed and non-empty. 00:48:45.230 --> 00:48:52.930 align:middle line:90% 00:48:52.930 --> 00:48:56.850 align:middle line:90% Each is closed non-empty. 00:48:56.850 --> 00:49:08.610 align:middle line:84% And the family or the sequence of balls 00:49:08.610 --> 00:49:13.330 align:middle line:90% Is nested by assumption. 00:49:13.330 --> 00:49:17.850 align:middle line:84% By assumption we assume that this thing here 00:49:17.850 --> 00:49:22.410 align:middle line:90% is our i plus 1, xi plus 1. 00:49:22.410 --> 00:49:27.150 align:middle line:84% That this thing here is contained in the previous. 00:49:27.150 --> 00:49:33.800 align:middle line:84% So by assumption-- so again, we use the lemma, 00:49:33.800 --> 00:49:35.840 align:middle line:90% and this is the Ci's. 00:49:35.840 --> 00:49:38.720 align:middle line:84% And the Ci's are closed, and they are indeed 00:49:38.720 --> 00:49:40.520 align:middle line:90% nested as they should be. 00:49:40.520 --> 00:49:43.440 align:middle line:84% And they are all non-empty just because they 00:49:43.440 --> 00:49:48.040 align:middle line:90% each contain their own centers. 00:49:48.040 --> 00:49:48.760 align:middle line:90% Yeah? 00:49:48.760 --> 00:49:50.880 align:middle line:84% AUDIENCE: So by making that assumption that they're nested-- 00:49:50.880 --> 00:49:51.160 align:middle line:90% TOBIAS COLDING: Yeah. 00:49:51.160 --> 00:49:53.480 align:middle line:84% AUDIENCE: --are we implicitly making the assumption 00:49:53.480 --> 00:49:56.852 align:middle line:90% about the structure like xi? 00:49:56.852 --> 00:49:58.060 align:middle line:90% TOBIAS COLDING: That's right. 00:49:58.060 --> 00:50:00.300 align:middle line:84% We're making some assumption about-- 00:50:00.300 --> 00:50:08.440 align:middle line:90% 00:50:08.440 --> 00:50:12.960 align:middle line:84% right, we're making some assumption about that-- 00:50:12.960 --> 00:50:16.280 align:middle line:84% when you make this assumption about nested, 00:50:16.280 --> 00:50:20.640 align:middle line:84% there's some assumption about the centers, the distance 00:50:20.640 --> 00:50:25.240 align:middle line:84% between centers, and there's also something about the radii. 00:50:25.240 --> 00:50:28.700 align:middle line:84% How large is one radii relative to the other? 00:50:28.700 --> 00:50:37.500 align:middle line:84% Because if you imagine that this here is Bri like this, 00:50:37.500 --> 00:50:42.400 align:middle line:84% and the picture would kind of contain the boundary, and then 00:50:42.400 --> 00:50:46.840 align:middle line:84% the next one-- but the next one may be out here on the side. 00:50:46.840 --> 00:50:52.240 align:middle line:84% But it's supposed to be contained in it. 00:50:52.240 --> 00:50:55.640 align:middle line:84% So you see that the center may have moved. 00:50:55.640 --> 00:50:59.460 align:middle line:84% This here was the xi and the next center may have moved. 00:50:59.460 --> 00:51:01.980 align:middle line:90% It may have shifted quite a bit. 00:51:01.980 --> 00:51:06.600 align:middle line:84% But of course, this here is the Ri radius, 00:51:06.600 --> 00:51:10.100 align:middle line:84% and this here is the Ri plus 1 radius. 00:51:10.100 --> 00:51:15.110 align:middle line:84% So you see that there's some of relation. 00:51:15.110 --> 00:51:19.320 align:middle line:90% 00:51:19.320 --> 00:51:26.040 align:middle line:84% So we have that it just follows from the lemma 00:51:26.040 --> 00:51:30.870 align:middle line:84% that the intersection of all these closed balls is nonempty. 00:51:30.870 --> 00:51:31.710 align:middle line:90% Yeah? 00:51:31.710 --> 00:51:35.230 align:middle line:84% AUDIENCE: Doesn't it just follow from the assumption we made 00:51:35.230 --> 00:51:37.152 align:middle line:90% and that Ri goes to 0? 00:51:37.152 --> 00:51:38.610 align:middle line:84% TOBIAS COLDING: Yeah, that's right. 00:51:38.610 --> 00:51:41.270 align:middle line:90% So it follows now from that i. 00:51:41.270 --> 00:51:44.150 align:middle line:84% That's why we're using that i equal to 0. 00:51:44.150 --> 00:51:48.090 align:middle line:84% Because suppose that you have two elements. 00:51:48.090 --> 00:51:51.870 align:middle line:84% Suppose that you have x here and y. 00:51:51.870 --> 00:51:56.170 align:middle line:84% Suppose that they are in the intersection of these guys here. 00:51:56.170 --> 00:52:01.654 align:middle line:90% 00:52:01.654 --> 00:52:06.950 align:middle line:84% Now look at the distance between x and y. 00:52:06.950 --> 00:52:09.890 align:middle line:90% Well, they are intersections. 00:52:09.890 --> 00:52:14.990 align:middle line:84% So this means that for each i, for each i-- 00:52:14.990 --> 00:52:26.000 align:middle line:84% so for each i, we have that x here, and y is in this guy here. 00:52:26.000 --> 00:52:29.140 align:middle line:90% 00:52:29.140 --> 00:52:34.940 align:middle line:84% So if it's in this ball here for an i, 00:52:34.940 --> 00:52:37.760 align:middle line:84% then you can use the triangle inequality. 00:52:37.760 --> 00:52:43.420 align:middle line:90% 00:52:43.420 --> 00:52:46.540 align:middle line:84% And this thing here is bounded by-- 00:52:46.540 --> 00:52:49.060 align:middle line:90% this here should have been Ri. 00:52:49.060 --> 00:52:51.540 align:middle line:90% This here is bounded by-- 00:52:51.540 --> 00:52:54.260 align:middle line:84% lesser or equal to because I have this bar 00:52:54.260 --> 00:52:59.940 align:middle line:90% --Ri plus Ri, so 2 Ri, 00:52:59.940 --> 00:53:05.980 align:middle line:84% But if this here holds for all i, this goes to 0. 00:53:05.980 --> 00:53:08.140 align:middle line:84% So this means that this implies-- 00:53:08.140 --> 00:53:13.820 align:middle line:84% as you say, this implies that the distance had to be 0. 00:53:13.820 --> 00:53:15.700 align:middle line:84% But this is part of what it means 00:53:15.700 --> 00:53:19.180 align:middle line:84% to be a metric space that if the distance is 0, 00:53:19.180 --> 00:53:22.140 align:middle line:84% then the two points are the same. 00:53:22.140 --> 00:53:26.220 align:middle line:84% So the whole [INAUDIBLE] was really 00:53:26.220 --> 00:53:30.300 align:middle line:90% quite easy from the lemma. 00:53:30.300 --> 00:53:32.300 align:middle line:84% And so now the question is-- so this 00:53:32.300 --> 00:53:35.660 align:middle line:84% is your question is, why is this relevant 00:53:35.660 --> 00:53:39.935 align:middle line:90% in this Bolzano-Weierstrass? 00:53:39.935 --> 00:53:42.900 align:middle line:84% I mean, they may be not so interesting, I mean, the lemma 00:53:42.900 --> 00:53:46.540 align:middle line:84% and the [INAUDIBLE] on its own, but there 00:53:46.540 --> 00:53:50.410 align:middle line:84% are tools to prove this Bolzano-Weierstrass theorem. 00:53:50.410 --> 00:53:53.180 align:middle line:90% 00:53:53.180 --> 00:54:03.360 align:middle line:84% So let me try to first explain the idea. 00:54:03.360 --> 00:54:07.500 align:middle line:90% And then we go to the proof. 00:54:07.500 --> 00:54:14.020 align:middle line:90% So we have this metric space. 00:54:14.020 --> 00:54:17.340 align:middle line:90% And we have a subset. 00:54:17.340 --> 00:54:20.606 align:middle line:90% And the subset here is compact. 00:54:20.606 --> 00:54:23.000 align:middle line:84% A subset is assumed to be compact. 00:54:23.000 --> 00:54:26.690 align:middle line:90% 00:54:26.690 --> 00:54:31.690 align:middle line:90% Then we have a sequence Xn. 00:54:31.690 --> 00:54:33.770 align:middle line:90% And the sequence here-- 00:54:33.770 --> 00:54:48.890 align:middle line:84% so this is a sequence contained in A. 00:54:48.890 --> 00:54:53.350 align:middle line:84% And we want to prove that there exists-- 00:54:53.350 --> 00:54:55.050 align:middle line:90% so I want to show-- 00:54:55.050 --> 00:55:06.410 align:middle line:84% want to find a convergent subsequence. 00:55:06.410 --> 00:55:09.490 align:middle line:90% 00:55:09.490 --> 00:55:12.250 align:middle line:84% Remember that here-- I mean, just that there's no 00:55:12.250 --> 00:55:13.570 align:middle line:90% assumption-- 00:55:13.570 --> 00:55:17.810 align:middle line:84% I'm not talking about some Cauchy complete set 00:55:17.810 --> 00:55:19.690 align:middle line:90% space or something like that. 00:55:19.690 --> 00:55:22.610 align:middle line:84% So in this case, I really have to construct the-- 00:55:22.610 --> 00:55:26.250 align:middle line:84% I have to construct the convergence sequence. 00:55:26.250 --> 00:55:29.490 align:middle line:84% But I also have to construct the limits. 00:55:29.490 --> 00:55:34.890 align:middle line:84% And so this is where in particular, like the limit, 00:55:34.890 --> 00:55:36.970 align:middle line:84% this is why we want this thing here 00:55:36.970 --> 00:55:41.730 align:middle line:90% to have a unique intersection. 00:55:41.730 --> 00:55:43.910 align:middle line:90% So that's what I want to do. 00:55:43.910 --> 00:55:46.970 align:middle line:90% So you have this space. 00:55:46.970 --> 00:55:49.170 align:middle line:84% You have the subset that's compact. 00:55:49.170 --> 00:55:52.410 align:middle line:90% And you are given a sequence. 00:55:52.410 --> 00:55:59.770 align:middle line:84% So and before I actually get to the proof, 00:55:59.770 --> 00:56:01.960 align:middle line:84% let me just try to explain the idea. 00:56:01.960 --> 00:56:10.010 align:middle line:90% 00:56:10.010 --> 00:56:14.840 align:middle line:84% So again, this is before the proof. 00:56:14.840 --> 00:56:23.360 align:middle line:90% 00:56:23.360 --> 00:56:25.380 align:middle line:84% So I want to make an observation. 00:56:25.380 --> 00:56:29.080 align:middle line:84% So if I am in this iteration, and suppose 00:56:29.080 --> 00:56:34.960 align:middle line:84% now that I take-- so I have A here, and I take some r, 00:56:34.960 --> 00:56:37.640 align:middle line:84% sigma of r is being really small, 00:56:37.640 --> 00:56:41.400 align:middle line:84% and then do what we just did a little while ago. 00:56:41.400 --> 00:56:44.360 align:middle line:90% We cover. 00:56:44.360 --> 00:56:50.320 align:middle line:84% We're looking at A here is covered. 00:56:50.320 --> 00:56:53.760 align:middle line:84% You can write as A as a union of these balls 00:56:53.760 --> 00:56:58.480 align:middle line:84% here, where the center is anything in A. 00:56:58.480 --> 00:57:03.570 align:middle line:84% So again, that cover A-- and so this means that finitely many-- 00:57:03.570 --> 00:57:06.120 align:middle line:90% 00:57:06.120 --> 00:57:18.750 align:middle line:84% that's finitely many cover A since A is compact. 00:57:18.750 --> 00:57:23.080 align:middle line:90% 00:57:23.080 --> 00:57:31.660 align:middle line:84% So I can write A here as contained in and then r-- 00:57:31.660 --> 00:57:34.037 align:middle line:90% 00:57:34.037 --> 00:57:35.120 align:middle line:90% now, I have to be careful. 00:57:35.120 --> 00:57:37.500 align:middle line:90% Because this is not the x. 00:57:37.500 --> 00:57:41.040 align:middle line:84% Probably, I should use a y here, sorry, just to not 00:57:41.040 --> 00:57:45.960 align:middle line:84% confuse it with the sequence that I have, OK. 00:57:45.960 --> 00:57:58.200 align:middle line:84% And so it's contained in the union of balls like this, Ym. 00:57:58.200 --> 00:58:03.160 align:middle line:84% Y is contained in the union of finitely many balls like this. 00:58:03.160 --> 00:58:06.320 align:middle line:84% OK, so far we haven't used our sequence. 00:58:06.320 --> 00:58:12.560 align:middle line:84% But now, of course, I can say that we were given a sequence. 00:58:12.560 --> 00:58:16.798 align:middle line:90% We now have finitely many balls. 00:58:16.798 --> 00:58:20.050 align:middle line:84% And there are infinitely many elements in the sequence. 00:58:20.050 --> 00:58:23.290 align:middle line:84% So it means that there is one of these balls, 00:58:23.290 --> 00:58:26.210 align:middle line:84% finitely many balls, so for one of them-- 00:58:26.210 --> 00:58:28.910 align:middle line:90% 00:58:28.910 --> 00:58:32.170 align:middle line:84% for one, maybe more than 1, maybe all of them. 00:58:32.170 --> 00:58:41.590 align:middle line:84% But for at least for one of these balls, 00:58:41.590 --> 00:58:56.220 align:middle line:84% there are infinitely many elements in the sequence. 00:58:56.220 --> 00:58:58.990 align:middle line:90% 00:58:58.990 --> 00:59:01.410 align:middle line:84% Let me be very precise what I mean by that. 00:59:01.410 --> 00:59:04.210 align:middle line:90% 00:59:04.210 --> 00:59:06.850 align:middle line:84% Of course, the sequence could be like a constant sequence. 00:59:06.850 --> 00:59:11.470 align:middle line:84% What I mean that for infinitely many n's Xn 00:59:11.470 --> 00:59:14.270 align:middle line:90% is in one of these balls. 00:59:14.270 --> 00:59:15.790 align:middle line:84% So let me just write that out, Just. 00:59:15.790 --> 00:59:18.030 align:middle line:90% To be totally clear. 00:59:18.030 --> 00:59:33.670 align:middle line:84% So meaning for one of these balls, 00:59:33.670 --> 00:59:44.570 align:middle line:84% there are infinitely many n's such that. 00:59:44.570 --> 00:59:48.150 align:middle line:90% 00:59:48.150 --> 00:59:53.150 align:middle line:90% Xn is in that ball. 00:59:53.150 --> 00:59:59.350 align:middle line:84% And let's say that this is y3, whatever. 00:59:59.350 --> 01:00:01.050 align:middle line:90% So there's infinitely many n's. 01:00:01.050 --> 01:00:03.990 align:middle line:90% 01:00:03.990 --> 01:00:08.430 align:middle line:84% So now, I can just think about that one ball. 01:00:08.430 --> 01:00:12.220 align:middle line:84% And I can just think about the sequence. 01:00:12.220 --> 01:00:14.300 align:middle line:84% Instead of the original sequence, 01:00:14.300 --> 01:00:18.880 align:middle line:84% I think about a subsequence that is contained in it, 01:00:18.880 --> 01:00:22.060 align:middle line:90% So this means that there is a-- 01:00:22.060 --> 01:00:34.500 align:middle line:84% therefore, there is a subsequence 01:00:34.500 --> 01:00:54.610 align:middle line:84% of the given sequence That is contained in this form. 01:00:54.610 --> 01:01:11.220 align:middle line:90% 01:01:11.220 --> 01:01:15.300 align:middle line:84% OK, now, I want to somehow repeat this process. 01:01:15.300 --> 01:01:17.700 align:middle line:90% I want to repeat this process. 01:01:17.700 --> 01:01:20.800 align:middle line:84% And again, I haven't quite started the proof here, 01:01:20.800 --> 01:01:22.600 align:middle line:84% but I'm just trying to explain the idea. 01:01:22.600 --> 01:01:27.380 align:middle line:84% So this is like a step in the proof 01:01:27.380 --> 01:01:30.660 align:middle line:84% that you then for the right radii 01:01:30.660 --> 01:01:34.796 align:middle line:90% has to use over and over again. 01:01:34.796 --> 01:01:40.200 align:middle line:84% And so I want to repeat this process. 01:01:40.200 --> 01:01:44.220 align:middle line:84% And for that, I need to remind you 01:01:44.220 --> 01:01:49.440 align:middle line:84% that if you take a metric space, if you have a metric space, 01:01:49.440 --> 01:02:11.010 align:middle line:84% so recall that if X here is a metric space, 01:02:11.010 --> 01:02:14.690 align:middle line:90% A here is a subset-- 01:02:14.690 --> 01:02:16.120 align:middle line:90% is a compact subset. 01:02:16.120 --> 01:02:24.770 align:middle line:90% 01:02:24.770 --> 01:02:40.570 align:middle line:84% A is a compact subset, and B here is a closed subset of X. 01:02:40.570 --> 01:02:48.170 align:middle line:84% Well then A intersected with B. If it's compact, 01:02:48.170 --> 01:02:51.490 align:middle line:90% it's also closed. 01:02:51.490 --> 01:02:53.410 align:middle line:90% We proved that. 01:02:53.410 --> 01:02:56.630 align:middle line:84% And so this is the intersection of two closed sets. 01:02:56.630 --> 01:02:58.250 align:middle line:90% So this is closed. 01:02:58.250 --> 01:03:00.610 align:middle line:90% This is contained in A. 01:03:00.610 --> 01:03:04.890 align:middle line:84% And now we proved last time that if you take-- 01:03:04.890 --> 01:03:07.300 align:middle line:90% so this here is a closed subset. 01:03:07.300 --> 01:03:07.930 align:middle line:90% This is closed. 01:03:07.930 --> 01:03:11.000 align:middle line:90% 01:03:11.000 --> 01:03:15.140 align:middle line:84% And it's contained in A. Since A is compact, 01:03:15.140 --> 01:03:19.700 align:middle line:84% if you take a closed subset of A, then this is also compact, 01:03:19.700 --> 01:03:21.790 align:middle line:90% so this is also compact. 01:03:21.790 --> 01:03:29.040 align:middle line:90% 01:03:29.040 --> 01:03:37.560 align:middle line:84% So now let's continue that string of thoughts 01:03:37.560 --> 01:03:39.040 align:middle line:90% from over there. 01:03:39.040 --> 01:03:43.910 align:middle line:90% So now we repeat this. 01:03:43.910 --> 01:03:48.280 align:middle line:90% 01:03:48.280 --> 01:03:52.440 align:middle line:90% So I have a set. 01:03:52.440 --> 01:03:54.880 align:middle line:84% Maybe I should draw it even bigger. 01:03:54.880 --> 01:03:59.760 align:middle line:90% I have here this is my space X. 01:03:59.760 --> 01:04:02.040 align:middle line:90% Then I have some subset. 01:04:02.040 --> 01:04:06.760 align:middle line:90% This here is A. A is compact. 01:04:06.760 --> 01:04:11.920 align:middle line:84% I covered it with balls here of radius-- 01:04:11.920 --> 01:04:14.540 align:middle line:84% and I now have that finitely many. 01:04:14.540 --> 01:04:19.760 align:middle line:84% These are balls of radius r around some y's. 01:04:19.760 --> 01:04:23.180 align:middle line:90% 01:04:23.180 --> 01:04:25.160 align:middle line:84% There's finitely many of these balls, 01:04:25.160 --> 01:04:34.600 align:middle line:84% and they cover A Then these sets here are, of course, not closed. 01:04:34.600 --> 01:04:41.900 align:middle line:84% And so I'm looking at this ball here, these finitely many. 01:04:41.900 --> 01:04:43.890 align:middle line:84% And I just take the closed balls. 01:04:43.890 --> 01:04:46.560 align:middle line:90% 01:04:46.560 --> 01:04:50.000 align:middle line:90% Now I want to cover-- 01:04:50.000 --> 01:04:53.040 align:middle line:84% and I found that for one of them, 01:04:53.040 --> 01:04:56.280 align:middle line:84% there was a subsequence that was contained, 01:04:56.280 --> 01:05:00.880 align:middle line:84% in fact, in the open ball, so therefore, also 01:05:00.880 --> 01:05:02.180 align:middle line:90% in the closed ball. 01:05:02.180 --> 01:05:04.810 align:middle line:84% There was a subsequence of the original sequence. 01:05:04.810 --> 01:05:08.430 align:middle line:90% That's a ball I'm focusing on. 01:05:08.430 --> 01:05:12.730 align:middle line:84% Now, this ball, I can then cover by balls of radius-- 01:05:12.730 --> 01:05:22.880 align:middle line:84% so I'm covering this-- so cover this by balls of radius. 01:05:22.880 --> 01:05:27.430 align:middle line:90% 01:05:27.430 --> 01:05:29.870 align:middle line:90% r over 4. 01:05:29.870 --> 01:05:33.870 align:middle line:84% So I'm going to write this thing here in the same way. 01:05:33.870 --> 01:05:36.540 align:middle line:84% I'm going to write this-- this here, I now know-- 01:05:36.540 --> 01:05:44.710 align:middle line:90% 01:05:44.710 --> 01:05:48.010 align:middle line:84% I'm only interested in things that are contained in A. 01:05:48.010 --> 01:05:51.750 align:middle line:84% So I know that if I intersect it with A, 01:05:51.750 --> 01:05:55.650 align:middle line:90% then this here is compact. 01:05:55.650 --> 01:05:57.290 align:middle line:90% I know that this is compact. 01:05:57.290 --> 01:05:59.450 align:middle line:84% That was because this here was closed. 01:05:59.450 --> 01:06:02.190 align:middle line:90% And A was compact. 01:06:02.190 --> 01:06:03.260 align:middle line:90% So this here is compact. 01:06:03.260 --> 01:06:09.670 align:middle line:90% 01:06:09.670 --> 01:06:17.470 align:middle line:84% So this means that if I can write that set here-- 01:06:17.470 --> 01:06:22.990 align:middle line:90% I can write that set as-- 01:06:22.990 --> 01:06:24.780 align:middle line:90% so I can write this set here. 01:06:24.780 --> 01:06:33.110 align:middle line:90% 01:06:33.110 --> 01:06:42.190 align:middle line:84% So I start with A that is compact. 01:06:42.190 --> 01:06:47.270 align:middle line:84% Then I find a ball so that a subsequence 01:06:47.270 --> 01:06:49.030 align:middle line:90% is contained in that ball. 01:06:49.030 --> 01:06:53.270 align:middle line:90% This is with that ball. 01:06:53.270 --> 01:06:54.890 align:middle line:90% Then I'm looking at-- 01:06:54.890 --> 01:06:57.250 align:middle line:84% so this is like the process I start with here. 01:06:57.250 --> 01:06:58.830 align:middle line:90% This is step one. 01:06:58.830 --> 01:07:01.660 align:middle line:84% Step two, I'm looking at this set. 01:07:01.660 --> 01:07:09.140 align:middle line:84% Three, I'm looking at now the closure of this ball intersected 01:07:09.140 --> 01:07:11.340 align:middle line:90% with A. This here is-- 01:07:11.340 --> 01:07:14.780 align:middle line:90% yeah, is compact. 01:07:14.780 --> 01:07:25.580 align:middle line:84% This here contains a subsequence of Xn. 01:07:25.580 --> 01:07:27.190 align:middle line:90% This thing here is compact. 01:07:27.190 --> 01:07:31.020 align:middle line:90% 01:07:31.020 --> 01:07:35.480 align:middle line:84% This means that I can cover it by balls of radius-- 01:07:35.480 --> 01:07:48.920 align:middle line:84% so cover it by balls of radius r over 4. 01:07:48.920 --> 01:07:51.980 align:middle line:84% I want smaller radius, as I'm going along. 01:07:51.980 --> 01:07:55.100 align:middle line:90% So I'm covering it with-- 01:07:55.100 --> 01:07:58.970 align:middle line:84% and in fact, I can cover it by finitely many balls. 01:07:58.970 --> 01:08:07.140 align:middle line:90% 01:08:07.140 --> 01:08:08.680 align:middle line:90% Why finitely many? 01:08:08.680 --> 01:08:10.600 align:middle line:90% This is because this is compact. 01:08:10.600 --> 01:08:14.900 align:middle line:84% I'm writing this here as the union of balls of this radius, 01:08:14.900 --> 01:08:18.220 align:middle line:84% where anything in this set is a center. 01:08:18.220 --> 01:08:22.600 align:middle line:84% And so I have the finitely many of those covered by compactness. 01:08:22.600 --> 01:08:25.974 align:middle line:90% 01:08:25.974 --> 01:08:31.260 align:middle line:84% And so now what I observe, is that if I take-- and now I just 01:08:31.260 --> 01:08:33.580 align:middle line:90% had to be a little bit careful. 01:08:33.580 --> 01:08:40.020 align:middle line:84% So now I'm observing that if I take the ball of radius-- 01:08:40.020 --> 01:08:45.939 align:middle line:84% the double ball, suppose I'm taking this ball here, 01:08:45.939 --> 01:08:49.979 align:middle line:84% so I get finitely-- sorry, no, no, before that. 01:08:49.979 --> 01:08:54.800 align:middle line:84% So I have that it's now finitely many balls cover this. 01:08:54.800 --> 01:09:00.830 align:middle line:84% So this means that a subsequence must lie in one of these balls. 01:09:00.830 --> 01:09:04.609 align:middle line:90% So this is the fifth step. 01:09:04.609 --> 01:09:21.569 align:middle line:84% A subsequence of the subsequence lies 01:09:21.569 --> 01:09:43.430 align:middle line:84% in one of these finitely many balls of radius r over 4. 01:09:43.430 --> 01:09:47.521 align:middle line:90% 01:09:47.521 --> 01:09:57.850 align:middle line:90% And now I take-- 01:09:57.850 --> 01:10:00.490 align:middle line:90% So the next thing is I do-- 01:10:00.490 --> 01:10:04.330 align:middle line:84% I want to apply the collar from before. 01:10:04.330 --> 01:10:14.770 align:middle line:84% So I started with my space X. I got this ball here of radius r, 01:10:14.770 --> 01:10:17.850 align:middle line:90% and then there was some center. 01:10:17.850 --> 01:10:25.901 align:middle line:84% Then the next thing is that I get a ball of radius r. 01:10:25.901 --> 01:10:30.810 align:middle line:90% 01:10:30.810 --> 01:10:36.210 align:middle line:84% And now this is with one of these other finitely many balls, 01:10:36.210 --> 01:10:40.810 align:middle line:84% where a subsequence is called it's like this, maybe like one 01:10:40.810 --> 01:10:43.730 align:middle line:90% or something like that. 01:10:43.730 --> 01:10:52.160 align:middle line:84% So the balls of radius r over 4, they cover this. 01:10:52.160 --> 01:10:56.048 align:middle line:84% So a subsequence action would have to lie here. 01:10:56.048 --> 01:10:58.900 align:middle line:90% 01:10:58.900 --> 01:11:02.280 align:middle line:84% But now I want to use the corner. 01:11:02.280 --> 01:11:07.160 align:middle line:84% If I need to use the corner, I need to have nested 01:11:07.160 --> 01:11:09.040 align:middle line:90% and I need to have closed sets. 01:11:09.040 --> 01:11:14.880 align:middle line:84% And so in order to make them nested, then I double-- 01:11:14.880 --> 01:11:17.200 align:middle line:90% I'm going to double this ball. 01:11:17.200 --> 01:11:19.200 align:middle line:90% Why do I double this ball? 01:11:19.200 --> 01:11:24.040 align:middle line:84% Well, I double this ball here because these guys here 01:11:24.040 --> 01:11:26.022 align:middle line:90% cover this. 01:11:26.022 --> 01:11:27.480 align:middle line:84% But it means, of course, that there 01:11:27.480 --> 01:11:29.600 align:middle line:90% could be something outside. 01:11:29.600 --> 01:11:31.800 align:middle line:90% There could be. 01:11:31.800 --> 01:11:35.800 align:middle line:84% So this is like a ball of radius r. 01:11:35.800 --> 01:11:39.860 align:middle line:84% And so then you have this ball of radius a quarter r. 01:11:39.860 --> 01:11:43.720 align:middle line:84% But the quarter balls, they may have some part outside 01:11:43.720 --> 01:11:45.180 align:middle line:90% and some part inside. 01:11:45.180 --> 01:11:49.280 align:middle line:90% So it's not this ball here. 01:11:49.280 --> 01:11:51.680 align:middle line:90% It's not contained in that ball. 01:11:51.680 --> 01:11:55.840 align:middle line:84% But if you went to the double ball, 01:11:55.840 --> 01:11:58.900 align:middle line:84% if you went to this ball of radius 2r, 01:11:58.900 --> 01:12:07.900 align:middle line:84% you see this point here, the center here, it would lie-- 01:12:07.900 --> 01:12:11.000 align:middle line:84% there would be some points in this ball here 01:12:11.000 --> 01:12:13.000 align:middle line:90% that would be within. 01:12:13.000 --> 01:12:15.560 align:middle line:84% It may not be the center, but there's some point here 01:12:15.560 --> 01:12:19.440 align:middle line:90% that's within r of the center. 01:12:19.440 --> 01:12:21.920 align:middle line:84% And then by the triangle inequality, 01:12:21.920 --> 01:12:25.640 align:middle line:84% any other points is actually within r 2, 01:12:25.640 --> 01:12:28.760 align:middle line:90% r over 2 of that point. 01:12:28.760 --> 01:12:31.520 align:middle line:90% So everything is comfortably-- 01:12:31.520 --> 01:12:35.600 align:middle line:84% so this thing here and even the half ball-- 01:12:35.600 --> 01:12:42.520 align:middle line:84% so I can conclude that even the ball of radius 01:12:42.520 --> 01:12:45.660 align:middle line:90% r over 2, even this one here. 01:12:45.660 --> 01:12:48.240 align:middle line:90% 01:12:48.240 --> 01:12:57.650 align:middle line:84% Is contained in the double ball like that. 01:12:57.650 --> 01:13:00.890 align:middle line:90% 01:13:00.890 --> 01:13:07.390 align:middle line:84% And in fact, if you make things less or equal to on both things, 01:13:07.390 --> 01:13:08.410 align:middle line:90% then you have that. 01:13:08.410 --> 01:13:11.985 align:middle line:90% 01:13:11.985 --> 01:13:13.590 align:middle line:90% Now what I have is-- 01:13:13.590 --> 01:13:21.190 align:middle line:84% at least here you have something that is nested. 01:13:21.190 --> 01:13:26.510 align:middle line:84% And you see that the radius went down. 01:13:26.510 --> 01:13:29.470 align:middle line:84% So these are nested, and they are closed. 01:13:29.470 --> 01:13:34.030 align:middle line:84% And so now what we do is-- so now 01:13:34.030 --> 01:13:38.290 align:middle line:84% we want to do this for a sequence. 01:13:38.290 --> 01:13:40.870 align:middle line:84% And then of course, we want to-- so now we 01:13:40.870 --> 01:13:44.530 align:middle line:84% want to construct a sequence of closed balls like this. 01:13:44.530 --> 01:13:47.040 align:middle line:90% 01:13:47.040 --> 01:13:50.340 align:middle line:84% And now when you pick your subsequence, 01:13:50.340 --> 01:13:55.820 align:middle line:84% you can't wait with picking the first element in subsequence. 01:13:55.820 --> 01:13:57.940 align:middle line:84% Remember, if you keep waiting, then you 01:13:57.940 --> 01:13:59.620 align:middle line:90% will end up getting nothing. 01:13:59.620 --> 01:14:02.540 align:middle line:84% So you pick-- at each stage, you pick 01:14:02.540 --> 01:14:04.740 align:middle line:90% one element in the subsequence. 01:14:04.740 --> 01:14:06.680 align:middle line:84% And then when you have the next ball, 01:14:06.680 --> 01:14:11.075 align:middle line:84% you pick the next element in the subsequent and so on. 01:14:11.075 --> 01:14:17.020 align:middle line:90% And so we will do this with-- 01:14:17.020 --> 01:14:19.480 align:middle line:84% so we're going to do this procedure here. 01:14:19.480 --> 01:14:22.880 align:middle line:84% So now we're getting back to the proof of the theorem. 01:14:22.880 --> 01:14:25.380 align:middle line:84% And again, remember the theorem was this-- maybe 01:14:25.380 --> 01:14:27.460 align:middle line:90% I erased it now by mistake. 01:14:27.460 --> 01:14:30.340 align:middle line:84% But the theorem was this Bolzano-Weierstrass theorem 01:14:30.340 --> 01:14:31.700 align:middle line:90% for a metric space. 01:14:31.700 --> 01:14:34.900 align:middle line:84% If you have a metric space, you have a compact subset, 01:14:34.900 --> 01:14:37.900 align:middle line:84% and you have a sequence in this compact subset. 01:14:37.900 --> 01:14:42.100 align:middle line:84% And you want to construct a convergent subsequence. 01:14:42.100 --> 01:14:45.700 align:middle line:90% So what you do is-- 01:14:45.700 --> 01:14:51.900 align:middle line:84% that's a proof of the Bolzano-Weierstrass 01:14:51.900 --> 01:14:52.850 align:middle line:90% for metric spaces. 01:14:52.850 --> 01:14:59.700 align:middle line:90% 01:14:59.700 --> 01:15:02.460 align:middle line:84% So what you do is-- so you have this. 01:15:02.460 --> 01:15:06.700 align:middle line:84% Again, A here is compact, and you have a sequence Xn. 01:15:06.700 --> 01:15:09.100 align:middle line:84% This is a sequence that is contained 01:15:09.100 --> 01:15:13.740 align:middle line:84% in A. What you do is that first you 01:15:13.740 --> 01:15:18.140 align:middle line:90% can just let r be equal to 1. 01:15:18.140 --> 01:15:20.820 align:middle line:90% And then you do this. 01:15:20.820 --> 01:15:24.940 align:middle line:90% You cover it by-- 01:15:24.940 --> 01:15:26.220 align:middle line:90% you cover it. 01:15:26.220 --> 01:15:29.180 align:middle line:84% You have here A. And you cover it 01:15:29.180 --> 01:15:33.940 align:middle line:84% by finitely many balls of radius 1. 01:15:33.940 --> 01:15:37.900 align:middle line:84% And for one of these balls, you have that there's infinitely 01:15:37.900 --> 01:15:40.020 align:middle line:90% many element in the sequence. 01:15:40.020 --> 01:15:43.970 align:middle line:84% So you're going to focus on that ball from here on and out. 01:15:43.970 --> 01:15:45.570 align:middle line:90% And you just pick-- 01:15:45.570 --> 01:15:49.010 align:middle line:84% in your subsequence, you now pick your first element 01:15:49.010 --> 01:15:54.730 align:middle line:84% in the subsequence to be some element in that ball. 01:15:54.730 --> 01:15:59.090 align:middle line:84% So now the picture is that you just look at that ball. 01:15:59.090 --> 01:16:02.810 align:middle line:84% Forget about the other ball and everything else. 01:16:02.810 --> 01:16:09.330 align:middle line:84% You now cover it by balls of radius a quarter. 01:16:09.330 --> 01:16:12.690 align:middle line:90% And you have that-- 01:16:12.690 --> 01:16:16.610 align:middle line:84% again, that's finitely many cover-- 01:16:16.610 --> 01:16:22.250 align:middle line:84% this ball intersected with A, finitely many cover it. 01:16:22.250 --> 01:16:27.650 align:middle line:84% And so one of them has infinitely many elements 01:16:27.650 --> 01:16:32.290 align:middle line:84% in this subsequence you consider. 01:16:32.290 --> 01:16:35.710 align:middle line:84% And so you just pick one element in that. 01:16:35.710 --> 01:16:38.730 align:middle line:90% That's the next one. 01:16:38.730 --> 01:16:41.390 align:middle line:90% And now you repeat this process. 01:16:41.390 --> 01:16:43.870 align:middle line:84% And so you see that for these balls-- 01:16:43.870 --> 01:16:47.690 align:middle line:84% but remember that-- so you're starting with the radius 1. 01:16:47.690 --> 01:16:50.310 align:middle line:90% But you have to double things. 01:16:50.310 --> 01:16:55.050 align:middle line:84% So the sequence of closed sets, the initial closed sets, 01:16:55.050 --> 01:16:57.450 align:middle line:90% will have radius 2. 01:16:57.450 --> 01:17:01.930 align:middle line:84% And the next one won't have radius a quarter, 01:17:01.930 --> 01:17:06.550 align:middle line:84% but it will have radius 2 times a quarter, so one half. 01:17:06.550 --> 01:17:09.630 align:middle line:84% And the next one will then be a ball. 01:17:09.630 --> 01:17:11.610 align:middle line:84% It will have a different center, and it 01:17:11.610 --> 01:17:13.330 align:middle line:90% will have radius one half. 01:17:13.330 --> 01:17:16.050 align:middle line:84% And then the next one you see, at each stage 01:17:16.050 --> 01:17:21.770 align:middle line:84% when I do it like this, then it goes down with a factor of 4 01:17:21.770 --> 01:17:23.070 align:middle line:90% at each stage. 01:17:23.070 --> 01:17:26.910 align:middle line:84% So this here would be 1 over 8 and so on. 01:17:26.910 --> 01:17:32.050 align:middle line:84% And so you see that this sequence, these closed balls, 01:17:32.050 --> 01:17:35.210 align:middle line:84% they are nested as they should be. 01:17:35.210 --> 01:17:38.370 align:middle line:90% And the radii goes to zero. 01:17:38.370 --> 01:17:41.560 align:middle line:84% So this means that there is an element that 01:17:41.560 --> 01:17:46.360 align:middle line:84% lies in all of these, and that is now the limit because 01:17:46.360 --> 01:17:48.800 align:middle line:90% of the subsequence. 01:17:48.800 --> 01:17:53.400 align:middle line:84% From here on and out, it would be in that ball and from it 01:17:53.400 --> 01:17:56.360 align:middle line:84% later on and out, it would be in the next ball. 01:17:56.360 --> 01:18:01.140 align:middle line:84% So that intersection of all of these balls will be your limit. 01:18:01.140 --> 01:18:02.430 align:middle line:90% Yeah? 01:18:02.430 --> 01:18:05.280 align:middle line:90% AUDIENCE: [INAUDIBLE] 01:18:05.280 --> 01:18:10.040 align:middle line:90% 01:18:10.040 --> 01:18:12.120 align:middle line:84% TOBIAS COLDING: So really what you're 01:18:12.120 --> 01:18:18.680 align:middle line:84% looking at that this thing here, it's intersected with A. 01:18:18.680 --> 01:18:23.560 align:middle line:84% So it's just that these balls here, so the intersection here, 01:18:23.560 --> 01:18:28.060 align:middle line:84% that these are the closed sets, and these are the compacts. 01:18:28.060 --> 01:18:30.600 align:middle line:84% They are each of them are non-empty. 01:18:30.600 --> 01:18:35.200 align:middle line:84% And they are the closed subsets of A. 01:18:35.200 --> 01:18:38.200 align:middle line:84% So you're really arguing on these guys here. 01:18:38.200 --> 01:18:40.760 align:middle line:90% You see what I mean? 01:18:40.760 --> 01:18:44.420 align:middle line:84% I know I stated it for balls in the column. 01:18:44.420 --> 01:18:47.800 align:middle line:84% But if you just think about it, the balls 01:18:47.800 --> 01:18:50.560 align:middle line:84% play no particular role, except for saying 01:18:50.560 --> 01:18:54.460 align:middle line:84% that the intersection of all of them contain just one point. 01:18:54.460 --> 01:18:57.180 align:middle line:84% And that was because the radii went to zero. 01:18:57.180 --> 01:19:01.120 align:middle line:84% The important points were that the sets were closed subsets. 01:19:01.120 --> 01:19:03.960 align:middle line:84% Because then there was the lemma was just all about closed 01:19:03.960 --> 01:19:04.500 align:middle line:90% subset. 01:19:04.500 --> 01:19:06.320 align:middle line:90% It had nothing to do with-- 01:19:06.320 --> 01:19:16.040 align:middle line:84% And so you see that these guys here are closed subsets of A. 01:19:16.040 --> 01:19:17.880 align:middle line:90% And so-- 01:19:17.880 --> 01:19:20.204 align:middle line:90% yeah? 01:19:20.204 --> 01:19:21.660 align:middle line:84% AUDIENCE: I don't quite understand 01:19:21.660 --> 01:19:23.640 align:middle line:90% where the [INAUDIBLE]. 01:19:23.640 --> 01:19:26.080 align:middle line:84% Is it, you're looking at the ball of radius 1 01:19:26.080 --> 01:19:29.320 align:middle line:84% and trying to see how to contain the ball of radius 1. 01:19:29.320 --> 01:19:32.495 align:middle line:84% TOBIAS COLDING: You mean why I went to double the ball, no? 01:19:32.495 --> 01:19:33.120 align:middle line:90% AUDIENCE: Yeah. 01:19:33.120 --> 01:19:37.430 align:middle line:84% TOBIAS COLDING: OK, so the reason I go to the double ball, 01:19:37.430 --> 01:19:40.710 align:middle line:84% is that I want to make sure that things are nested. 01:19:40.710 --> 01:19:48.150 align:middle line:84% And if I just did this procedure-- suppose so again, 01:19:48.150 --> 01:19:50.350 align:middle line:90% I take this ball here. 01:19:50.350 --> 01:19:51.650 align:middle line:90% This is the first step. 01:19:51.650 --> 01:19:54.790 align:middle line:90% I have these balls of radius 1. 01:19:54.790 --> 01:19:57.070 align:middle line:84% And I find one of them so that there's 01:19:57.070 --> 01:19:59.270 align:middle line:90% infinite many in the sequence. 01:19:59.270 --> 01:20:02.950 align:middle line:84% Then I want to have something finer. 01:20:02.950 --> 01:20:05.590 align:middle line:90% And so I go to-- 01:20:05.590 --> 01:20:07.790 align:middle line:84% I chose because I think-- you didn't quite 01:20:07.790 --> 01:20:08.930 align:middle line:90% have to do a quarter. 01:20:08.930 --> 01:20:10.930 align:middle line:84% You could actually have gotten away with a half, 01:20:10.930 --> 01:20:13.650 align:middle line:84% but I thought it wasn't as clear if I did it. 01:20:13.650 --> 01:20:15.510 align:middle line:84% So I did it with a quarter because then I 01:20:15.510 --> 01:20:18.070 align:middle line:90% thought it looked more obvious. 01:20:18.070 --> 01:20:23.170 align:middle line:84% So if you now take this ball here, and as you say, 01:20:23.170 --> 01:20:28.150 align:middle line:84% you always have to intersect with A, just 01:20:28.150 --> 01:20:29.470 align:middle line:90% to make things compact. 01:20:29.470 --> 01:20:36.510 align:middle line:84% So I cover it with balls of radius a quarter. 01:20:36.510 --> 01:20:42.250 align:middle line:84% But some of these balls, part of it may lie outside this ball 01:20:42.250 --> 01:20:42.750 align:middle line:90% here. 01:20:42.750 --> 01:20:45.590 align:middle line:90% Because I couldn't cover-- 01:20:45.590 --> 01:20:48.070 align:middle line:84% you see what I mean, if I take a ball here. 01:20:48.070 --> 01:20:51.970 align:middle line:84% I can't cover it by finitely many balls. 01:20:51.970 --> 01:20:54.090 align:middle line:84% Just if it was like Euclidean space, 01:20:54.090 --> 01:20:56.950 align:middle line:84% if I insist that the ball had to be-- this smaller ball 01:20:56.950 --> 01:21:00.170 align:middle line:84% had to be inside this ball, then there wouldn't be finitely many. 01:21:00.170 --> 01:21:04.870 align:middle line:84% So I have to look at balls where some of these balls 01:21:04.870 --> 01:21:07.550 align:middle line:90% is kind of outside the surface. 01:21:07.550 --> 01:21:10.390 align:middle line:84% But now, of course, so that wouldn't really 01:21:10.390 --> 01:21:13.210 align:middle line:84% work because it's not like this ball here, 01:21:13.210 --> 01:21:15.050 align:middle line:84% let alone the double of this ball. 01:21:15.050 --> 01:21:16.910 align:middle line:90% It's not contained in this. 01:21:16.910 --> 01:21:19.390 align:middle line:84% So that's why I doubled that ball. 01:21:19.390 --> 01:21:24.830 align:middle line:84% Then everything is comfortably contained [INAUDIBLE]. 01:21:24.830 --> 01:21:26.950 align:middle line:90% Any other questions? 01:21:26.950 --> 01:21:27.830 align:middle line:90% Yeah? 01:21:27.830 --> 01:21:30.090 align:middle line:84% AUDIENCE: Since they're all closed ball-- or wait, no. 01:21:30.090 --> 01:21:32.860 align:middle line:90% These are open balls? 01:21:32.860 --> 01:21:34.620 align:middle line:84% TOBIAS COLDING: When I cover things, 01:21:34.620 --> 01:21:36.840 align:middle line:84% they are open because I have to have-- 01:21:36.840 --> 01:21:41.380 align:middle line:84% when I have a compact set, I have to cover it by open subset. 01:21:41.380 --> 01:21:44.720 align:middle line:84% But then once I'm looking at the nested things, 01:21:44.720 --> 01:21:47.860 align:middle line:84% I have to go to the closure, because otherwise the lemma 01:21:47.860 --> 01:21:49.560 align:middle line:90% and the corner would not apply. 01:21:49.560 --> 01:21:50.460 align:middle line:90% AUDIENCE: OK. 01:21:50.460 --> 01:21:51.410 align:middle line:90% TOBIAS COLDING: OK. 01:21:51.410 --> 01:21:53.660 align:middle line:84% AUDIENCE: And [INAUDIBLE], if they were closed though, 01:21:53.660 --> 01:21:59.202 align:middle line:84% you would be able to cover like E1 with a-- 01:21:59.202 --> 01:22:00.660 align:middle line:84% TOBIAS COLDING: And I'm just saying 01:22:00.660 --> 01:22:04.620 align:middle line:84% that if the open balls cover it, then of course, 01:22:04.620 --> 01:22:08.140 align:middle line:84% their closure also cover it because the closure is a larger 01:22:08.140 --> 01:22:08.860 align:middle line:90% set. 01:22:08.860 --> 01:22:09.976 align:middle line:90% AUDIENCE: Yeah. 01:22:09.976 --> 01:22:12.580 align:middle line:84% TOBIAS COLDING: But it wouldn't be a priori clear 01:22:12.580 --> 01:22:16.480 align:middle line:84% that-- it's not like they're closed set. 01:22:16.480 --> 01:22:19.480 align:middle line:84% So that it's not working, the definition. 01:22:19.480 --> 01:22:20.860 align:middle line:90% Yeah? 01:22:20.860 --> 01:22:22.700 align:middle line:84% AUDIENCE: I'm not sure if you have proof 01:22:22.700 --> 01:22:27.300 align:middle line:84% that up there we have that if you have a compact subset, 01:22:27.300 --> 01:22:29.645 align:middle line:84% and you take the intersection with a closed subset. 01:22:29.645 --> 01:22:31.020 align:middle line:84% Yeah, we proved that it's closed. 01:22:31.020 --> 01:22:32.168 align:middle line:90% Can we prove it is compact? 01:22:32.168 --> 01:22:33.960 align:middle line:84% TOBIAS COLDING: We prove that it's compact. 01:22:33.960 --> 01:22:37.740 align:middle line:84% And I can prove it again because it's very easy. 01:22:37.740 --> 01:22:55.100 align:middle line:84% So if you have a metric space, this here is compact. 01:22:55.100 --> 01:22:58.540 align:middle line:84% And now you take a subset here of this here. 01:22:58.540 --> 01:23:01.420 align:middle line:90% And this here is closed. 01:23:01.420 --> 01:23:04.980 align:middle line:90% It's a closed subset of X. 01:23:04.980 --> 01:23:10.860 align:middle line:84% Then you look at X take away B. You take any cover of this. 01:23:10.860 --> 01:23:15.360 align:middle line:84% Together with this, they cover A. But then finitely many-- 01:23:15.360 --> 01:23:19.020 align:middle line:84% but this here have nothing in B. So actually finitely many 01:23:19.020 --> 01:23:21.700 align:middle line:90% of the other one cover B here. 01:23:21.700 --> 01:23:24.620 align:middle line:90% Any other questions? 01:23:24.620 --> 01:23:27.230 align:middle line:90% OK, great, thanks. 01:23:27.230 --> 01:23:30.000 align:middle line:90%