WEBVTT 00:00:00.000 --> 00:00:02.490 align:middle line:90% [SQUEAKING] 00:00:02.490 --> 00:00:04.482 align:middle line:90% [RUSTLING] 00:00:04.482 --> 00:00:06.972 align:middle line:90% [CLICKING] 00:00:06.972 --> 00:00:13.460 align:middle line:90% 00:00:13.460 --> 00:00:19.230 align:middle line:84% TOBIAS COLDING: So last time, we looked at power series. 00:00:19.230 --> 00:00:23.210 align:middle line:90% 00:00:23.210 --> 00:00:27.620 align:middle line:90% So you have a sequence a n-- 00:00:27.620 --> 00:00:28.495 align:middle line:90% is a sequence. 00:00:28.495 --> 00:00:31.100 align:middle line:90% 00:00:31.100 --> 00:00:39.170 align:middle line:84% And then for each x, you form the new series-- 00:00:39.170 --> 00:00:40.210 align:middle line:90% you form the series. 00:00:40.210 --> 00:00:46.910 align:middle line:90% 00:00:46.910 --> 00:00:51.600 align:middle line:84% And the series is with an a n times x to the power n. 00:00:51.600 --> 00:00:55.230 align:middle line:84% And then you're summing-- you decide somewhere to start, 00:00:55.230 --> 00:00:59.280 align:middle line:84% and then you're summing to infinity like that. 00:00:59.280 --> 00:01:02.370 align:middle line:84% So something like this-- that's a power series. 00:01:02.370 --> 00:01:06.630 align:middle line:84% And we looked at a particular case where 00:01:06.630 --> 00:01:13.530 align:middle line:84% the a n's-- so example here is where a n is equal to 1 over n 00:01:13.530 --> 00:01:15.480 align:middle line:90% factorial. 00:01:15.480 --> 00:01:20.380 align:middle line:84% And so we got the function E of x, 00:01:20.380 --> 00:01:26.970 align:middle line:84% which is the sum from n equal to 0 to infinity of x to the power 00:01:26.970 --> 00:01:31.500 align:middle line:90% n divided by n factorial. 00:01:31.500 --> 00:01:40.330 align:middle line:90% And then we used this to-- 00:01:40.330 --> 00:01:46.860 align:middle line:84% we observed that E is bigger or equal to 0 00:01:46.860 --> 00:01:51.570 align:middle line:90% for x bigger or equal to 0. 00:01:51.570 --> 00:02:00.770 align:middle line:84% And we also observed that E of 0 is equal to 1. 00:02:00.770 --> 00:02:08.570 align:middle line:84% And we defined literally to be e this function E of 1. 00:02:08.570 --> 00:02:16.550 align:middle line:84% And then we defined another function E to the q, 00:02:16.550 --> 00:02:23.130 align:middle line:84% where q here is now a rational number. 00:02:23.130 --> 00:02:25.590 align:middle line:90% And it was defined-- 00:02:25.590 --> 00:02:30.740 align:middle line:84% in particular, E of 0 was defined to be equal to 1. 00:02:30.740 --> 00:02:39.170 align:middle line:84% E of 1 was just defined to be [INAUDIBLE]. 00:02:39.170 --> 00:02:43.310 align:middle line:84% Then we also-- we assumed, which we will prove today, 00:02:43.310 --> 00:02:55.580 align:middle line:84% that capital E of x plus y is equal to E of x times E of 1. 00:02:55.580 --> 00:02:56.990 align:middle line:90% So we assume that. 00:02:56.990 --> 00:03:04.230 align:middle line:90% 00:03:04.230 --> 00:03:05.860 align:middle line:90% And then we defined-- 00:03:05.860 --> 00:03:10.950 align:middle line:84% so this function based on the little e-- 00:03:10.950 --> 00:03:16.530 align:middle line:84% in particular, e to some power m if m is an integer-- 00:03:16.530 --> 00:03:20.735 align:middle line:90% is just e times itself m times. 00:03:20.735 --> 00:03:25.620 align:middle line:90% 00:03:25.620 --> 00:03:38.550 align:middle line:84% And then we defined e to a power 1/n here to be the positive 00:03:38.550 --> 00:03:55.360 align:middle line:84% number, as it was defined to be a positive number so that alpha 00:03:55.360 --> 00:03:59.608 align:middle line:84% raised to the power n is equal to e. 00:03:59.608 --> 00:04:02.890 align:middle line:90% 00:04:02.890 --> 00:04:09.820 align:middle line:84% And we also defined e to a negative integer. 00:04:09.820 --> 00:04:12.310 align:middle line:84% And this was just to be defined to 1 00:04:12.310 --> 00:04:19.810 align:middle line:84% over-- so we defined e to minus m to 1 over e to the n. 00:04:19.810 --> 00:04:23.650 align:middle line:90% So this way, we defined-- 00:04:23.650 --> 00:04:29.230 align:middle line:84% this function here-- the power series was defined for all x. 00:04:29.230 --> 00:04:33.880 align:middle line:84% And we defined another function little e of q 00:04:33.880 --> 00:04:37.030 align:middle line:90% on all rational numbers q. 00:04:37.030 --> 00:04:40.090 align:middle line:90% And then we saw-- 00:04:40.090 --> 00:04:41.710 align:middle line:90% we assumed this here. 00:04:41.710 --> 00:04:45.980 align:middle line:84% We saw that these two functions-- 00:04:45.980 --> 00:04:51.910 align:middle line:90% so E of x is equal to e. 00:04:51.910 --> 00:04:55.100 align:middle line:84% This function here is only defined on rational numbers, 00:04:55.100 --> 00:05:01.550 align:middle line:84% so they're equal for rational numbers. 00:05:01.550 --> 00:05:04.090 align:middle line:90% So we saw that. 00:05:04.090 --> 00:05:08.800 align:middle line:90% And then in the psets-- 00:05:08.800 --> 00:05:12.610 align:middle line:84% pset 5-- I think it's for both of them-- 00:05:12.610 --> 00:05:13.940 align:middle line:90% you will show-- 00:05:13.940 --> 00:05:16.250 align:middle line:84% Or maybe it's one for 4 and one for 5 00:05:16.250 --> 00:05:18.086 align:middle line:90% to show that E is continuous. 00:05:18.086 --> 00:05:23.950 align:middle line:90% 00:05:23.950 --> 00:05:30.230 align:middle line:84% And you will also show that E is the-- 00:05:30.230 --> 00:05:32.350 align:middle line:84% so it would also show the following-- 00:05:32.350 --> 00:05:37.703 align:middle line:90% 00:05:37.703 --> 00:05:40.180 align:middle line:90% that if you have-- 00:05:40.180 --> 00:05:45.816 align:middle line:84% if f and g are two continuous function-- 00:05:45.816 --> 00:05:54.580 align:middle line:90% 00:05:54.580 --> 00:05:57.220 align:middle line:90% and so they are defined-- 00:05:57.220 --> 00:06:03.340 align:middle line:84% both of them-- both f and g are defined on the real numbers. 00:06:03.340 --> 00:06:08.530 align:middle line:84% And you assume that if you restrict them 00:06:08.530 --> 00:06:14.020 align:middle line:84% to the rational numbers, then the two functions are equal. 00:06:14.020 --> 00:06:17.410 align:middle line:84% So if they're both continuous and they 00:06:17.410 --> 00:06:20.440 align:middle line:84% agree on the rational numbers, then actually, 00:06:20.440 --> 00:06:22.150 align:middle line:90% they must agree everywhere-- 00:06:22.150 --> 00:06:28.016 align:middle line:90% 00:06:28.016 --> 00:06:39.610 align:middle line:84% on all of R. So in other words, this one here-- 00:06:39.610 --> 00:06:41.560 align:middle line:90% this is a function-- 00:06:41.560 --> 00:06:46.490 align:middle line:84% by pset 5, it's a function that's continuous on all of R. 00:06:46.490 --> 00:06:49.270 align:middle line:90% It agrees with this function. 00:06:49.270 --> 00:06:54.230 align:middle line:84% And so this is the only extension 00:06:54.230 --> 00:06:58.370 align:middle line:84% of this function to be continuous and defined 00:06:58.370 --> 00:07:01.430 align:middle line:90% on all of R. 00:07:01.430 --> 00:07:05.270 align:middle line:84% So now-- but we still needed to prove-- 00:07:05.270 --> 00:07:08.810 align:middle line:84% we still needed to prove this relationship here 00:07:08.810 --> 00:07:10.490 align:middle line:90% that we assumed. 00:07:10.490 --> 00:07:12.810 align:middle line:84% So let's try to prove this one here. 00:07:12.810 --> 00:07:17.270 align:middle line:84% And I will just do the main case of it. 00:07:17.270 --> 00:07:19.960 align:middle line:84% And so we're going to prove-- so theorem-- 00:07:19.960 --> 00:07:22.640 align:middle line:90% 00:07:22.640 --> 00:07:30.080 align:middle line:84% to prove that E of x times y is equal to E of x times E 00:07:30.080 --> 00:07:31.670 align:middle line:90% to the y. 00:07:31.670 --> 00:07:43.670 align:middle line:84% And I am going to show this proof in the case 00:07:43.670 --> 00:07:50.610 align:middle line:84% where these are non-negative numbers, both of them. 00:07:50.610 --> 00:07:52.420 align:middle line:90% The general case is-- 00:07:52.420 --> 00:07:55.090 align:middle line:84% once you have this, the general case is not very difficult, 00:07:55.090 --> 00:07:59.040 align:middle line:90% but I won't do the general case. 00:07:59.040 --> 00:08:02.200 align:middle line:84% Now, how are we going to prove that? 00:08:02.200 --> 00:08:08.320 align:middle line:84% So to prove that, we're going to use another theorem. 00:08:08.320 --> 00:08:10.630 align:middle line:90% So the other theorem is that-- 00:08:10.630 --> 00:08:12.125 align:middle line:90% so we take two-- 00:08:12.125 --> 00:08:15.600 align:middle line:90% 00:08:15.600 --> 00:08:19.450 align:middle line:90% suppose you take two series. 00:08:19.450 --> 00:08:24.720 align:middle line:84% Each of them are of non-negative numbers. 00:08:24.720 --> 00:08:28.680 align:middle line:90% So the setup is-- 00:08:28.680 --> 00:08:31.500 align:middle line:84% so again, this is going to go into the proof 00:08:31.500 --> 00:08:33.570 align:middle line:90% of that theorem over there. 00:08:33.570 --> 00:08:39.090 align:middle line:84% But the setup for this theorem is more general. 00:08:39.090 --> 00:08:48.355 align:middle line:84% I take a series a n, and I take another serious bn. 00:08:48.355 --> 00:08:51.500 align:middle line:90% 00:08:51.500 --> 00:08:58.640 align:middle line:84% And I'm assuming that both a n and bn for all n 00:08:58.640 --> 00:09:05.060 align:middle line:90% are non-negative, so for all n. 00:09:05.060 --> 00:09:08.250 align:middle line:84% So these are series of non-negative numbers. 00:09:08.250 --> 00:09:12.260 align:middle line:90% And then I form another series. 00:09:12.260 --> 00:09:22.670 align:middle line:84% We will form a third series, and that third series 00:09:22.670 --> 00:09:27.880 align:middle line:84% is going to be defined from the two previous series. 00:09:27.880 --> 00:09:29.820 align:middle line:90% And the third series-- 00:09:29.820 --> 00:09:34.140 align:middle line:90% let's call it cn like this. 00:09:34.140 --> 00:09:46.590 align:middle line:84% And now cn here is going to be the sum of a i b n minus i, 00:09:46.590 --> 00:09:49.860 align:middle line:90% i equal to 0 through n. 00:09:49.860 --> 00:09:54.600 align:middle line:84% So this has the effect that the index 00:09:54.600 --> 00:10:00.180 align:middle line:84% of this plus the index of that is always adding up to n. 00:10:00.180 --> 00:10:05.325 align:middle line:84% So the sum of the two indices is exactly n. 00:10:05.325 --> 00:10:08.160 align:middle line:90% 00:10:08.160 --> 00:10:19.390 align:middle line:84% So that's-- so obviously-- so note that the cn's here are, 00:10:19.390 --> 00:10:20.910 align:middle line:90% of course, also non-negative. 00:10:20.910 --> 00:10:22.850 align:middle line:84% They're just sums of non-negative numbers. 00:10:22.850 --> 00:10:26.032 align:middle line:90% 00:10:26.032 --> 00:10:31.650 align:middle line:84% So this is a new series of non-negative numbers. 00:10:31.650 --> 00:10:45.520 align:middle line:84% And then the theorem is that if this series here-- 00:10:45.520 --> 00:11:01.510 align:middle line:84% if this one here and this one here are convergent, 00:11:01.510 --> 00:11:09.710 align:middle line:84% then this series here is also convergent. 00:11:09.710 --> 00:11:15.860 align:middle line:90% 00:11:15.860 --> 00:11:20.215 align:middle line:84% And we also have that this sum here-- 00:11:20.215 --> 00:11:23.699 align:middle line:90% 00:11:23.699 --> 00:11:28.880 align:middle line:84% that this sum is just the product of the two sums. 00:11:28.880 --> 00:11:43.242 align:middle line:84% So it's the product of the sum here, like that. 00:11:43.242 --> 00:11:46.230 align:middle line:90% 00:11:46.230 --> 00:11:50.290 align:middle line:84% So again, we have these two series of non-negative numbers. 00:11:50.290 --> 00:11:53.620 align:middle line:84% From those two series, you form a third series. 00:11:53.620 --> 00:11:55.540 align:middle line:90% And the third series-- 00:11:55.540 --> 00:11:58.440 align:middle line:84% the idea is that the third series is kind 00:11:58.440 --> 00:12:00.780 align:middle line:90% of a product of the other two. 00:12:00.780 --> 00:12:05.070 align:middle line:84% And the cn's in the third series is 00:12:05.070 --> 00:12:09.195 align:middle line:84% given where you take the sum of the products 00:12:09.195 --> 00:12:15.690 align:middle line:84% of the elements in the other two, so that the index is at n. 00:12:15.690 --> 00:12:20.890 align:middle line:84% And then if the two series you started with are convergent, 00:12:20.890 --> 00:12:26.010 align:middle line:84% then this product series is also convergent, 00:12:26.010 --> 00:12:28.950 align:middle line:84% and the sum of the product series 00:12:28.950 --> 00:12:34.852 align:middle line:84% is really the product of the sum of the other two series. 00:12:34.852 --> 00:12:37.360 align:middle line:84% Now-- so that's why we want to prove. 00:12:37.360 --> 00:12:39.750 align:middle line:84% And then once we have this we're going 00:12:39.750 --> 00:12:43.840 align:middle line:84% to use it to prove this theorem here. 00:12:43.840 --> 00:12:50.000 align:middle line:84% And so now, to prove it, let me just first observe that-- 00:12:50.000 --> 00:12:52.323 align:middle line:84% so let me observe over here that-- 00:12:52.323 --> 00:12:59.500 align:middle line:90% 00:12:59.500 --> 00:13:02.750 align:middle line:84% so let me make a couple of definitions, 00:13:02.750 --> 00:13:05.110 align:middle line:90% and then making an observation. 00:13:05.110 --> 00:13:13.900 align:middle line:84% Let me start with making some definitions that 00:13:13.900 --> 00:13:15.030 align:middle line:90% is useful in the proof. 00:13:15.030 --> 00:13:20.200 align:middle line:90% 00:13:20.200 --> 00:13:30.520 align:middle line:84% So I'm going to just find S n a just like we usually 00:13:30.520 --> 00:13:39.470 align:middle line:84% do where we're summing here from i equal to 0 up to n a i. 00:13:39.470 --> 00:13:46.010 align:middle line:84% The superscript comes from the a series. 00:13:46.010 --> 00:13:55.940 align:middle line:84% And then likewise, I define S n b to be the sum here of the b 00:13:55.940 --> 00:13:56.720 align:middle line:90% i's. 00:13:56.720 --> 00:14:09.590 align:middle line:84% And then I'm defining S n c to be the sum of the ci's up to n, 00:14:09.590 --> 00:14:11.600 align:middle line:90% like that. 00:14:11.600 --> 00:14:14.270 align:middle line:84% And so of course, you have that-- we 00:14:14.270 --> 00:14:22.680 align:middle line:84% know that a i, b i, and ci are all non-negative numbers. 00:14:22.680 --> 00:14:25.590 align:middle line:90% So this means that convergence-- 00:14:25.590 --> 00:14:33.130 align:middle line:84% so convergence of any of these series-- 00:14:33.130 --> 00:14:41.515 align:middle line:90% 00:14:41.515 --> 00:14:43.990 align:middle line:84% so convergence, say, of this one is the same 00:14:43.990 --> 00:14:46.910 align:middle line:84% as saying that these numbers here are bounded 00:14:46.910 --> 00:14:48.680 align:middle line:90% That's by monotone convergence. 00:14:48.680 --> 00:14:50.650 align:middle line:84% So convergence of any of these series 00:14:50.650 --> 00:15:07.600 align:middle line:84% is equivalent to that the corresponding S n a, b, or c 00:15:07.600 --> 00:15:09.645 align:middle line:90% is bounded. 00:15:09.645 --> 00:15:14.980 align:middle line:90% 00:15:14.980 --> 00:15:17.480 align:middle line:90% So that's just an observation. 00:15:17.480 --> 00:15:20.680 align:middle line:84% And again, this is the monotone convergence theorem. 00:15:20.680 --> 00:15:23.660 align:middle line:90% This is by monotone convergence. 00:15:23.660 --> 00:15:37.015 align:middle line:90% 00:15:37.015 --> 00:15:40.280 align:middle line:84% So that's the monotone convergence theorem. 00:15:40.280 --> 00:15:45.860 align:middle line:84% The other thing-- see, I thought of something else 00:15:45.860 --> 00:15:47.635 align:middle line:90% I wanted to note. 00:15:47.635 --> 00:15:52.370 align:middle line:90% 00:15:52.370 --> 00:15:56.600 align:middle line:90% So let's try to prove that. 00:15:56.600 --> 00:16:05.810 align:middle line:84% So to prove it, let me just try to think about it for a second 00:16:05.810 --> 00:16:06.540 align:middle line:90% here. 00:16:06.540 --> 00:16:10.130 align:middle line:90% So if I'm looking at a1-- 00:16:10.130 --> 00:16:16.490 align:middle line:90% say, a2, plus, up to a n. 00:16:16.490 --> 00:16:20.330 align:middle line:84% Suppose I'm looking at this here. 00:16:20.330 --> 00:16:27.230 align:middle line:90% This is like S n a. 00:16:27.230 --> 00:16:30.980 align:middle line:84% And I have that S n a, and suppose 00:16:30.980 --> 00:16:33.855 align:middle line:90% I'm now looking at the S n b. 00:16:33.855 --> 00:16:39.300 align:middle line:90% 00:16:39.300 --> 00:16:42.561 align:middle line:90% This here is the S n b. 00:16:42.561 --> 00:16:45.510 align:middle line:90% 00:16:45.510 --> 00:16:51.400 align:middle line:84% Now, if you just multiply out the product of these two, 00:16:51.400 --> 00:16:52.830 align:middle line:90% then you would get-- 00:16:52.830 --> 00:16:54.940 align:middle line:90% and I should have started at 0. 00:16:54.940 --> 00:16:56.190 align:middle line:90% Sorry. 00:16:56.190 --> 00:17:02.700 align:middle line:84% Let's start at 0 because that's what we do up there, like that. 00:17:02.700 --> 00:17:11.190 align:middle line:84% So now I multiply out, and you see that you're getting a0, b0. 00:17:11.190 --> 00:17:14.670 align:middle line:84% And you're getting-- then you can kind of-- if you try to do 00:17:14.670 --> 00:17:15.390 align:middle line:90% it-- 00:17:15.390 --> 00:17:18.579 align:middle line:84% so let's try to multiply this out, 00:17:18.579 --> 00:17:21.609 align:middle line:84% but let's do it in an ordered way 00:17:21.609 --> 00:17:23.700 align:middle line:84% And when I'm talking about "ordered way," 00:17:23.700 --> 00:17:28.023 align:middle line:84% I'm not talking about just multiplying this one on that. 00:17:28.023 --> 00:17:29.440 align:middle line:84% You could say that's very ordered, 00:17:29.440 --> 00:17:36.870 align:middle line:84% but it's not related to the c series. 00:17:36.870 --> 00:17:41.960 align:middle line:84% So I want to just group things so that they remind us 00:17:41.960 --> 00:17:43.440 align:middle line:90% of the c series. 00:17:43.440 --> 00:17:44.790 align:middle line:90% But if I do that-- 00:17:44.790 --> 00:17:48.710 align:middle line:90% so we're starting with this-- 00:17:48.710 --> 00:17:52.200 align:middle line:90% is this multiplied by this. 00:17:52.200 --> 00:17:58.572 align:middle line:84% Then I take this one here, multiplied by that, so a0 b1. 00:17:58.572 --> 00:18:01.550 align:middle line:90% 00:18:01.550 --> 00:18:06.035 align:middle line:84% This here is the only one where the index-- 00:18:06.035 --> 00:18:09.180 align:middle line:84% the subscripts are non-negative integers. 00:18:09.180 --> 00:18:14.330 align:middle line:84% This is the only one where the sum of the two subscripts is 0. 00:18:14.330 --> 00:18:16.250 align:middle line:90% If I'm looking at where the sum 00:18:16.250 --> 00:18:18.150 align:middle line:90% Of the two subscripts is 1-- 00:18:18.150 --> 00:18:21.180 align:middle line:84% this is one of them, but there is another one, 00:18:21.180 --> 00:18:25.080 align:middle line:84% and that's where I take this multiplied by that. 00:18:25.080 --> 00:18:30.440 align:middle line:84% So I group those together, so a1 b0. 00:18:30.440 --> 00:18:33.820 align:middle line:84% And now I want to collect the ones-- 00:18:33.820 --> 00:18:38.140 align:middle line:84% so those are the only two where the sum of the subscripts is 1. 00:18:38.140 --> 00:18:42.720 align:middle line:84% Then if I'm looking at, when I'm multiplying out, 00:18:42.720 --> 00:18:46.410 align:middle line:84% where the sum of the subscript is 2, 00:18:46.410 --> 00:18:51.300 align:middle line:84% well, I could start with this one, but then I have to pick b2. 00:18:51.300 --> 00:18:55.650 align:middle line:90% So I take plus a0 b2. 00:18:55.650 --> 00:18:59.190 align:middle line:90% Then I could take a1 b1. 00:18:59.190 --> 00:19:03.330 align:middle line:84% That has also sum of subscripts 2. 00:19:03.330 --> 00:19:05.920 align:middle line:90% And the last one is-- 00:19:05.920 --> 00:19:10.660 align:middle line:84% If I take a2, but then the b have to be the 0 subscript. 00:19:10.660 --> 00:19:13.680 align:middle line:90% 00:19:13.680 --> 00:19:17.550 align:middle line:84% And now I proceed like this, so I'm grouping it like this. 00:19:17.550 --> 00:19:20.380 align:middle line:90% 00:19:20.380 --> 00:19:24.070 align:middle line:84% Now, of course, there's going to be a little bit of an issue. 00:19:24.070 --> 00:19:28.620 align:middle line:84% So for a while, everything is-- you see that this thing here-- 00:19:28.620 --> 00:19:34.060 align:middle line:84% this here is what we call c0, this here is c1, 00:19:34.060 --> 00:19:38.050 align:middle line:90% and this here is c2. 00:19:38.050 --> 00:19:41.770 align:middle line:84% And then we presumably also get c3. 00:19:41.770 --> 00:19:50.980 align:middle line:84% But then there is a point where we run into trouble if we 00:19:50.980 --> 00:19:54.565 align:middle line:90% want to really do it like this. 00:19:54.565 --> 00:19:57.020 align:middle line:90% So what we get is that-- 00:19:57.020 --> 00:19:59.740 align:middle line:84% so when you do this multiplication, what 00:19:59.740 --> 00:20:01.790 align:middle line:90% you will see is that-- 00:20:01.790 --> 00:20:09.550 align:middle line:84% so you see like this-- a0 plus a1, a n-- 00:20:09.550 --> 00:20:17.230 align:middle line:84% what you will see-- times b0, b1, up to bn. 00:20:17.230 --> 00:20:19.900 align:middle line:84% What you will see-- that this pattern here 00:20:19.900 --> 00:20:22.570 align:middle line:84% is continuing for a while, and you're 00:20:22.570 --> 00:20:29.960 align:middle line:84% getting that this is equal to c0, c1, plus c2, up to cn. 00:20:29.960 --> 00:20:32.930 align:middle line:90% 00:20:32.930 --> 00:20:38.900 align:middle line:84% In the cn, this is, again, where the sum of these indices 00:20:38.900 --> 00:20:41.490 align:middle line:84% could be this one here multiplied by that. 00:20:41.490 --> 00:20:46.790 align:middle line:84% The sum of the subscripts will be n. 00:20:46.790 --> 00:20:50.180 align:middle line:84% But if you're now looking at the next one, the one that 00:20:50.180 --> 00:20:53.180 align:middle line:84% corresponds to c n plus 1, you're 00:20:53.180 --> 00:20:58.460 align:middle line:84% not getting all of the c n plus 1 because one of the c n plus 1 00:20:58.460 --> 00:21:03.570 align:middle line:84% would be like a to the n plus 1 times b0. 00:21:03.570 --> 00:21:06.610 align:middle line:84% But you don't have that one here. 00:21:06.610 --> 00:21:10.280 align:middle line:84% So what you get is that you're getting all of these, 00:21:10.280 --> 00:21:12.256 align:middle line:90% plus something additional. 00:21:12.256 --> 00:21:18.815 align:middle line:90% 00:21:18.815 --> 00:21:22.950 align:middle line:84% And the additional is-- because everything is non-negative, 00:21:22.950 --> 00:21:27.710 align:middle line:84% the additional thing is a non-negative number. 00:21:27.710 --> 00:21:30.680 align:middle line:84% So what we have established here is that we have-- 00:21:30.680 --> 00:21:37.850 align:middle line:90% so this has shown that S n c-- 00:21:37.850 --> 00:21:40.040 align:middle line:84% this here is-- let me just write it like this-- 00:21:40.040 --> 00:21:42.640 align:middle line:90% this here is S n c. 00:21:42.640 --> 00:21:57.340 align:middle line:84% So you see that this thing here was S n a, 00:21:57.340 --> 00:22:00.280 align:middle line:90% and this here was S n b. 00:22:00.280 --> 00:22:03.070 align:middle line:84% So what you have is-- so this inequality 00:22:03.070 --> 00:22:08.350 align:middle line:84% just tells you that S n a times S 00:22:08.350 --> 00:22:13.630 align:middle line:84% n b is equal to S n c plus something 00:22:13.630 --> 00:22:15.140 align:middle line:90% positive-- non-negative. 00:22:15.140 --> 00:22:20.230 align:middle line:90% 00:22:20.230 --> 00:22:24.100 align:middle line:84% So in other words, this number here 00:22:24.100 --> 00:22:27.950 align:middle line:84% is less than or equal to the product of these two. 00:22:27.950 --> 00:22:35.390 align:middle line:84% So this implies that S n c is less than or equal to this here. 00:22:35.390 --> 00:22:39.920 align:middle line:84% But now you see you already have something, because these here 00:22:39.920 --> 00:22:42.630 align:middle line:90% were increasing-- 00:22:42.630 --> 00:22:46.290 align:middle line:84% they were increasing because the a n was all non-negative. 00:22:46.290 --> 00:22:49.010 align:middle line:84% So these here are increasing, and they 00:22:49.010 --> 00:22:52.340 align:middle line:90% are less than or equal to-- 00:22:52.340 --> 00:22:55.230 align:middle line:84% the series was supposed to be convergent, 00:22:55.230 --> 00:22:57.080 align:middle line:90% so you have that-- sorry. 00:22:57.080 --> 00:23:05.000 align:middle line:84% So here I'm using that the S n a is monotone increasing up 00:23:05.000 --> 00:23:06.010 align:middle line:90% to the limit here. 00:23:06.010 --> 00:23:12.260 align:middle line:90% 00:23:12.260 --> 00:23:17.120 align:middle line:84% And likewise, the S n b here were monotone increasing up 00:23:17.120 --> 00:23:17.770 align:middle line:90% to the limit. 00:23:17.770 --> 00:23:21.920 align:middle line:90% 00:23:21.920 --> 00:23:23.500 align:middle line:90% So this thing here-- 00:23:23.500 --> 00:23:25.810 align:middle line:84% so this is smaller than the limit. 00:23:25.810 --> 00:23:28.360 align:middle line:84% This is smaller than the limit of that. 00:23:28.360 --> 00:23:36.830 align:middle line:84% So this thing here is less than or equal to the product 00:23:36.830 --> 00:23:37.865 align:middle line:90% of these guys. 00:23:37.865 --> 00:23:47.160 align:middle line:90% 00:23:47.160 --> 00:23:49.920 align:middle line:84% And so you see-- and that's supposed 00:23:49.920 --> 00:23:51.850 align:middle line:84% to be just a real number, this here. 00:23:51.850 --> 00:23:55.710 align:middle line:84% So you see that these guys here are now bounded. 00:23:55.710 --> 00:23:57.210 align:middle line:90% So this means that-- 00:23:57.210 --> 00:23:58.500 align:middle line:90% because they're monotone. 00:23:58.500 --> 00:24:01.750 align:middle line:84% It means that that series is also convergent. 00:24:01.750 --> 00:24:05.850 align:middle line:90% 00:24:05.850 --> 00:24:16.680 align:middle line:84% So what we have concluded here is that this series here 00:24:16.680 --> 00:24:17.846 align:middle line:90% is convergent. 00:24:17.846 --> 00:24:21.530 align:middle line:90% 00:24:21.530 --> 00:24:26.960 align:middle line:84% And we have that this infinite sum here 00:24:26.960 --> 00:24:32.000 align:middle line:84% is less than or equal to the product of the other ones. 00:24:32.000 --> 00:24:34.210 align:middle line:90% This is what this here gave us. 00:24:34.210 --> 00:24:39.650 align:middle line:90% 00:24:39.650 --> 00:24:41.840 align:middle line:90% So this is what we have so far. 00:24:41.840 --> 00:24:54.000 align:middle line:84% Now, we want to get the reverse inequality holds. 00:24:54.000 --> 00:24:56.120 align:middle line:84% So we want the reverse inequality 00:24:56.120 --> 00:24:59.420 align:middle line:84% of what is over there so that we can say that those two 00:24:59.420 --> 00:25:02.155 align:middle line:90% things here are actually equal. 00:25:02.155 --> 00:25:04.550 align:middle line:84% Now, let's try to see how do we prove 00:25:04.550 --> 00:25:06.450 align:middle line:90% the reverse of this inequality. 00:25:06.450 --> 00:25:12.140 align:middle line:84% The reverse, we are proving by examining more closely what 00:25:12.140 --> 00:25:13.165 align:middle line:90% that product was. 00:25:13.165 --> 00:25:19.700 align:middle line:90% 00:25:19.700 --> 00:25:24.390 align:middle line:84% So you see-- so what is this additional thing here? 00:25:24.390 --> 00:25:29.900 align:middle line:84% So again-- so we have-- we're looking at this thing here. 00:25:29.900 --> 00:25:41.820 align:middle line:90% 00:25:41.820 --> 00:25:44.220 align:middle line:84% And what we got is that this here was 00:25:44.220 --> 00:25:51.690 align:middle line:90% equal to the sum here of cn. 00:25:51.690 --> 00:25:55.182 align:middle line:84% But then we have some additional terms. 00:25:55.182 --> 00:25:57.660 align:middle line:90% And the additional terms is-- 00:25:57.660 --> 00:25:59.970 align:middle line:84% the thing is that if you start looking 00:25:59.970 --> 00:26:01.810 align:middle line:90% at the additional terms-- 00:26:01.810 --> 00:26:05.220 align:middle line:84% so what are the additional terms? 00:26:05.220 --> 00:26:11.170 align:middle line:90% You're getting a1 bn-- 00:26:11.170 --> 00:26:15.780 align:middle line:84% that's one where the sum here is n plus 1. 00:26:15.780 --> 00:26:19.540 align:middle line:84% And then you also get-- so this is coming from this-- 00:26:19.540 --> 00:26:22.030 align:middle line:84% sorry, that one here multiplied by this. 00:26:22.030 --> 00:26:30.340 align:middle line:84% You would also get a2 b n minus 2. 00:26:30.340 --> 00:26:37.180 align:middle line:84% You get an a3 b-- sorry, not n minus 2, n minus 1-- 00:26:37.180 --> 00:26:40.620 align:middle line:90% b3 n minus 2, et cetera. 00:26:40.620 --> 00:26:43.430 align:middle line:90% 00:26:43.430 --> 00:26:51.175 align:middle line:84% But all of these here-- and then-- so you get that, 00:26:51.175 --> 00:26:52.900 align:middle line:90% but you don't get-- 00:26:52.900 --> 00:26:55.605 align:middle line:84% again, you don't get the one that is missing here-- 00:26:55.605 --> 00:27:00.160 align:middle line:90% 00:27:00.160 --> 00:27:04.100 align:middle line:90% is a0 b n plus 1. 00:27:04.100 --> 00:27:06.050 align:middle line:90% This one is missing. 00:27:06.050 --> 00:27:11.080 align:middle line:84% And the other one that is missing is this one here. 00:27:11.080 --> 00:27:14.170 align:middle line:84% But otherwise, you get all of the ones that 00:27:14.170 --> 00:27:19.510 align:middle line:84% was in the c n plus 1, and you don't get anything more 00:27:19.510 --> 00:27:22.200 align:middle line:90% And so this pattern continues. 00:27:22.200 --> 00:27:26.610 align:middle line:90% So the next term would be a2-- 00:27:26.610 --> 00:27:28.020 align:middle line:90% sorry-- yeah. 00:27:28.020 --> 00:27:30.870 align:middle line:84% And then you would also get something where 00:27:30.870 --> 00:27:33.250 align:middle line:90% the sum of the indices was 2. 00:27:33.250 --> 00:27:37.960 align:middle line:90% That's like a2 bn. 00:27:37.960 --> 00:27:40.510 align:middle line:84% But here, you would be missing more. 00:27:40.510 --> 00:27:45.340 align:middle line:84% You would be missing a0 b n plus 2. 00:27:45.340 --> 00:27:48.970 align:middle line:90% You're missing a1 b n plus 1. 00:27:48.970 --> 00:27:51.820 align:middle line:90% But you don't get anything more. 00:27:51.820 --> 00:27:55.570 align:middle line:84% And so the thing is, like when you continue like this, 00:27:55.570 --> 00:28:00.040 align:middle line:84% the last one you get is when you're multiplying that by that. 00:28:00.040 --> 00:28:03.480 align:middle line:90% And that's a n times bn. 00:28:03.480 --> 00:28:08.936 align:middle line:84% Those here are the sum of the indices is 2-- 00:28:08.936 --> 00:28:10.720 align:middle line:90% 2n, sorry-- it's 2n. 00:28:10.720 --> 00:28:13.270 align:middle line:90% But you're missing a lot now. 00:28:13.270 --> 00:28:17.006 align:middle line:84% You only have one of many, many terms. 00:28:17.006 --> 00:28:19.630 align:middle line:84% But it is a term that all appear, 00:28:19.630 --> 00:28:22.100 align:middle line:84% and this one would appear in c 2n. 00:28:22.100 --> 00:28:27.370 align:middle line:84% And so in this way, what you have is that this here is less 00:28:27.370 --> 00:28:32.980 align:middle line:90% than S to the power 2n c. 00:28:32.980 --> 00:28:36.980 align:middle line:84% They are all in there, but you will be missing terms, 00:28:36.980 --> 00:28:39.730 align:middle line:84% but all terms are non-negative So this means 00:28:39.730 --> 00:28:41.740 align:middle line:90% that you have that inequality. 00:28:41.740 --> 00:28:44.500 align:middle line:90% So you see that now we have-- 00:28:44.500 --> 00:28:50.800 align:middle line:84% now what we have here is that we have-- 00:28:50.800 --> 00:28:57.730 align:middle line:84% what we established before was the inequality 00:28:57.730 --> 00:29:08.080 align:middle line:84% that S n c was less than S n a times S n b. 00:29:08.080 --> 00:29:11.360 align:middle line:84% But what we have now here is that this thing here, 00:29:11.360 --> 00:29:15.836 align:middle line:84% this product, is less than S 2n c. 00:29:15.836 --> 00:29:19.880 align:middle line:90% 00:29:19.880 --> 00:29:25.130 align:middle line:84% This one here is squeezed between this one here and that 00:29:25.130 --> 00:29:28.850 align:middle line:84% one here, but both these here converge to the same. 00:29:28.850 --> 00:29:33.175 align:middle line:84% They both converge to this infinite sum. 00:29:33.175 --> 00:29:36.190 align:middle line:90% 00:29:36.190 --> 00:29:38.310 align:middle line:90% So these converge to the same. 00:29:38.310 --> 00:29:43.310 align:middle line:84% This means that the limit of this here must also be the same. 00:29:43.310 --> 00:29:50.840 align:middle line:84% And so this shows that this here is the reverse inequality. 00:29:50.840 --> 00:29:54.840 align:middle line:90% So proof-- proof over there. 00:29:54.840 --> 00:30:03.608 align:middle line:90% 00:30:03.608 --> 00:30:07.250 align:middle line:84% So this is that if you take two series 00:30:07.250 --> 00:30:12.570 align:middle line:84% and you form the product, then that also 00:30:12.570 --> 00:30:16.020 align:middle line:84% converges if the two series is non-negative numbers. 00:30:16.020 --> 00:30:23.170 align:middle line:84% So let me just make a corollary of this, that, of course, 00:30:23.170 --> 00:30:26.760 align:middle line:90% if you take a series here-- 00:30:26.760 --> 00:30:31.000 align:middle line:84% and so I'll just do a trivial corollary, 00:30:31.000 --> 00:30:34.710 align:middle line:84% but you can see how you can use this. 00:30:34.710 --> 00:30:37.960 align:middle line:84% And this is used in proving the general case. 00:30:37.960 --> 00:30:39.370 align:middle line:90% So we will only-- 00:30:39.370 --> 00:30:42.990 align:middle line:90% yeah, [INAUDIBLE]. 00:30:42.990 --> 00:30:45.280 align:middle line:84% So suppose I take two series here. 00:30:45.280 --> 00:30:48.390 align:middle line:90% 00:30:48.390 --> 00:30:52.000 align:middle line:84% So this is a trivial corollary of it. 00:30:52.000 --> 00:30:56.820 align:middle line:84% Suppose I take two series here, and I know that this one here-- 00:30:56.820 --> 00:31:00.130 align:middle line:84% that they are both absolutely convergent. 00:31:00.130 --> 00:31:08.640 align:middle line:90% 00:31:08.640 --> 00:31:12.410 align:middle line:84% If they are both absolutely convergent, 00:31:12.410 --> 00:31:17.570 align:middle line:84% then if you take this series here-- 00:31:17.570 --> 00:31:20.730 align:middle line:90% cn-- sorry, ci-- 00:31:20.730 --> 00:31:23.990 align:middle line:90% 00:31:23.990 --> 00:31:28.590 align:middle line:84% well, this one here I claim is also absolutely convergent. 00:31:28.590 --> 00:31:41.120 align:middle line:90% 00:31:41.120 --> 00:31:45.518 align:middle line:84% And again, this trivial corollary 00:31:45.518 --> 00:31:50.720 align:middle line:84% is the key ingredient in proving a more general statement that 00:31:50.720 --> 00:31:54.750 align:middle line:84% would be used, say, to prove this inequality here 00:31:54.750 --> 00:31:59.580 align:middle line:84% when x and y are not-- assumed to be non-negative. 00:31:59.580 --> 00:32:00.870 align:middle line:90% But I won't get into that. 00:32:00.870 --> 00:32:05.035 align:middle line:84% But I'll just make this observation, and get this form. 00:32:05.035 --> 00:32:08.360 align:middle line:90% 00:32:08.360 --> 00:32:12.750 align:middle line:84% So I claim that this thing here is also absolutely convergent. 00:32:12.750 --> 00:32:16.270 align:middle line:84% So again, I'm not assuming now a sign on these, 00:32:16.270 --> 00:32:19.140 align:middle line:84% but I am assuming that they're absolutely convergent. 00:32:19.140 --> 00:32:22.020 align:middle line:90% 00:32:22.020 --> 00:32:26.670 align:middle line:84% And so if they're absolutely convergent of proof, 00:32:26.670 --> 00:32:31.740 align:middle line:90% so absolutely convergent-- 00:32:31.740 --> 00:32:34.530 align:middle line:90% 00:32:34.530 --> 00:32:40.980 align:middle line:84% this just means that the series here, this one here, 00:32:40.980 --> 00:32:47.620 align:middle line:84% and this one here are convergent. 00:32:47.620 --> 00:32:51.300 align:middle line:90% 00:32:51.300 --> 00:32:55.020 align:middle line:84% That's what absolute convergence means. 00:32:55.020 --> 00:33:01.740 align:middle line:84% But now if you're looking at ci, the ci series-- 00:33:01.740 --> 00:33:09.100 align:middle line:84% this here is the sum here, i equal to 0 to n of a i-- 00:33:09.100 --> 00:33:16.000 align:middle line:90% sorry-- a i b n minus 1. 00:33:16.000 --> 00:33:19.270 align:middle line:90% [INAUDIBLE] 00:33:19.270 --> 00:33:24.970 align:middle line:84% But this means that ci absolute value here 00:33:24.970 --> 00:33:36.070 align:middle line:84% is less than this thing here, absolute value, like that. 00:33:36.070 --> 00:33:38.590 align:middle line:84% You can take the absolute value sign 00:33:38.590 --> 00:33:40.490 align:middle line:90% in under the summation sign. 00:33:40.490 --> 00:33:46.420 align:middle line:84% So this is less than the sum here from i equal to 0 to n, 00:33:46.420 --> 00:33:52.210 align:middle line:84% and then a i times b n minus i, like that. 00:33:52.210 --> 00:33:55.930 align:middle line:90% 00:33:55.930 --> 00:34:04.616 align:middle line:84% But now you see that this here is the corresponding c 00:34:04.616 --> 00:34:11.010 align:middle line:90% so you could now define-- 00:34:11.010 --> 00:34:15.030 align:middle line:84% so you start with this sequence a i, 00:34:15.030 --> 00:34:22.270 align:middle line:84% and I now define a new sequence b i bar. 00:34:22.270 --> 00:34:26.679 align:middle line:84% And I do it by just taking the absolute value like that. 00:34:26.679 --> 00:34:29.860 align:middle line:84% So I know that this new series is convergent. 00:34:29.860 --> 00:34:40.380 align:middle line:84% Likewise, I defined b i bar to be the b i absolute value. 00:34:40.380 --> 00:34:43.500 align:middle line:84% And I know that this is convergent. 00:34:43.500 --> 00:34:51.100 align:middle line:84% And then I defined a sequence here, ci bar, 00:34:51.100 --> 00:34:54.463 align:middle line:84% which is just the absolute value of the ci's. 00:34:54.463 --> 00:34:57.930 align:middle line:90% 00:34:57.930 --> 00:35:00.790 align:middle line:90% And I know that this one here-- 00:35:00.790 --> 00:35:06.430 align:middle line:84% so I know that this here is convergent. 00:35:06.430 --> 00:35:09.670 align:middle line:90% That's by assumption. 00:35:09.670 --> 00:35:13.900 align:middle line:84% This one here is convergent by assumption. 00:35:13.900 --> 00:35:19.780 align:middle line:84% And this inequality here proves that this here-- 00:35:19.780 --> 00:35:23.440 align:middle line:84% that the c i bar is also convergent, 00:35:23.440 --> 00:35:26.980 align:middle line:84% because this one here is the one where 00:35:26.980 --> 00:35:30.560 align:middle line:84% you form the products of these two convergent sequences. 00:35:30.560 --> 00:35:33.140 align:middle line:84% So this one here is also convergent. 00:35:33.140 --> 00:35:42.495 align:middle line:84% So this here is also convergent by this theorem. 00:35:42.495 --> 00:35:47.290 align:middle line:90% 00:35:47.290 --> 00:35:49.870 align:middle line:84% Anyway, that was just an observation 00:35:49.870 --> 00:35:53.630 align:middle line:84% that we were getting this statement out of it. 00:35:53.630 --> 00:35:56.300 align:middle line:90% 00:35:56.300 --> 00:35:58.550 align:middle line:90% Now, this is also related. 00:35:58.550 --> 00:36:02.060 align:middle line:84% So why do we have non-negative here, 00:36:02.060 --> 00:36:05.180 align:middle line:84% or why do we talk about absolute convergence? 00:36:05.180 --> 00:36:10.740 align:middle line:84% So this is a simple fact that if you take a series, 00:36:10.740 --> 00:36:13.600 align:middle line:84% if you're trying to sum infinite numbers-- 00:36:13.600 --> 00:36:15.350 align:middle line:84% we now know how to sum an infinite number. 00:36:15.350 --> 00:36:18.140 align:middle line:90% But if you're trying to do it-- 00:36:18.140 --> 00:36:20.930 align:middle line:84% if the numbers that you're summing 00:36:20.930 --> 00:36:23.840 align:middle line:84% are all non-negative, then actually, 00:36:23.840 --> 00:36:26.990 align:middle line:90% the ordering doesn't matter. 00:36:26.990 --> 00:36:29.810 align:middle line:84% You can check the fifth element first and then 00:36:29.810 --> 00:36:34.290 align:middle line:84% the seventh and then the 119 and then come back and take 1. 00:36:34.290 --> 00:36:37.490 align:middle line:84% It doesn't matter what order you do it in. 00:36:37.490 --> 00:36:42.560 align:middle line:84% But if the numbers are not non-negative 00:36:42.560 --> 00:36:46.070 align:middle line:84% and the series is not absolutely convergent, now 00:36:46.070 --> 00:36:49.190 align:middle line:84% it certainly matters because you could end up 00:36:49.190 --> 00:36:51.990 align:middle line:84% taking all the negative numbers first, 00:36:51.990 --> 00:36:58.680 align:middle line:84% and then you get really, really far down towards minus infinity. 00:36:58.680 --> 00:37:02.120 align:middle line:84% Then you take a few of the positive elements, 00:37:02.120 --> 00:37:04.100 align:middle line:84% and then you take more of the negative. 00:37:04.100 --> 00:37:08.230 align:middle line:84% And so then you would actually stay very far down 00:37:08.230 --> 00:37:12.190 align:middle line:84% toward minus infinity, and you would never 00:37:12.190 --> 00:37:14.660 align:middle line:90% be able to come up again. 00:37:14.660 --> 00:37:20.950 align:middle line:84% So this is a basic fact, that if you take a series and the series 00:37:20.950 --> 00:37:24.010 align:middle line:84% are of non-negative numbers, then actually-- 00:37:24.010 --> 00:37:27.940 align:middle line:84% and it's converging, then you can rearrange the ordering, 00:37:27.940 --> 00:37:30.220 align:middle line:84% and it will still converge, and it will still 00:37:30.220 --> 00:37:32.290 align:middle line:90% converge to the same number. 00:37:32.290 --> 00:37:35.770 align:middle line:84% If you're trying to sum infinite numbers 00:37:35.770 --> 00:37:42.070 align:middle line:84% and they are not non-negative and they 00:37:42.070 --> 00:37:47.025 align:middle line:84% don't converge absolutely, then the ordering does matter. 00:37:47.025 --> 00:37:51.370 align:middle line:84% And this is related to why we are assuming that all of these 00:37:51.370 --> 00:37:54.110 align:middle line:84% a i, b i's are non-negative numbers-- 00:37:54.110 --> 00:37:55.490 align:middle line:90% [INAUDIBLE] 00:37:55.490 --> 00:37:57.320 align:middle line:90% But we won't get into this. 00:37:57.320 --> 00:37:58.280 align:middle line:90% You can read about it. 00:37:58.280 --> 00:37:59.700 align:middle line:90% It's a sort of standard fact. 00:37:59.700 --> 00:38:03.410 align:middle line:84% It's certainly somewhere in the book, but it's not fun. 00:38:03.410 --> 00:38:04.880 align:middle line:84% AUDIENCE: How do we reconcile that 00:38:04.880 --> 00:38:08.720 align:middle line:84% with our typical properties, like the addition operator? 00:38:08.720 --> 00:38:13.820 align:middle line:84% TOBIAS COLDING: Yeah, so it's related to-- 00:38:13.820 --> 00:38:18.020 align:middle line:90% 00:38:18.020 --> 00:38:20.460 align:middle line:84% again, this is not part of the syllabus, 00:38:20.460 --> 00:38:24.335 align:middle line:84% but I think it's a useful thing to keep in mind because it also 00:38:24.335 --> 00:38:27.440 align:middle line:84% explains why we're doing it in a certain way. 00:38:27.440 --> 00:38:34.140 align:middle line:84% Suppose you take the alternating harmonic series, 00:38:34.140 --> 00:38:36.230 align:middle line:90% Remember that this thing here-- 00:38:36.230 --> 00:38:39.470 align:middle line:90% that's the harmonic series. 00:38:39.470 --> 00:38:41.790 align:middle line:90% This here is divergent. 00:38:41.790 --> 00:38:45.580 align:middle line:90% 00:38:45.580 --> 00:38:46.860 align:middle line:90% I actually proved-- 00:38:46.860 --> 00:38:51.500 align:middle line:84% I proved, or at least I outlined in class, why this series here 00:38:51.500 --> 00:38:53.150 align:middle line:90% is convergent. 00:38:53.150 --> 00:38:57.110 align:middle line:90% And this is-- and it's just-- 00:38:57.110 --> 00:39:00.300 align:middle line:84% again, the idea here is that if you 00:39:00.300 --> 00:39:02.700 align:middle line:84% take-- so you have something which is called 00:39:02.700 --> 00:39:05.460 align:middle line:90% the alternating series test. 00:39:05.460 --> 00:39:06.885 align:middle line:90% So this is a-- 00:39:06.885 --> 00:39:08.260 align:middle line:84% I'm coming back to your question. 00:39:08.260 --> 00:39:10.952 align:middle line:90% Just a second. 00:39:10.952 --> 00:39:13.120 align:middle line:84% And this is closely related to that. 00:39:13.120 --> 00:39:20.160 align:middle line:84% So the alternating series test is that if you take a sequence 00:39:20.160 --> 00:39:29.400 align:middle line:84% a n of non-negative numbers and a n converges to 0-- 00:39:29.400 --> 00:39:34.440 align:middle line:84% let's say that a n converges to 0 in a monotone way. 00:39:34.440 --> 00:39:37.090 align:middle line:90% 00:39:37.090 --> 00:39:40.210 align:middle line:84% So an example here of this would, of course, 00:39:40.210 --> 00:39:44.010 align:middle line:90% be where a n is 1/n. 00:39:44.010 --> 00:39:47.140 align:middle line:90% That's a monotone sequence. 00:39:47.140 --> 00:39:50.190 align:middle line:84% So if you take a monotone sequence like this-- but 00:39:50.190 --> 00:39:54.370 align:middle line:84% it doesn't need to go very fast to 0. 00:39:54.370 --> 00:39:57.710 align:middle line:84% This one does not converge to 0 very fast. 00:39:57.710 --> 00:40:00.020 align:middle line:84% But if you have that and you're now 00:40:00.020 --> 00:40:06.320 align:middle line:84% looking at the alternating series based on it, 00:40:06.320 --> 00:40:08.960 align:middle line:90% this one here is convergent. 00:40:08.960 --> 00:40:10.500 align:middle line:90% This one here is convergent. 00:40:10.500 --> 00:40:13.850 align:middle line:90% 00:40:13.850 --> 00:40:16.280 align:middle line:90% And why is it convergent? 00:40:16.280 --> 00:40:20.000 align:middle line:84% It's convergent because of the following thing. 00:40:20.000 --> 00:40:23.360 align:middle line:84% And this is exactly what I did when 00:40:23.360 --> 00:40:27.560 align:middle line:84% I tried to explain why this here is convergent-- 00:40:27.560 --> 00:40:31.800 align:middle line:84% that you're starting somewhere, let's say starting at 0. 00:40:31.800 --> 00:40:35.420 align:middle line:84% Then you're looking at the first element. 00:40:35.420 --> 00:40:37.500 align:middle line:90% If you're summing here-- 00:40:37.500 --> 00:40:41.300 align:middle line:84% let's say you're summing from 1 to infinity. 00:40:41.300 --> 00:40:44.610 align:middle line:84% If you're summing from 1-- then the first element here-- 00:40:44.610 --> 00:40:46.110 align:middle line:90% this is minus 1. 00:40:46.110 --> 00:40:48.630 align:middle line:84% This is non-negative, so this is a negative number. 00:40:48.630 --> 00:40:50.030 align:middle line:90% So you're going down. 00:40:50.030 --> 00:40:51.260 align:middle line:90% You're starting at 0. 00:40:51.260 --> 00:40:53.720 align:middle line:90% You're going down 00:40:53.720 --> 00:40:59.440 align:middle line:84% The next element-- that will be positive because it's 2 here. 00:40:59.440 --> 00:41:02.920 align:middle line:84% So it's positive-- it would go up. 00:41:02.920 --> 00:41:06.420 align:middle line:84% But it's monotone, so you don't go as much 00:41:06.420 --> 00:41:08.220 align:middle line:90% up as you're going down. 00:41:08.220 --> 00:41:10.900 align:middle line:84% So you're first-- first you're going down, 00:41:10.900 --> 00:41:12.070 align:middle line:90% then you're going up. 00:41:12.070 --> 00:41:15.240 align:middle line:84% But you're going less up,e then you're going less down, 00:41:15.240 --> 00:41:17.880 align:middle line:84% then you're going less up, et cetera. 00:41:17.880 --> 00:41:20.585 align:middle line:84% And that actually is forcing it to converge. 00:41:20.585 --> 00:41:23.454 align:middle line:90% 00:41:23.454 --> 00:41:26.205 align:middle line:84% And so now why is this related to your question about, 00:41:26.205 --> 00:41:28.030 align:middle line:84% well, how do you reconcile these? 00:41:28.030 --> 00:41:30.990 align:middle line:84% So you see, if you're looking at something like this 00:41:30.990 --> 00:41:34.910 align:middle line:90% and you now took all of the-- 00:41:34.910 --> 00:41:40.890 align:middle line:84% if you first took many, many of the positive elements and then 00:41:40.890 --> 00:41:43.920 align:middle line:84% one or two of the negative and then many 00:41:43.920 --> 00:41:48.630 align:middle line:84% of the positive elements again, then actually, you could-- 00:41:48.630 --> 00:41:50.970 align:middle line:84% for this series, if you rearrange 00:41:50.970 --> 00:41:52.710 align:middle line:84% the order of the things you're summing, 00:41:52.710 --> 00:41:56.390 align:middle line:84% and then you could either get it to diverge, 00:41:56.390 --> 00:41:59.260 align:middle line:84% or you could get it to converge to pretty much anything you 00:41:59.260 --> 00:41:59.760 align:middle line:90% want. 00:41:59.760 --> 00:42:02.590 align:middle line:90% 00:42:02.590 --> 00:42:05.980 align:middle line:84% So this hopefully explains why when 00:42:05.980 --> 00:42:11.500 align:middle line:84% the series is not non-negative numbers, why the ordering really 00:42:11.500 --> 00:42:12.140 align:middle line:90% matters. 00:42:12.140 --> 00:42:16.780 align:middle line:84% And again, this is very closely related to this 00:42:16.780 --> 00:42:18.350 align:middle line:90% with the product of things. 00:42:18.350 --> 00:42:19.880 align:middle line:90% Why it's related? 00:42:19.880 --> 00:42:25.090 align:middle line:84% Because we're getting this remainder term. 00:42:25.090 --> 00:42:28.570 align:middle line:84% We've got something that was like the remainder, 00:42:28.570 --> 00:42:30.910 align:middle line:84% and it's the remainder that you want 00:42:30.910 --> 00:42:34.330 align:middle line:84% to be small in order to say that things are converging 00:42:34.330 --> 00:42:35.540 align:middle line:90% to the same thing. 00:42:35.540 --> 00:42:41.110 align:middle line:84% And you wouldn't have that necessarily in general 00:42:41.110 --> 00:42:45.010 align:middle line:84% unless you're assuming that the terms were non-negative. 00:42:45.010 --> 00:42:49.600 align:middle line:84% So let's come back to-- that was just a little bit of [? side, ?] 00:42:49.600 --> 00:42:53.300 align:middle line:84% but I hope it makes things a bit more clear. 00:42:53.300 --> 00:42:55.920 align:middle line:84% Let's come back to this theorem here. 00:42:55.920 --> 00:42:57.750 align:middle line:90% How do we prove this? 00:42:57.750 --> 00:43:00.260 align:middle line:84% So we're going to prove it from the theorem 00:43:00.260 --> 00:43:04.520 align:middle line:90% that we did with that if you-- 00:43:04.520 --> 00:43:09.860 align:middle line:84% so we're going to use this theorem we just proved, 00:43:09.860 --> 00:43:11.070 align:middle line:90% which is this theorem here. 00:43:11.070 --> 00:43:12.695 align:middle line:84% So we're going to use this theorem here 00:43:12.695 --> 00:43:16.730 align:middle line:90% to prove that theorem up there. 00:43:16.730 --> 00:43:21.333 align:middle line:84% So now it's just a matter of identifying the right series. 00:43:21.333 --> 00:43:26.540 align:middle line:90% 00:43:26.540 --> 00:43:35.750 align:middle line:84% So this is what we want to prove, 00:43:35.750 --> 00:43:39.510 align:middle line:84% this is what we want to show, and this one here-- 00:43:39.510 --> 00:43:47.210 align:middle line:84% this is the infinite sum i equal to 0 to infinity x plus y 00:43:47.210 --> 00:43:48.740 align:middle line:90% to the power n-- 00:43:48.740 --> 00:43:53.110 align:middle line:90% sorry, over-- i-- 00:43:53.110 --> 00:43:57.280 align:middle line:90% 00:43:57.280 --> 00:44:03.790 align:middle line:84% let's call it the sum from n, like that. 00:44:03.790 --> 00:44:06.090 align:middle line:90% That's what that series is. 00:44:06.090 --> 00:44:08.710 align:middle line:90% 00:44:08.710 --> 00:44:11.140 align:middle line:90% And this one here-- 00:44:11.140 --> 00:44:17.230 align:middle line:84% that's the sum x n, n factorial, and this one here 00:44:17.230 --> 00:44:25.450 align:middle line:90% is sum y n, n factorial. 00:44:25.450 --> 00:44:33.860 align:middle line:90% Now, so if I want cn-- 00:44:33.860 --> 00:44:39.640 align:middle line:84% so the idea is that if I'm writing cn as x plus y 00:44:39.640 --> 00:44:43.150 align:middle line:84% to the power n over n factorial-- 00:44:43.150 --> 00:44:46.390 align:middle line:84% suppose I write cn like this, and I 00:44:46.390 --> 00:44:53.150 align:middle line:84% write a n x n over n factorial, and I 00:44:53.150 --> 00:44:59.600 align:middle line:90% write bn y n over n factorial. 00:44:59.600 --> 00:45:03.390 align:middle line:90% Suppose I define them like this. 00:45:03.390 --> 00:45:06.180 align:middle line:84% I already know that this here is convergent 00:45:06.180 --> 00:45:08.310 align:middle line:90% and that here is convergent. 00:45:08.310 --> 00:45:09.800 align:middle line:84% By the way, of course, I also know 00:45:09.800 --> 00:45:12.780 align:middle line:90% that this here is convergent. 00:45:12.780 --> 00:45:16.970 align:middle line:90% But I now have that-- 00:45:16.970 --> 00:45:19.250 align:middle line:90% but I don't have-- 00:45:19.250 --> 00:45:23.660 align:middle line:84% I haven't proved-- but I claim now 00:45:23.660 --> 00:45:29.470 align:middle line:84% that this cn is really formed in this way. 00:45:29.470 --> 00:45:30.420 align:middle line:90% So let's see. 00:45:30.420 --> 00:45:38.240 align:middle line:84% So cn here-- if I define cn to be this thing here, 00:45:38.240 --> 00:45:40.590 align:middle line:90% by the binomial formula-- 00:45:40.590 --> 00:45:43.220 align:middle line:90% so x plus y-- 00:45:43.220 --> 00:45:46.130 align:middle line:84% if you take this here to the power n, 00:45:46.130 --> 00:45:55.980 align:middle line:84% this thing here is the sum here of x i times y n minus i. 00:45:55.980 --> 00:45:58.830 align:middle line:90% And then the coefficient is-- 00:45:58.830 --> 00:46:07.770 align:middle line:84% sorry-- x i times y to the n minus y, i equal to 0 through n. 00:46:07.770 --> 00:46:11.770 align:middle line:84% And the coefficient is n choose i. 00:46:11.770 --> 00:46:13.280 align:middle line:90% That's the binomial formula. 00:46:13.280 --> 00:46:29.916 align:middle line:90% 00:46:29.916 --> 00:46:35.670 align:middle line:90% Now, what is n choose i? 00:46:35.670 --> 00:46:40.740 align:middle line:90% Well, n choose i-- 00:46:40.740 --> 00:46:53.550 align:middle line:84% this is just n times n minus 1 down to n minus i plus 1. 00:46:53.550 --> 00:46:58.790 align:middle line:84% This gives you n elements, and then divided by i factorial. 00:46:58.790 --> 00:47:02.900 align:middle line:90% 00:47:02.900 --> 00:47:07.230 align:middle line:84% But you can write this thing here as n factorial. 00:47:07.230 --> 00:47:10.770 align:middle line:84% If you do that, then you included too many, 00:47:10.770 --> 00:47:17.100 align:middle line:84% so you have to divide by n minus i factorial. 00:47:17.100 --> 00:47:20.930 align:middle line:84% And then, of course, you still have the i factorial 00:47:20.930 --> 00:47:25.790 align:middle line:84% in the denominator, so you can write it like that 00:47:25.790 --> 00:47:29.450 align:middle line:90% So this means that the cn here-- 00:47:29.450 --> 00:47:36.830 align:middle line:84% our cn is-- so we defined cn like this-- x to the power 00:47:36.830 --> 00:47:41.261 align:middle line:84% plus y to the power n over n factorial. 00:47:41.261 --> 00:47:45.180 align:middle line:90% 00:47:45.180 --> 00:47:48.750 align:middle line:84% And if we use the binomial formula, 00:47:48.750 --> 00:47:50.680 align:middle line:90% then it was equal to that. 00:47:50.680 --> 00:47:57.480 align:middle line:84% So this here was equal to 1 over n factorial sum i equal to 0 00:47:57.480 --> 00:48:07.770 align:middle line:84% to n x i y n minus i, and then n choose i-- like that. 00:48:07.770 --> 00:48:13.800 align:middle line:84% Now I write in this little equation here. 00:48:13.800 --> 00:48:19.800 align:middle line:84% And so I get that 1 over n factorial summing from i 00:48:19.800 --> 00:48:28.140 align:middle line:84% equal to 0 through n, and then n factorial, n minus i 00:48:28.140 --> 00:48:35.040 align:middle line:84% factorial times i factorial, x to the i times 00:48:35.040 --> 00:48:37.920 align:middle line:90% y to the n minus i. 00:48:37.920 --> 00:48:42.220 align:middle line:84% And now you see that the n factorials here 00:48:42.220 --> 00:48:44.410 align:middle line:90% cancel with the n factorials. 00:48:44.410 --> 00:48:50.770 align:middle line:84% So you're left with i equal to 0 through n. 00:48:50.770 --> 00:48:52.970 align:middle line:90% And now I can group it. 00:48:52.970 --> 00:48:54.080 align:middle line:90% I can group it. 00:48:54.080 --> 00:49:04.270 align:middle line:84% I can write it as x i divided by i factorial times y 00:49:04.270 --> 00:49:09.290 align:middle line:90% over n minus i factorial. 00:49:09.290 --> 00:49:14.530 align:middle line:84% And this is to the power n minus i, like that. 00:49:14.530 --> 00:49:23.130 align:middle line:84% But now you see that this here is exactly when we defined 00:49:23.130 --> 00:49:25.970 align:middle line:90% a n like this and bn like this. 00:49:25.970 --> 00:49:31.030 align:middle line:84% This here is exactly the sum from i equal to 0 00:49:31.030 --> 00:49:36.600 align:middle line:90% through n of a i b n minus i. 00:49:36.600 --> 00:49:39.270 align:middle line:90% 00:49:39.270 --> 00:49:45.870 align:middle line:84% And so you see that what we have is that this thing here-- 00:49:45.870 --> 00:49:49.560 align:middle line:90% so E of x plus y-- 00:49:49.560 --> 00:49:52.740 align:middle line:84% with our definition of the c's, this 00:49:52.740 --> 00:49:56.345 align:middle line:84% was just defined to be equal to this limit. 00:49:56.345 --> 00:49:58.890 align:middle line:90% 00:49:58.890 --> 00:50:04.620 align:middle line:84% But the cn's is also given by the product of these two 00:50:04.620 --> 00:50:06.180 align:middle line:90% other series. 00:50:06.180 --> 00:50:12.030 align:middle line:84% But this means that it must be the product of the two, 00:50:12.030 --> 00:50:14.670 align:middle line:90% like that. 00:50:14.670 --> 00:50:21.970 align:middle line:84% It's the product of the two sums, infinite sums, like that. 00:50:21.970 --> 00:50:27.520 align:middle line:84% But this thing here is, of course, just E x, 00:50:27.520 --> 00:50:31.270 align:middle line:90% and this is just E y. 00:50:31.270 --> 00:50:32.370 align:middle line:90% So this proved it. 00:50:32.370 --> 00:50:37.041 align:middle line:90% 00:50:37.041 --> 00:50:41.500 align:middle line:84% And again, this here was just assuming that the x and y 00:50:41.500 --> 00:50:42.770 align:middle line:90% are non-negative. 00:50:42.770 --> 00:50:45.070 align:middle line:84% But given what I said-- also, this 00:50:45.070 --> 00:50:48.040 align:middle line:84% corollary-- it's not very hard to prove 00:50:48.040 --> 00:50:49.520 align:middle line:90% the general case from that. 00:50:49.520 --> 00:50:52.490 align:middle line:90% 00:50:52.490 --> 00:50:56.530 align:middle line:90% So now-- but I won't do that. 00:50:56.530 --> 00:51:05.180 align:middle line:84% So let's just talk a little bit more about continuous functions. 00:51:05.180 --> 00:51:08.210 align:middle line:84% So let me one more time just remind you about it, 00:51:08.210 --> 00:51:18.105 align:middle line:84% and then let me give you a wild example of a function. 00:51:18.105 --> 00:51:22.225 align:middle line:90% 00:51:22.225 --> 00:51:27.100 align:middle line:84% So first let me remind you with continuous. 00:51:27.100 --> 00:51:30.250 align:middle line:84% So continuous [INAUDIBLE] is a really important concept, also, 00:51:30.250 --> 00:51:32.320 align:middle line:90% in this class. 00:51:32.320 --> 00:51:37.160 align:middle line:84% So I have a function, and it doesn't need to be defined. 00:51:37.160 --> 00:51:43.370 align:middle line:84% It's defined, say A on some subset of R into R. 00:51:43.370 --> 00:51:45.810 align:middle line:90% And then we say that-- 00:51:45.810 --> 00:51:50.490 align:middle line:84% and then we have some point x0 in the domain. 00:51:50.490 --> 00:52:12.783 align:middle line:84% And we say that f here is continuous at x0 00:52:12.783 --> 00:52:13.700 align:middle line:90% if the following hold. 00:52:13.700 --> 00:52:17.210 align:middle line:90% That's for all epsilon. 00:52:17.210 --> 00:52:27.020 align:middle line:84% For all epsilon greater than 0, there exists delta greater 00:52:27.020 --> 00:52:38.320 align:middle line:84% than 0 such that if x is in the domain and x minus x0 00:52:38.320 --> 00:52:47.140 align:middle line:84% is less than delta, then the images 00:52:47.140 --> 00:52:50.730 align:middle line:90% deviate at most by epsilon. 00:52:50.730 --> 00:52:54.500 align:middle line:84% So this is, again, what it says to be continuous. 00:52:54.500 --> 00:52:57.370 align:middle line:84% And here, unlike the previous couple 00:52:57.370 --> 00:53:01.250 align:middle line:84% of times when I defined it, here it's really defined. 00:53:01.250 --> 00:53:05.150 align:middle line:84% You see it makes total sense on any subset of R. 00:53:05.150 --> 00:53:09.250 align:middle line:84% It doesn't need to be an interval, even. 00:53:09.250 --> 00:53:11.150 align:middle line:84% So it doesn't need to be all of R, 00:53:11.150 --> 00:53:12.720 align:middle line:84% but it does need to be an interval. 00:53:12.720 --> 00:53:17.830 align:middle line:90% 00:53:17.830 --> 00:53:23.720 align:middle line:84% Now, so this is what it means for a function to be continuous. 00:53:23.720 --> 00:53:27.045 align:middle line:84% Let me give you an example of a sort of somewhat wild function. 00:53:27.045 --> 00:53:30.550 align:middle line:90% 00:53:30.550 --> 00:53:35.730 align:middle line:84% And so continuous is a very nice property. 00:53:35.730 --> 00:53:40.340 align:middle line:84% It's continuous if it's continuous at all points. 00:53:40.340 --> 00:53:44.690 align:middle line:84% Usually, continuous is a nice property to have. 00:53:44.690 --> 00:53:48.090 align:middle line:90% Often, it's the bare minimum. 00:53:48.090 --> 00:53:49.850 align:middle line:90% It's often not enough. 00:53:49.850 --> 00:53:52.940 align:middle line:84% You want something more than continuous. 00:53:52.940 --> 00:54:01.850 align:middle line:84% But of course, other times, you may have much less. 00:54:01.850 --> 00:54:03.093 align:middle line:90% So here's an example. 00:54:03.093 --> 00:54:07.970 align:middle line:90% 00:54:07.970 --> 00:54:10.620 align:middle line:90% So this is an example. 00:54:10.620 --> 00:54:14.600 align:middle line:84% So it's a function defined on all of R-- 00:54:14.600 --> 00:54:27.590 align:middle line:84% example of a function defined on all of R 00:54:27.590 --> 00:54:32.236 align:middle line:90% that is nowhere continuous. 00:54:32.236 --> 00:54:42.540 align:middle line:90% 00:54:42.540 --> 00:54:46.140 align:middle line:84% By "nowhere continuous," I mean that there's not a single point 00:54:46.140 --> 00:54:49.375 align:middle line:90% where it's continuous. 00:54:49.375 --> 00:54:52.920 align:middle line:84% And so this is the function f of x. 00:54:52.920 --> 00:54:55.680 align:middle line:90% And the function is-- 00:54:55.680 --> 00:55:05.880 align:middle line:84% if x here is a rational point, then f of x is 0. 00:55:05.880 --> 00:55:15.420 align:middle line:84% And if x here is an irrational point, then [? I'm ?] setting it 00:55:15.420 --> 00:55:16.730 align:middle line:90% equal to 1. 00:55:16.730 --> 00:55:19.410 align:middle line:90% 00:55:19.410 --> 00:55:21.420 align:middle line:84% So it's a pretty simple function. 00:55:21.420 --> 00:55:26.820 align:middle line:84% And so I claim that this is not continuous at any point. 00:55:26.820 --> 00:55:28.865 align:middle line:90% And so the thing is that-- 00:55:28.865 --> 00:55:35.694 align:middle line:90% 00:55:35.694 --> 00:55:38.290 align:middle line:84% you see, if you take any points-- let's just 00:55:38.290 --> 00:55:40.230 align:middle line:90% take a rational point first. 00:55:40.230 --> 00:55:44.530 align:middle line:90% 00:55:44.530 --> 00:55:49.690 align:middle line:84% So suppose you take a rational point here. 00:55:49.690 --> 00:55:52.515 align:middle line:90% This here is-- x0 is rational. 00:55:52.515 --> 00:56:00.280 align:middle line:90% 00:56:00.280 --> 00:56:03.080 align:middle line:90% Then you have that-- 00:56:03.080 --> 00:56:07.430 align:middle line:84% if you are looking at square root of 2, 00:56:07.430 --> 00:56:11.240 align:middle line:84% we know that this here is not rational. 00:56:11.240 --> 00:56:12.105 align:middle line:90% We proved that. 00:56:12.105 --> 00:56:17.920 align:middle line:90% 00:56:17.920 --> 00:56:20.690 align:middle line:84% We showed that this is not rational. 00:56:20.690 --> 00:56:27.070 align:middle line:84% And so in fact, if you take 1 over-- 00:56:27.070 --> 00:56:30.340 align:middle line:84% sorry, square root of 2 divided by n, 00:56:30.340 --> 00:56:33.850 align:middle line:84% where n here is sufficiently large, 00:56:33.850 --> 00:56:37.190 align:middle line:84% then you can make this thing here very close to 0. 00:56:37.190 --> 00:56:39.820 align:middle line:90% 00:56:39.820 --> 00:56:42.220 align:middle line:84% And then you could add this thing here. 00:56:42.220 --> 00:56:49.530 align:middle line:84% So you could look at x0, and you could look at square root of 2 00:56:49.530 --> 00:56:53.880 align:middle line:90% over n for n really large. 00:56:53.880 --> 00:56:56.980 align:middle line:84% If n is really large, you can make this thing here, 00:56:56.980 --> 00:57:01.810 align:middle line:84% so this is square root of 2 plus x0. 00:57:01.810 --> 00:57:05.880 align:middle line:84% So you could make this really, really close to x0. 00:57:05.880 --> 00:57:09.210 align:middle line:84% This thing here would be irrational, 00:57:09.210 --> 00:57:13.350 align:middle line:84% and so the value of the function here would be 1. 00:57:13.350 --> 00:57:15.533 align:middle line:84% Let's make this a little bit more precise. 00:57:15.533 --> 00:57:19.980 align:middle line:90% 00:57:19.980 --> 00:57:23.430 align:middle line:84% And so-- let me maybe say it just 00:57:23.430 --> 00:57:25.030 align:middle line:90% before I make it more precise. 00:57:25.030 --> 00:57:28.060 align:middle line:84% So let's just draw the graph here. 00:57:28.060 --> 00:57:31.960 align:middle line:90% So we have here x0. 00:57:31.960 --> 00:57:33.800 align:middle line:90% We have here this-- 00:57:33.800 --> 00:57:36.610 align:middle line:84% this here is this irrational number, 00:57:36.610 --> 00:57:38.530 align:middle line:84% which you could think about, but it 00:57:38.530 --> 00:57:39.920 align:middle line:90% doesn't need to be this number. 00:57:39.920 --> 00:57:42.470 align:middle line:84% It could be anything that is irrational. 00:57:42.470 --> 00:57:44.510 align:middle line:90% This one here is irrational. 00:57:44.510 --> 00:57:48.940 align:middle line:84% And by making n super large, this here super close 00:57:48.940 --> 00:57:54.010 align:middle line:84% to x0, the value of the function here is 0. 00:57:54.010 --> 00:57:58.170 align:middle line:84% The value of the function here is 1. 00:57:58.170 --> 00:58:01.480 align:middle line:90% 00:58:01.480 --> 00:58:05.050 align:middle line:84% And so you see that you have these points as 00:58:05.050 --> 00:58:08.210 align:middle line:84% close to this point here as you want, 00:58:08.210 --> 00:58:11.580 align:middle line:84% but the value of the function is very different 00:58:11.580 --> 00:58:22.210 align:middle line:84% So again, just to spell it out, given an epsilon-- 00:58:22.210 --> 00:58:27.940 align:middle line:90% 00:58:27.940 --> 00:58:37.730 align:middle line:84% so you see that if epsilon here was smaller than 1, 00:58:37.730 --> 00:58:45.180 align:middle line:84% then you would have that f of x minus f of x0-- 00:58:45.180 --> 00:58:48.780 align:middle line:90% this here is a rational point. 00:58:48.780 --> 00:58:51.960 align:middle line:90% So this here is-- 00:58:51.960 --> 00:58:57.140 align:middle line:84% this is 0, so this is just f of x. 00:58:57.140 --> 00:59:04.670 align:middle line:84% And if this here is irrational-- so if x here is irrational, 00:59:04.670 --> 00:59:07.580 align:middle line:90% then this here is 1. 00:59:07.580 --> 00:59:14.720 align:middle line:84% So you would have that you can choose this irrational point. 00:59:14.720 --> 00:59:35.500 align:middle line:84% So for any delta, we can choose x, which is irrational, 00:59:35.500 --> 00:59:42.040 align:middle line:84% and so that x minus x0 is smaller than delta, but we have 00:59:42.040 --> 00:59:49.100 align:middle line:84% that f of x here is actually equal to 1. 00:59:49.100 --> 00:59:50.980 align:middle line:84% But this difference here was supposed 00:59:50.980 --> 00:59:53.755 align:middle line:90% to be smaller than delta. 00:59:53.755 --> 00:59:55.360 align:middle line:90% That is not the case. 00:59:55.360 --> 00:59:58.100 align:middle line:90% 00:59:58.100 --> 01:00:02.030 align:middle line:84% If epsilon is smaller than 1, it's definitely not. 01:00:02.030 --> 01:00:02.980 align:middle line:90% It's 1. 01:00:02.980 --> 01:00:08.590 align:middle line:90% 01:00:08.590 --> 01:00:13.460 align:middle line:84% And so similarly, if you take-- so this was proving-- 01:00:13.460 --> 01:00:37.340 align:middle line:84% so this showed-- so this proof proved that f is discontinuous 01:00:37.340 --> 01:00:40.380 align:middle line:90% at all rational points. 01:00:40.380 --> 01:00:45.920 align:middle line:90% 01:00:45.920 --> 01:00:48.140 align:middle line:84% Now let's prove that it's discontinuous 01:00:48.140 --> 01:00:51.095 align:middle line:90% at all irrational points. 01:00:51.095 --> 01:01:00.860 align:middle line:84% So if x [INAUDIBLE] 0 is irrational, 01:01:00.860 --> 01:01:07.610 align:middle line:84% then we have, of course, that f of x0 is equal to 1. 01:01:07.610 --> 01:01:11.880 align:middle line:84% But we also know that arbitrary close-- 01:01:11.880 --> 01:01:26.040 align:middle line:84% so for any delta greater than 0, there 01:01:26.040 --> 01:01:39.540 align:middle line:84% exists a rational point, a rational number x, 01:01:39.540 --> 01:01:45.750 align:middle line:84% so that x minus x0 is smaller than delta. 01:01:45.750 --> 01:01:52.530 align:middle line:84% But now, again, we have that f of x minus f of x0-- 01:01:52.530 --> 01:01:56.610 align:middle line:84% this thing here is 1, this thing here is 0, 01:01:56.610 --> 01:02:00.900 align:middle line:90% so this difference here is 1. 01:02:00.900 --> 01:02:04.750 align:middle line:84% And so you couldn't make it strictly less than 1 01:02:04.750 --> 01:02:07.365 align:middle line:84% by having this delta sufficiently small. 01:02:07.365 --> 01:02:09.910 align:middle line:84% So this proof is sort of same thing. 01:02:09.910 --> 01:02:16.830 align:middle line:84% It proves that the function is also discontinuous 01:02:16.830 --> 01:02:19.295 align:middle line:90% at all irrational points. 01:02:19.295 --> 01:02:21.800 align:middle line:90% 01:02:21.800 --> 01:02:26.713 align:middle line:84% Now, let's talk a little bit more-- 01:02:26.713 --> 01:02:29.970 align:middle line:90% 01:02:29.970 --> 01:02:32.060 align:middle line:84% so let's talk a little bit more about some 01:02:32.060 --> 01:02:38.450 align:middle line:84% of the algebraic rules that we discussed last time. 01:02:38.450 --> 01:02:40.945 align:middle line:84% And I wanted to single out two of them. 01:02:40.945 --> 01:02:50.130 align:middle line:90% 01:02:50.130 --> 01:02:52.590 align:middle line:90% So we had the following. 01:02:52.590 --> 01:03:09.900 align:middle line:84% If f and g are continuous functions and c is a constant-- 01:03:09.900 --> 01:03:12.740 align:middle line:90% 01:03:12.740 --> 01:03:16.700 align:middle line:84% then last time, we talked about the following. 01:03:16.700 --> 01:03:22.290 align:middle line:84% We talked about that f plus g, the sum of these two functions, 01:03:22.290 --> 01:03:23.626 align:middle line:90% is also continuous. 01:03:23.626 --> 01:03:30.450 align:middle line:90% 01:03:30.450 --> 01:03:34.380 align:middle line:84% We talked about that if you take a function 01:03:34.380 --> 01:03:36.580 align:middle line:84% and you multiply it by a constant, 01:03:36.580 --> 01:03:39.690 align:middle line:90% this is also continuous. 01:03:39.690 --> 01:03:43.980 align:middle line:84% We talked about that if you're looking 01:03:43.980 --> 01:03:47.040 align:middle line:84% at the product of these functions, that's 01:03:47.040 --> 01:03:48.870 align:middle line:90% also continuous. 01:03:48.870 --> 01:03:52.620 align:middle line:84% The ones we mentioned, but we didn't actually-- 01:03:52.620 --> 01:03:57.600 align:middle line:84% this here was so elementary, this one here I outlined, 01:03:57.600 --> 01:04:00.730 align:middle line:84% and let's talk about the last two properties. 01:04:00.730 --> 01:04:05.230 align:middle line:84% One is close to what we had for sequences. 01:04:05.230 --> 01:04:09.640 align:middle line:84% And that was that if f here does not vanish, 01:04:09.640 --> 01:04:18.130 align:middle line:84% it's nowhere vanishing, then 1/f here is also continuous. 01:04:18.130 --> 01:04:21.340 align:middle line:84% And the last one, which is sort of different from anything 01:04:21.340 --> 01:04:29.320 align:middle line:84% we had for our sequences, is that if you 01:04:29.320 --> 01:04:33.070 align:middle line:90% have now a function-- 01:04:33.070 --> 01:04:36.040 align:middle line:84% so you have these two functions, f and g. 01:04:36.040 --> 01:04:38.710 align:middle line:84% But now I'm thinking about f here 01:04:38.710 --> 01:04:46.930 align:middle line:84% as going from, say, some subset of R into R, 01:04:46.930 --> 01:04:50.120 align:middle line:84% but it may not [INAUDIBLE] into some set B. 01:04:50.120 --> 01:04:54.850 align:middle line:84% This is a subset of R. This is another subset of R. 01:04:54.850 --> 01:05:03.380 align:middle line:84% And then I have g here goes from this set into R. 01:05:03.380 --> 01:05:06.080 align:middle line:90% It needs to be-- 01:05:06.080 --> 01:05:11.420 align:middle line:84% so you need this here to be defined on the images of f. 01:05:11.420 --> 01:05:18.650 align:middle line:84% If you have that, and these two functions are continuous, then 01:05:18.650 --> 01:05:21.660 align:middle line:90% if you form the composition-- 01:05:21.660 --> 01:05:24.680 align:middle line:90% so this is g of f of x-- 01:05:24.680 --> 01:05:29.426 align:middle line:84% this new function h here is also continuous. 01:05:29.426 --> 01:05:38.690 align:middle line:90% 01:05:38.690 --> 01:05:43.520 align:middle line:90% And so I want to-- 01:05:43.520 --> 01:05:46.580 align:middle line:84% in the few remaining minutes, I want 01:05:46.580 --> 01:05:49.140 align:middle line:84% to talk about these two properties. 01:05:49.140 --> 01:05:52.490 align:middle line:90% 01:05:52.490 --> 01:05:54.860 align:middle line:84% And I will start with the last one, 01:05:54.860 --> 01:05:59.090 align:middle line:84% because that's somewhat different from stuff 01:05:59.090 --> 01:06:01.880 align:middle line:90% we talked about last time-- 01:06:01.880 --> 01:06:05.400 align:middle line:90% I mean for sequences. 01:06:05.400 --> 01:06:11.840 align:middle line:90% 01:06:11.840 --> 01:06:19.990 align:middle line:84% So let's try to prove that this composition here is continuous, 01:06:19.990 --> 01:06:21.810 align:middle line:90% so proof of 5. 01:06:21.810 --> 01:06:30.490 align:middle line:90% 01:06:30.490 --> 01:06:41.290 align:middle line:84% So given epsilon, well, we know that the function-- 01:06:41.290 --> 01:06:43.310 align:middle line:90% you sort of to start backwards. 01:06:43.310 --> 01:06:45.560 align:middle line:90% Given epsilon, you know-- 01:06:45.560 --> 01:06:46.060 align:middle line:90% sorry. 01:06:46.060 --> 01:06:50.020 align:middle line:90% 01:06:50.020 --> 01:06:52.400 align:middle line:84% Let me just say that I want to prove it. 01:06:52.400 --> 01:07:01.900 align:middle line:84% Let's show that h, which is the composition, 01:07:01.900 --> 01:07:09.040 align:middle line:90% is continuous at x0. 01:07:09.040 --> 01:07:12.980 align:middle line:90% 01:07:12.980 --> 01:07:15.630 align:middle line:90% And so I want to prove that. 01:07:15.630 --> 01:07:19.190 align:middle line:84% And for convenience, I'm just going 01:07:19.190 --> 01:07:25.060 align:middle line:84% to call the image of x0 under f, and we'll call that y. 01:07:25.060 --> 01:07:29.180 align:middle line:90% 01:07:29.180 --> 01:07:43.015 align:middle line:84% So now, given epsilon bigger than 0, 01:07:43.015 --> 01:07:53.100 align:middle line:90% since g is continuous at y0-- 01:07:53.100 --> 01:07:57.170 align:middle line:90% note that this is y0. 01:07:57.170 --> 01:08:09.680 align:middle line:84% So since g is continuous at y0, there exists a delta-- 01:08:09.680 --> 01:08:12.210 align:middle line:90% let's call it delta g-- 01:08:12.210 --> 01:08:17.250 align:middle line:90% so that if y minus y0-- 01:08:17.250 --> 01:08:20.430 align:middle line:90% if this is less than delta g-- 01:08:20.430 --> 01:08:32.729 align:middle line:84% then the images-- sorry, under g not under f, but under g. 01:08:32.729 --> 01:08:41.220 align:middle line:84% So then the images g of y minus g of y0-- 01:08:41.220 --> 01:08:45.319 align:middle line:84% the images under g is smaller than epsilon. 01:08:45.319 --> 01:08:51.782 align:middle line:90% 01:08:51.782 --> 01:08:54.930 align:middle line:84% So this is just because g is continuous, 01:08:54.930 --> 01:08:57.663 align:middle line:84% and I'm using that it's continuous at y0. 01:08:57.663 --> 01:09:00.330 align:middle line:90% 01:09:00.330 --> 01:09:04.926 align:middle line:84% Now I need to use that f is continuous. 01:09:04.926 --> 01:09:10.399 align:middle line:90% 01:09:10.399 --> 01:09:14.810 align:middle line:84% So now I'm thinking about this delta g 01:09:14.810 --> 01:09:18.720 align:middle line:90% as my epsilon [INAUDIBLE] in f. 01:09:18.720 --> 01:09:51.800 align:middle line:84% So from delta g and the continuity of f, 01:09:51.800 --> 01:10:06.320 align:middle line:84% there exists delta of f such that, if x minus x0 01:10:06.320 --> 01:10:12.390 align:middle line:84% is less than delta of f, then the images 01:10:12.390 --> 01:10:17.640 align:middle line:90% f of x minus f of x0-- 01:10:17.640 --> 01:10:20.041 align:middle line:90% the images is less than delta g. 01:10:20.041 --> 01:10:23.610 align:middle line:90% 01:10:23.610 --> 01:10:27.270 align:middle line:90% But now you see-- now I have-- 01:10:27.270 --> 01:10:30.130 align:middle line:90% this thing here-- f of x-- 01:10:30.130 --> 01:10:33.005 align:middle line:84% this thing here-- this is f of x. 01:10:33.005 --> 01:10:35.670 align:middle line:90% 01:10:35.670 --> 01:10:42.550 align:middle line:84% And this thing here, f of x0, was y0. 01:10:42.550 --> 01:10:44.385 align:middle line:90% So this is the same as this. 01:10:44.385 --> 01:10:47.310 align:middle line:90% 01:10:47.310 --> 01:10:49.510 align:middle line:90% But now you see that-- 01:10:49.510 --> 01:10:52.900 align:middle line:84% so I have these two points that they are-- 01:10:52.900 --> 01:10:57.100 align:middle line:84% and where one of them is y0, and then I have this other point, 01:10:57.100 --> 01:11:03.600 align:middle line:84% and their difference is bounded by delta g. 01:11:03.600 --> 01:11:05.770 align:middle line:90% So I'm in iteration. 01:11:05.770 --> 01:11:08.710 align:middle line:84% So this means that the images of these two-- 01:11:08.710 --> 01:11:13.780 align:middle line:84% so this implies that the images of this thing here under g 01:11:13.780 --> 01:11:23.950 align:middle line:84% and this here on g-- so g of f of x minus g of f-- 01:11:23.950 --> 01:11:27.730 align:middle line:90% sorry, g of y0-- 01:11:27.730 --> 01:11:30.625 align:middle line:84% this thing here is less than epsilon. 01:11:30.625 --> 01:11:34.930 align:middle line:90% 01:11:34.930 --> 01:11:39.370 align:middle line:84% This is just because that was how delta g was chosen. 01:11:39.370 --> 01:11:43.390 align:middle line:84% And you see that one of them-- one of these two points 01:11:43.390 --> 01:11:49.090 align:middle line:84% have to be y0, but that's exactly what it is over there. 01:11:49.090 --> 01:11:51.010 align:middle line:90% So I have this-- 01:11:51.010 --> 01:11:54.910 align:middle line:90% but you see, this thing here-- 01:11:54.910 --> 01:11:56.610 align:middle line:90% but this thing here is just-- 01:11:56.610 --> 01:12:04.470 align:middle line:90% 01:12:04.470 --> 01:12:09.720 align:middle line:90% but you see that g of f of x-- 01:12:09.720 --> 01:12:11.950 align:middle line:90% this is just the composition. 01:12:11.950 --> 01:12:16.320 align:middle line:90% So this is what we call h of x. 01:12:16.320 --> 01:12:27.660 align:middle line:84% g of y0, where y0 was defined to be f of x0-- 01:12:27.660 --> 01:12:34.680 align:middle line:84% so this thing here is g of f of x0. 01:12:34.680 --> 01:12:39.330 align:middle line:84% But that's just the composition at x0. 01:12:39.330 --> 01:12:42.550 align:middle line:84% So you see, what we have now is-- 01:12:42.550 --> 01:12:50.610 align:middle line:84% So therefore, we have that if x minus x0-- 01:12:50.610 --> 01:12:55.890 align:middle line:84% if this thing here is less than this delta f, 01:12:55.890 --> 01:13:00.520 align:middle line:90% then we have this thing here. 01:13:00.520 --> 01:13:09.580 align:middle line:84% But this thing here is h of x minus h of x0 01:13:09.580 --> 01:13:11.860 align:middle line:90% is less than epsilon. 01:13:11.860 --> 01:13:15.490 align:middle line:84% And so this proves that the composition of these two 01:13:15.490 --> 01:13:18.280 align:middle line:90% functions is continuous. 01:13:18.280 --> 01:13:22.420 align:middle line:84% And so again, when you have two functions 01:13:22.420 --> 01:13:24.910 align:middle line:84% and you look at the composition, it's 01:13:24.910 --> 01:13:30.770 align:middle line:84% really important to first check that the second one is defined. 01:13:30.770 --> 01:13:35.230 align:middle line:84% So the second one should be defined where 01:13:35.230 --> 01:13:37.945 align:middle line:90% you end up under the first map. 01:13:37.945 --> 01:13:50.590 align:middle line:90% 01:13:50.590 --> 01:13:52.720 align:middle line:84% So now let me just say a few words 01:13:52.720 --> 01:13:57.700 align:middle line:90% about the other one, number 4. 01:13:57.700 --> 01:14:10.280 align:middle line:84% But the number 4 is closer to things we had for sequences. 01:14:10.280 --> 01:14:12.415 align:middle line:90% So we have now proven [? 4. ?] 01:14:12.415 --> 01:14:17.780 align:middle line:90% 01:14:17.780 --> 01:14:23.420 align:middle line:84% So what we have is we have this function, it's never 0, 01:14:23.420 --> 01:14:29.270 align:middle line:84% and we have some x0 in the domain, 01:14:29.270 --> 01:14:32.016 align:middle line:84% and we know that f here is continuous. 01:14:32.016 --> 01:14:37.730 align:middle line:90% 01:14:37.730 --> 01:14:40.890 align:middle line:84% We only need it to be continuous at x0. 01:14:40.890 --> 01:15:00.580 align:middle line:84% And then I want to show that 1/f of x is continuous at x0. 01:15:00.580 --> 01:15:03.250 align:middle line:90% And so I'm looking at-- 01:15:03.250 --> 01:15:06.825 align:middle line:84% so I want to estimate this difference here. 01:15:06.825 --> 01:15:10.720 align:middle line:90% 01:15:10.720 --> 01:15:15.400 align:middle line:84% And I want to say that if I pick an epsilon, then 01:15:15.400 --> 01:15:17.660 align:middle line:90% I can pick x as close-- 01:15:17.660 --> 01:15:22.720 align:middle line:84% if I pick x to be within delta, where delta is sufficiently 01:15:22.720 --> 01:15:25.960 align:middle line:84% small, of x0, then this difference here 01:15:25.960 --> 01:15:28.450 align:middle line:90% can be smaller than epsilon. 01:15:28.450 --> 01:15:34.270 align:middle line:84% So now, let me just estimate this thing here. 01:15:34.270 --> 01:15:37.610 align:middle line:90% And so this thing here is just-- 01:15:37.610 --> 01:15:40.310 align:middle line:84% so now I just put it on the common denominator. 01:15:40.310 --> 01:15:46.030 align:middle line:84% So I have f of x0 minus f of x divided 01:15:46.030 --> 01:15:56.410 align:middle line:90% by f of x0 f of x [INAUDIBLE]. 01:15:56.410 --> 01:15:59.790 align:middle line:84% And now it's sort of the usual problem, 01:15:59.790 --> 01:16:02.090 align:middle line:84% but now we're maybe not so worried 01:16:02.090 --> 01:16:04.610 align:middle line:90% about that usual problem. 01:16:04.610 --> 01:16:08.370 align:middle line:84% The usual problem is this x0 is fixed, 01:16:08.370 --> 01:16:10.140 align:middle line:90% so this is just some number. 01:16:10.140 --> 01:16:12.270 align:middle line:84% So that doesn't really matter too much. 01:16:12.270 --> 01:16:17.340 align:middle line:84% This one here is not fixed, so that's the usual problem. 01:16:17.340 --> 01:16:19.790 align:middle line:84% This is a problem we've seen before. 01:16:19.790 --> 01:16:23.570 align:middle line:84% We saw it in 3 up there, and we saw it in the similar things 01:16:23.570 --> 01:16:26.750 align:middle line:90% for sequences. 01:16:26.750 --> 01:16:45.290 align:middle line:84% But now the thing is that since f of x0 is not equal to 0, then 01:16:45.290 --> 01:16:47.100 align:middle line:90% you can just take-- 01:16:47.100 --> 01:16:49.135 align:middle line:90% so then now you can first take-- 01:16:49.135 --> 01:16:52.110 align:middle line:90% 01:16:52.110 --> 01:17:03.570 align:middle line:84% so now we're going to pick the delta we want in two steps. 01:17:03.570 --> 01:17:06.700 align:middle line:84% The first step is since this here is not 0, 01:17:06.700 --> 01:17:17.430 align:middle line:84% then we know that there exists a delta 1 greater than 0 so 01:17:17.430 --> 01:17:23.340 align:middle line:84% that if x minus x0 is smaller than delta 1, 01:17:23.340 --> 01:17:29.460 align:middle line:84% then the image between f of x and f of x0-- 01:17:29.460 --> 01:17:39.150 align:middle line:84% I can make that smaller than f of x0 over 2. 01:17:39.150 --> 01:17:45.366 align:middle line:84% You see, I'm just using this one here first as my epsilon. 01:17:45.366 --> 01:17:47.430 align:middle line:90% Once I have that-- 01:17:47.430 --> 01:17:52.980 align:middle line:84% but this here implies, of course, that f of x-- 01:17:52.980 --> 01:17:58.060 align:middle line:84% see, I have f of x in the denominator, 01:17:58.060 --> 01:18:01.160 align:middle line:84% and I want to bound this here from above. 01:18:01.160 --> 01:18:03.660 align:middle line:90% 01:18:03.660 --> 01:18:04.910 align:middle line:90% I want to bound it from above. 01:18:04.910 --> 01:18:09.190 align:middle line:84% So I just need to get a lower bound for f of x. 01:18:09.190 --> 01:18:12.970 align:middle line:90% So f of x here-- 01:18:12.970 --> 01:18:17.113 align:middle line:90% so I have that f of x plus-- 01:18:17.113 --> 01:18:20.140 align:middle line:90% 01:18:20.140 --> 01:18:23.530 align:middle line:84% I can write-- sorry, I can write. 01:18:23.530 --> 01:18:35.293 align:middle line:84% Using this, I can write f of x0 as f of x minus-- 01:18:35.293 --> 01:18:38.680 align:middle line:90% 01:18:38.680 --> 01:18:43.570 align:middle line:90% sorry, f of x0 like this-- 01:18:43.570 --> 01:18:47.775 align:middle line:90% minus f of x plus f of x. 01:18:47.775 --> 01:18:49.275 align:middle line:84% Obviously, I can write it like that. 01:18:49.275 --> 01:18:51.790 align:middle line:90% 01:18:51.790 --> 01:18:58.800 align:middle line:84% And so this means that f of x0, slapping an absolute value 01:18:58.800 --> 01:19:03.420 align:middle line:84% sign on it-- this here is less than f of x0 01:19:03.420 --> 01:19:07.835 align:middle line:90% minus f of x plus f of x. 01:19:07.835 --> 01:19:10.590 align:middle line:90% 01:19:10.590 --> 01:19:14.195 align:middle line:84% And so now I can move this one here over on the other side. 01:19:14.195 --> 01:19:20.640 align:middle line:90% 01:19:20.640 --> 01:19:23.850 align:middle line:84% So I have that-- so if I move this one here 01:19:23.850 --> 01:19:29.820 align:middle line:84% over on the other side, I get that f of x absolute value 01:19:29.820 --> 01:19:35.100 align:middle line:84% is bigger or equal to f of x0, absolute value, 01:19:35.100 --> 01:19:44.900 align:middle line:84% minus this difference here, like that. 01:19:44.900 --> 01:19:50.170 align:middle line:84% But if I make x and x0 delta 1 close to each other, 01:19:50.170 --> 01:19:53.170 align:middle line:84% then this difference here is at-- 01:19:53.170 --> 01:19:55.370 align:middle line:84% it's actually less than this thing here. 01:19:55.370 --> 01:19:57.560 align:middle line:90% So there's still another 1/2. 01:19:57.560 --> 01:20:04.960 align:middle line:84% You see, this one here must be bigger or equal to 1/2 of f of x 01:20:04.960 --> 01:20:12.160 align:middle line:84% here, because this here takes up strictly less than half of this 01:20:12.160 --> 01:20:16.550 align:middle line:84% thing here, so there's still another 1/2 left 01:20:16.550 --> 01:20:19.280 align:middle line:84% And so this gives us a lower bound for this. 01:20:19.280 --> 01:20:20.540 align:middle line:90% And now we are out of time. 01:20:20.540 --> 01:20:22.880 align:middle line:84% but this gives us a lower bound for this, 01:20:22.880 --> 01:20:28.610 align:middle line:84% as long as we make sure that x and x0 is delta 1 close. 01:20:28.610 --> 01:20:30.220 align:middle line:84% And once you have that, you can just 01:20:30.220 --> 01:20:32.030 align:middle line:90% treat this here as a constant. 01:20:32.030 --> 01:20:34.630 align:middle line:84% And so now it's clear that you then 01:20:34.630 --> 01:20:39.490 align:middle line:84% make another choice of delta, possibly even smaller 01:20:39.490 --> 01:20:41.440 align:middle line:90% than this one here. 01:20:41.440 --> 01:20:45.130 align:middle line:84% And then for this smallest of these two numbers, then 01:20:45.130 --> 01:20:47.710 align:middle line:84% you can now conclude that this fraction here 01:20:47.710 --> 01:20:48.970 align:middle line:90% is smaller than epsilon. 01:20:48.970 --> 01:20:52.580 align:middle line:90% 01:20:52.580 --> 01:20:54.442 align:middle line:90% Any questions? 01:20:54.442 --> 01:20:55.395 align:middle line:90% Yeah? 01:20:55.395 --> 01:20:57.395 align:middle line:84% AUDIENCE: I had a question going back to what we 01:20:57.395 --> 01:20:58.850 align:middle line:90% were talking about [INAUDIBLE]. 01:20:58.850 --> 01:21:01.340 align:middle line:84% I was a little bit confused because we're defining 01:21:01.340 --> 01:21:03.270 align:middle line:90% it to be non-continuous. 01:21:03.270 --> 01:21:07.100 align:middle line:84% But by definition of continuity, if you 01:21:07.100 --> 01:21:12.470 align:middle line:84% picked two x's near each other that were both rational, 01:21:12.470 --> 01:21:16.310 align:middle line:84% and their f of x is both 0, then wouldn't that be continuous? 01:21:16.310 --> 01:21:18.380 align:middle line:84% TOBIAS COLDING: So first of all, let 01:21:18.380 --> 01:21:22.397 align:middle line:84% me say that if you were to picture this function-- 01:21:22.397 --> 01:21:24.480 align:middle line:84% of course, it wouldn't be very easy to picture it. 01:21:24.480 --> 01:21:28.760 align:middle line:84% But you would sort of-- if you were to draw it, then actually, 01:21:28.760 --> 01:21:29.600 align:middle line:90% it would look-- 01:21:29.600 --> 01:21:31.730 align:middle line:84% it wouldn't look like a function anymore, 01:21:31.730 --> 01:21:35.700 align:middle line:84% because you would have it on all the rational numbers here, 01:21:35.700 --> 01:21:39.230 align:middle line:84% it would be 0, and all the irrational numbers here, it 01:21:39.230 --> 01:21:40.020 align:middle line:90% would be 1. 01:21:40.020 --> 01:21:42.540 align:middle line:84% So if you picture it, it would look like two lines. 01:21:42.540 --> 01:21:45.830 align:middle line:84% But of course, it wouldn't be two lines because it's-- 01:21:45.830 --> 01:21:47.780 align:middle line:84% but the points are just so close together 01:21:47.780 --> 01:21:49.780 align:middle line:84% that you couldn't really draw it. 01:21:49.780 --> 01:21:51.680 align:middle line:90% And so the thing is that-- 01:21:51.680 --> 01:21:54.280 align:middle line:90% but it also means that if you-- 01:21:54.280 --> 01:21:56.140 align:middle line:84% but this picture here illustrates 01:21:56.140 --> 01:21:59.900 align:middle line:90% what you're saying, that for-- 01:21:59.900 --> 01:22:02.240 align:middle line:84% if you take rational numbers, well, 01:22:02.240 --> 01:22:05.140 align:middle line:84% all the rational numbers that is really-- 01:22:05.140 --> 01:22:07.720 align:middle line:84% any rational number, whether it's close or not-- it 01:22:07.720 --> 01:22:09.800 align:middle line:90% will all be 0 on those. 01:22:09.800 --> 01:22:14.360 align:middle line:84% So that's not where the problem with continuity occurs. 01:22:14.360 --> 01:22:17.410 align:middle line:84% The problem occurs because you also have 01:22:17.410 --> 01:22:19.430 align:middle line:90% irrational numbers close to it. 01:22:19.430 --> 01:22:22.990 align:middle line:84% And it has to be for all points close to this 01:22:22.990 --> 01:22:25.730 align:middle line:84% that the value has to be almost the same. 01:22:25.730 --> 01:22:28.930 align:middle line:84% So it's really for the irrational numbers 01:22:28.930 --> 01:22:32.590 align:middle line:84% that the problem occurs with continuity-- 01:22:32.590 --> 01:22:33.760 align:middle line:90% in irrational numbers. 01:22:33.760 --> 01:22:36.800 align:middle line:84% And vice versa for an irrational number-- 01:22:36.800 --> 01:22:39.710 align:middle line:84% if you're looking at continuity around an irrational, 01:22:39.710 --> 01:22:42.590 align:middle line:84% the problem is not from other irrational numbers, 01:22:42.590 --> 01:22:45.270 align:middle line:84% but is from the rational numbers. 01:22:45.270 --> 01:22:48.000 align:middle line:90%