WEBVTT 00:00:00.000 --> 00:00:00.962 align:middle line:90% [SQUEAKING] 00:00:00.962 --> 00:00:01.924 align:middle line:90% 00:00:01.924 --> 00:00:02.886 align:middle line:90% [RUSTLING] 00:00:02.886 --> 00:00:04.329 align:middle line:90% 00:00:04.329 --> 00:00:05.291 align:middle line:90% [CLICKING] 00:00:05.291 --> 00:00:12.520 align:middle line:90% 00:00:12.520 --> 00:00:14.590 align:middle line:84% CASEY RODRIGUEZ: All right, so we 00:00:14.590 --> 00:00:19.560 align:middle line:84% proved these two theorems last time, and we used them for-- 00:00:19.560 --> 00:00:21.880 align:middle line:84% and we had a couple of applications of them. 00:00:21.880 --> 00:00:24.060 align:middle line:84% So the first theorem, simple theorem, 00:00:24.060 --> 00:00:28.650 align:middle line:84% was that a sequence converges to x if and only if the limit as n 00:00:28.650 --> 00:00:32.340 align:middle line:84% goes to infinity of the absolute value of xn minus x goes to 0. 00:00:32.340 --> 00:00:35.670 align:middle line:84% And then we also had the squeeze theorem, that if you have two-- 00:00:35.670 --> 00:00:38.610 align:middle line:84% if you have three sequences-- a sub n, 00:00:38.610 --> 00:00:42.480 align:middle line:84% b sub n, and x sub n-- so that x sub n is in between a sub 00:00:42.480 --> 00:00:46.470 align:middle line:84% n and b sub n, and the a sub n and b sub n converge 00:00:46.470 --> 00:00:51.130 align:middle line:84% to the same thing, then the sequence x sub n converges, 00:00:51.130 --> 00:00:53.470 align:middle line:84% and it converges to the [? common ?] 00:00:53.470 --> 00:00:56.050 align:middle line:90% limit of a sub n and b sub n. 00:00:56.050 --> 00:01:00.100 align:middle line:84% So x sub n gets squeezed in between a sub n and b sub n. 00:01:00.100 --> 00:01:09.730 align:middle line:90% 00:01:09.730 --> 00:01:16.450 align:middle line:84% Last time we used this to prove a kind of special limit 00:01:16.450 --> 00:01:24.660 align:middle line:84% that this equals 0 if the absolute value of c 00:01:24.660 --> 00:01:25.440 align:middle line:90% is less than 1. 00:01:25.440 --> 00:01:29.370 align:middle line:84% So I had proved it for c positive, and maybe 00:01:29.370 --> 00:01:32.850 align:middle line:84% c less than 1-- but same proof works with absolute values, 00:01:32.850 --> 00:01:35.970 align:middle line:84% just because the absolute value of c to the n 00:01:35.970 --> 00:01:42.300 align:middle line:84% equals the absolute value of c raised to the n-th power. 00:01:42.300 --> 00:01:46.370 align:middle line:84% So let's use this to do a few more 00:01:46.370 --> 00:01:48.840 align:middle line:90% special limits if you like. 00:01:48.840 --> 00:01:54.020 align:middle line:84% But first, I'm going to state the binomial theorem. 00:01:54.020 --> 00:01:59.630 align:middle line:84% I'm not going to prove it, but it's a simple exercise 00:01:59.630 --> 00:02:01.100 align:middle line:90% using induction. 00:02:01.100 --> 00:02:05.250 align:middle line:90% 00:02:05.250 --> 00:02:12.730 align:middle line:84% And the binomial theorem says that [? for all ?] [? N, the ?] 00:02:12.730 --> 00:02:21.130 align:middle line:84% natural numbers, x, y in R, x plus y raised to the n is equal 00:02:21.130 --> 00:02:27.875 align:middle line:84% to the sum from k equals 0 to n of n choose k, 00:02:27.875 --> 00:02:32.080 align:middle line:90% x to the n minus k y to the k. 00:02:32.080 --> 00:02:33.490 align:middle line:90% And here n choose k-- 00:02:33.490 --> 00:02:37.240 align:middle line:90% 00:02:37.240 --> 00:02:41.020 align:middle line:84% this is equal to n factorial over k factorial divided 00:02:41.020 --> 00:02:44.140 align:middle line:90% by n minus-- 00:02:44.140 --> 00:02:47.470 align:middle line:90% times n minus k factorial. 00:02:47.470 --> 00:02:48.400 align:middle line:90% OK. 00:02:48.400 --> 00:02:53.260 align:middle line:84% So again, you can prove this by inducting on n. 00:02:53.260 --> 00:02:53.820 align:middle line:90% All right. 00:02:53.820 --> 00:02:56.380 align:middle line:90% 00:02:56.380 --> 00:02:56.880 align:middle line:90% OK. 00:02:56.880 --> 00:03:03.670 align:middle line:90% 00:03:03.670 --> 00:03:06.010 align:middle line:84% We have a theorem, so we'll prove a few special limits. 00:03:06.010 --> 00:03:08.560 align:middle line:90% 00:03:08.560 --> 00:03:10.780 align:middle line:84% I have a real number that's positive. 00:03:10.780 --> 00:03:14.356 align:middle line:90% 00:03:14.356 --> 00:03:20.710 align:middle line:84% Then the limit as n goes to infinity of 1 00:03:20.710 --> 00:03:24.115 align:middle line:90% over n the p equals 0. 00:03:24.115 --> 00:03:31.420 align:middle line:90% 00:03:31.420 --> 00:03:36.670 align:middle line:84% If p is positive, then limit as n 00:03:36.670 --> 00:03:41.350 align:middle line:84% goes to infinity of p to the 1 over n equals 1. 00:03:41.350 --> 00:03:44.500 align:middle line:90% 00:03:44.500 --> 00:03:45.625 align:middle line:90% And the third is-- 00:03:45.625 --> 00:03:52.121 align:middle line:90% 00:03:52.121 --> 00:03:55.040 align:middle line:84% it's just that a certain limit exists-- 00:03:55.040 --> 00:04:00.320 align:middle line:84% limit as n goes to infinity of n to the 1 over n equals 1. 00:04:00.320 --> 00:04:01.520 align:middle line:90% OK? 00:04:01.520 --> 00:04:04.400 align:middle line:84% Let me make a small comment here. 00:04:04.400 --> 00:04:09.610 align:middle line:84% So far in our discussion of the real numbers, 00:04:09.610 --> 00:04:15.630 align:middle line:84% we've only defined what it means to take a real number 00:04:15.630 --> 00:04:22.850 align:middle line:90% to an integer power, but-- 00:04:22.850 --> 00:04:26.810 align:middle line:90% and n-th roots. 00:04:26.810 --> 00:04:29.660 align:middle line:84% So we have n-th powers and n-th roots. 00:04:29.660 --> 00:04:32.270 align:middle line:84% So using that, we can then define 00:04:32.270 --> 00:04:35.698 align:middle line:84% how to take a real number to a rational power-- 00:04:35.698 --> 00:04:37.490 align:middle line:84% although there needs to be something that's 00:04:37.490 --> 00:04:39.240 align:middle line:84% checked to make sure this is well-defined, 00:04:39.240 --> 00:04:43.550 align:middle line:84% because you can always write a rational number not uniquely 00:04:43.550 --> 00:04:44.900 align:middle line:90% as one integer over another. 00:04:44.900 --> 00:04:47.950 align:middle line:90% 00:04:47.950 --> 00:05:02.280 align:middle line:84% So we can define a positive real number to a rational power, 00:05:02.280 --> 00:05:06.200 align:middle line:84% but using really only the elementary facts 00:05:06.200 --> 00:05:08.700 align:middle line:84% about the real numbers, the fact that it has the least upper 00:05:08.700 --> 00:05:14.627 align:middle line:84% bound property, and the fact that the rationales are somehow 00:05:14.627 --> 00:05:16.210 align:middle line:84% dense in the real numbers, we can then 00:05:16.210 --> 00:05:21.430 align:middle line:84% define what it means to take a positive real number 00:05:21.430 --> 00:05:24.400 align:middle line:90% to a positive real number power. 00:05:24.400 --> 00:05:25.570 align:middle line:90% OK? 00:05:25.570 --> 00:05:29.320 align:middle line:84% All of that is just to say that to define a positive real 00:05:29.320 --> 00:05:33.730 align:middle line:84% number to a positive real number or to a real number of power 00:05:33.730 --> 00:05:36.550 align:middle line:84% doesn't require the introduction of the exponential 00:05:36.550 --> 00:05:40.190 align:middle line:84% or logarithm-- although they, in the end too-- 00:05:40.190 --> 00:05:43.000 align:middle line:84% they both agree, once you do have the exponential 00:05:43.000 --> 00:05:45.710 align:middle line:90% and the logarithm. 00:05:45.710 --> 00:05:48.370 align:middle line:84% So all of that is just to say that 00:05:48.370 --> 00:05:49.900 align:middle line:90% what we've done up to now-- 00:05:49.900 --> 00:05:51.820 align:middle line:84% these things actually do make sense. 00:05:51.820 --> 00:05:54.340 align:middle line:84% You don't need the exponential and the logarithm 00:05:54.340 --> 00:05:58.480 align:middle line:84% to make sense of a positive real number 00:05:58.480 --> 00:06:01.360 align:middle line:90% to a real number of power. 00:06:01.360 --> 00:06:02.440 align:middle line:90% OK? 00:06:02.440 --> 00:06:06.700 align:middle line:84% But we're just going to use the basic properties of exponents 00:06:06.700 --> 00:06:09.590 align:middle line:84% throughout all this, so we don't-- 00:06:09.590 --> 00:06:13.120 align:middle line:84% we haven't even talked about continuity, or derivatives, 00:06:13.120 --> 00:06:14.620 align:middle line:84% or anything like that, so we'll just 00:06:14.620 --> 00:06:19.400 align:middle line:84% use elementary means to be able to prove these statements. 00:06:19.400 --> 00:06:19.900 align:middle line:90% OK? 00:06:19.900 --> 00:06:23.910 align:middle line:90% 00:06:23.910 --> 00:06:26.730 align:middle line:84% So for the first one, we'll prove this actually just 00:06:26.730 --> 00:06:31.250 align:middle line:84% using the definition of the limit, which, remember, 00:06:31.250 --> 00:06:33.290 align:middle line:84% means for every epsilon positive there, 00:06:33.290 --> 00:06:35.540 align:middle line:84% we should be able to find a capital number M, such 00:06:35.540 --> 00:06:37.880 align:middle line:84% that if n is bigger than or equal to capital M, 00:06:37.880 --> 00:06:43.080 align:middle line:84% 1 over n to the p is less than epsilon. 00:06:43.080 --> 00:06:47.390 align:middle line:90% So let epsilon be positive. 00:06:47.390 --> 00:06:50.240 align:middle line:90% 00:06:50.240 --> 00:06:58.290 align:middle line:84% Choose M, a natural number, such that M 00:06:58.290 --> 00:07:00.710 align:middle line:90% is greater than or equal to-- 00:07:00.710 --> 00:07:05.200 align:middle line:84% is greater than 1 over epsilon to the 1 over p. 00:07:05.200 --> 00:07:07.700 align:middle line:90% 00:07:07.700 --> 00:07:08.200 align:middle line:90% OK? 00:07:08.200 --> 00:07:13.160 align:middle line:90% 00:07:13.160 --> 00:07:16.440 align:middle line:84% And if n is bigger than or equal to M, 00:07:16.440 --> 00:07:21.155 align:middle line:84% 1 over n to the p minus 0, which equals 1 over n 00:07:21.155 --> 00:07:24.350 align:middle line:84% to the p-- this is less than or equal to 1 over capital 00:07:24.350 --> 00:07:27.180 align:middle line:90% M to the p. 00:07:27.180 --> 00:07:30.690 align:middle line:90% So here, again, I'm using-- 00:07:30.690 --> 00:07:32.880 align:middle line:84% an elementary fact that we all know 00:07:32.880 --> 00:07:36.930 align:middle line:84% is that, if I have a positive power here, then-- 00:07:36.930 --> 00:07:39.660 align:middle line:84% and little n is bigger than or equal to capital M 00:07:39.660 --> 00:07:42.780 align:middle line:84% then into the p would be bigger than or equal to capital M 00:07:42.780 --> 00:07:43.860 align:middle line:90% to the p. 00:07:43.860 --> 00:07:46.720 align:middle line:84% We do know that for integer exponents, 00:07:46.720 --> 00:07:53.100 align:middle line:84% but just trust me that one can define non-integer exponents, 00:07:53.100 --> 00:07:57.000 align:middle line:84% and that inequality remains true as long as the power 00:07:57.000 --> 00:08:00.103 align:middle line:90% that you're using is positive. 00:08:00.103 --> 00:08:01.770 align:middle line:84% So I have 1 over n to the p is less than 00:08:01.770 --> 00:08:03.870 align:middle line:84% or equal to 1 over capital M to the p. 00:08:03.870 --> 00:08:09.340 align:middle line:84% And by our choice of capital M, this is less than epsilon. 00:08:09.340 --> 00:08:13.630 align:middle line:90% So that proves number one. 00:08:13.630 --> 00:08:18.580 align:middle line:90% So number two we'll do-- 00:08:18.580 --> 00:08:22.130 align:middle line:84% we really only need to do two cases. 00:08:22.130 --> 00:08:25.150 align:middle line:90% One is p. 00:08:25.150 --> 00:08:26.455 align:middle line:90% So p equals 1 is fine. 00:08:26.455 --> 00:08:31.450 align:middle line:90% 00:08:31.450 --> 00:08:33.340 align:middle line:84% This is clear, because then I just get 1 00:08:33.340 --> 00:08:36.309 align:middle line:90% for the whole sequence. 00:08:36.309 --> 00:08:37.630 align:middle line:90% So that's one case. 00:08:37.630 --> 00:08:42.375 align:middle line:90% Now let me do p bigger than 1. 00:08:42.375 --> 00:08:42.875 align:middle line:90% OK. 00:08:42.875 --> 00:08:46.790 align:middle line:90% 00:08:46.790 --> 00:08:50.900 align:middle line:84% And if p is bigger than 1, the absolute value 00:08:50.900 --> 00:08:56.870 align:middle line:84% of p to the 1 over n minus 1-- this is just p to the 1 00:08:56.870 --> 00:08:58.790 align:middle line:90% over n minus 1. 00:08:58.790 --> 00:09:03.000 align:middle line:84% And so I want to show that this quantity here goes to 0. 00:09:03.000 --> 00:09:03.500 align:middle line:90% OK? 00:09:03.500 --> 00:09:11.590 align:middle line:90% 00:09:11.590 --> 00:09:15.820 align:middle line:84% So we have an inequality which we proved actually 00:09:15.820 --> 00:09:18.340 align:middle line:84% in the second lecture, I think, using induction, 00:09:18.340 --> 00:09:20.950 align:middle line:84% but you can actually get from binomial theorem as well. 00:09:20.950 --> 00:09:23.740 align:middle line:90% 00:09:23.740 --> 00:09:26.530 align:middle line:84% So let me just recall this right here, 00:09:26.530 --> 00:09:29.320 align:middle line:84% that we had this inequality that, if x 00:09:29.320 --> 00:09:31.600 align:middle line:84% is bigger than or equal to minus 1, 00:09:31.600 --> 00:09:36.700 align:middle line:84% then 1 plus x raised to the n is bigger than or equal to 1 00:09:36.700 --> 00:09:38.710 align:middle line:90% plus nx. 00:09:38.710 --> 00:09:41.340 align:middle line:90% OK? 00:09:41.340 --> 00:09:46.760 align:middle line:84% And so we use this inequality now with x equals p minus 1. 00:09:46.760 --> 00:09:58.140 align:middle line:84% So p-- this is equal to 1 plus p to the 1 over n minus 1 00:09:58.140 --> 00:10:01.250 align:middle line:90% raised to the n-th power. 00:10:01.250 --> 00:10:07.760 align:middle line:84% I'm sorry. x is p to the 1 over n minus 1, not p minus 1. 00:10:07.760 --> 00:10:09.170 align:middle line:90% And now we use this inequality. 00:10:09.170 --> 00:10:13.925 align:middle line:84% This is bigger than or equal to this thing times n plus 1. 00:10:13.925 --> 00:10:21.360 align:middle line:90% 00:10:21.360 --> 00:10:24.810 align:middle line:90% So then I'll go over here. 00:10:24.810 --> 00:10:27.080 align:middle line:84% So now subtracting the 1 and dividing by n 00:10:27.080 --> 00:10:32.390 align:middle line:84% tells me that p to the 1 over n minus 1 00:10:32.390 --> 00:10:37.250 align:middle line:84% is less than or equal to p minus 1 over n. 00:10:37.250 --> 00:10:40.310 align:middle line:84% And as we noted here, since p is bigger than 1, 00:10:40.310 --> 00:10:43.780 align:middle line:84% this is bigger than or equal to 0. 00:10:43.780 --> 00:10:45.350 align:middle line:90% OK? 00:10:45.350 --> 00:10:49.280 align:middle line:84% And now we apply the squeeze theorem. 00:10:49.280 --> 00:10:54.470 align:middle line:84% By squeeze theorem, this is just going to 0. 00:10:54.470 --> 00:10:56.780 align:middle line:90% p minus 1 divided by n-- 00:10:56.780 --> 00:10:58.310 align:middle line:90% that's just a number over n. 00:10:58.310 --> 00:11:00.470 align:middle line:84% This goes to 0 as n goes to infinity. 00:11:00.470 --> 00:11:03.920 align:middle line:84% And in, fact that's contained in number 1 for p 00:11:03.920 --> 00:11:09.820 align:middle line:84% equals 1, and also by our limit facts 00:11:09.820 --> 00:11:12.340 align:middle line:84% that we proved from last time, that the limit respects 00:11:12.340 --> 00:11:14.990 align:middle line:84% algebraic operations of the real numbers. 00:11:14.990 --> 00:11:18.640 align:middle line:84% So by the squeeze theorem, since this converges to 0 00:11:18.640 --> 00:11:28.970 align:middle line:84% and this converges to 0, this converges 00:11:28.970 --> 00:11:38.890 align:middle line:90% to 0, which implies that. 00:11:38.890 --> 00:11:41.680 align:middle line:90% OK? 00:11:41.680 --> 00:11:45.520 align:middle line:84% So that deals with the case p is bigger than 1. 00:11:45.520 --> 00:11:50.380 align:middle line:84% To deal with p less than 1, we use the case p bigger than 1, 00:11:50.380 --> 00:11:53.850 align:middle line:84% and again, the fact that limits respect algebraic operations. 00:11:53.850 --> 00:12:01.310 align:middle line:90% 00:12:01.310 --> 00:12:04.230 align:middle line:90% Then we write the limit as p-- 00:12:04.230 --> 00:12:08.280 align:middle line:84% ugly looking p-- write that better-- 00:12:08.280 --> 00:12:11.160 align:middle line:84% the limit as n goes to infinity up to the 1/n. 00:12:11.160 --> 00:12:15.180 align:middle line:84% Now p is less than 1, so 1/p is bigger than 1. 00:12:15.180 --> 00:12:20.730 align:middle line:84% This is equal to the limit as n goes to infinity of 1 over 1 00:12:20.730 --> 00:12:23.010 align:middle line:90% over p to the 1/n. 00:12:23.010 --> 00:12:25.560 align:middle line:90% 00:12:25.560 --> 00:12:29.490 align:middle line:84% And now 1/p is bigger than 1-- raised to the 1/n power 00:12:29.490 --> 00:12:33.340 align:middle line:84% converges to 1 by the case we did before. 00:12:33.340 --> 00:12:39.130 align:middle line:84% And so 1 over this converges to 1/1 equals 1. 00:12:39.130 --> 00:12:39.630 align:middle line:90% OK? 00:12:39.630 --> 00:12:50.380 align:middle line:90% 00:12:50.380 --> 00:12:56.370 align:middle line:84% OK, so let me just remark that, although we 00:12:56.370 --> 00:13:00.860 align:middle line:90% did prove this inequality for-- 00:13:00.860 --> 00:13:02.960 align:middle line:84% by induction, it actually follows 00:13:02.960 --> 00:13:06.560 align:middle line:90% from the binomial theorem. 00:13:06.560 --> 00:13:08.240 align:middle line:84% And we'll use the binomial theorem 00:13:08.240 --> 00:13:13.700 align:middle line:84% to get a little bit of a different inequality 00:13:13.700 --> 00:13:15.350 align:middle line:90% that we'll use for number 3. 00:13:15.350 --> 00:13:18.210 align:middle line:90% 00:13:18.210 --> 00:13:19.860 align:middle line:84% So for number 3, we want to prove limit 00:13:19.860 --> 00:13:25.810 align:middle line:84% as n goes to infinity of n over 1 over n equals 1. 00:13:25.810 --> 00:13:29.710 align:middle line:84% So rather than keep writing n to the 1 over n minus 1, 00:13:29.710 --> 00:13:32.020 align:middle line:84% which I want to show convergence to 0 now, 00:13:32.020 --> 00:13:32.980 align:middle line:90% I'm going to write xn. 00:13:32.980 --> 00:13:41.700 align:middle line:84% So let xn equal n to the 1/n minus 00:13:41.700 --> 00:13:46.610 align:middle line:84% 1, which we note is bigger than or equal to 0 for all n. 00:13:46.610 --> 00:13:47.240 align:middle line:90% OK? 00:13:47.240 --> 00:13:58.350 align:middle line:84% And so my goal I want to show is limit as x as n 00:13:58.350 --> 00:14:01.320 align:middle line:84% goes to infinity of x sub n equals 0, 00:14:01.320 --> 00:14:04.260 align:middle line:84% because then that proves that this converges to 0. 00:14:04.260 --> 00:14:06.270 align:middle line:84% And since this is equal to its absolute value, 00:14:06.270 --> 00:14:11.163 align:middle line:84% that means that n to the 1/n converges to 1. 00:14:11.163 --> 00:14:12.580 align:middle line:84% And the way we're going to do that 00:14:12.580 --> 00:14:16.600 align:middle line:84% is using an inequality we get from the binomial theorem-- 00:14:16.600 --> 00:14:25.250 align:middle line:90% 00:14:25.250 --> 00:14:29.200 align:middle line:90% and using this trick here. 00:14:29.200 --> 00:14:38.230 align:middle line:84% Now, if we look at plus x to the n raised to the n, 00:14:38.230 --> 00:14:41.460 align:middle line:90% this is just-- 00:14:41.460 --> 00:14:44.650 align:middle line:84% that's just n-- let me move it over a little bit-- 00:14:44.650 --> 00:14:48.590 align:middle line:84% plus x to the n raised the n. x to the n-- 00:14:48.590 --> 00:14:50.980 align:middle line:90% x sub n is n to the 1/n minus 1. 00:14:50.980 --> 00:14:51.480 align:middle line:90% Add 1. 00:14:51.480 --> 00:14:52.680 align:middle line:90% I get into n to the 1/n-- 00:14:52.680 --> 00:14:54.510 align:middle line:90% raised to n, I get n. 00:14:54.510 --> 00:14:58.140 align:middle line:84% Now, by the binomial theorem, this 00:14:58.140 --> 00:15:08.870 align:middle line:84% is equal to sum from k equals 0 to n, n choose k. 00:15:08.870 --> 00:15:13.880 align:middle line:84% And I have 1 raised to the n minus k power, x 00:15:13.880 --> 00:15:16.340 align:middle line:90% to to the n raised of the k. 00:15:16.340 --> 00:15:19.180 align:middle line:90% 00:15:19.180 --> 00:15:20.540 align:middle line:90% OK? 00:15:20.540 --> 00:15:24.170 align:middle line:84% Now, this is a sum of non-negative things 00:15:24.170 --> 00:15:27.290 align:middle line:84% because, x sub n is always non-negative. 00:15:27.290 --> 00:15:31.790 align:middle line:84% And these coefficients are just quotients of factorials, 00:15:31.790 --> 00:15:34.460 align:middle line:84% so they're always non-negative as well. 00:15:34.460 --> 00:15:38.150 align:middle line:84% This sum is always bigger than or equal to 1 term 00:15:38.150 --> 00:15:39.990 align:middle line:90% from the sum. 00:15:39.990 --> 00:15:45.350 align:middle line:84% So it's bigger than or equal to k equals 2 [INAUDIBLE] y2-- 00:15:45.350 --> 00:15:48.440 align:middle line:90% you'll see-- x sub n squared. 00:15:48.440 --> 00:15:51.550 align:middle line:90% 00:15:51.550 --> 00:15:55.650 align:middle line:84% Now, n choose 2-- this is equal to n 00:15:55.650 --> 00:16:02.820 align:middle line:84% factorial over 2 factorial n minus 2 factorial x to the n 00:16:02.820 --> 00:16:09.861 align:middle line:84% squares x of n squared, which equals n times n minus 1 over 2 00:16:09.861 --> 00:16:12.930 align:middle line:90% x of n squared. 00:16:12.930 --> 00:16:13.430 align:middle line:90% All right? 00:16:13.430 --> 00:16:15.680 align:middle line:84% Now, I started off with n and I proved 00:16:15.680 --> 00:16:18.380 align:middle line:84% it was bigger than or equal to this quantity here. 00:16:18.380 --> 00:16:26.930 align:middle line:90% 00:16:26.930 --> 00:16:31.310 align:middle line:84% So now I divide what's in front of the x of in 00:16:31.310 --> 00:16:33.630 align:middle line:90% and take square roots. 00:16:33.630 --> 00:16:40.360 align:middle line:84% So then that implies that, for n in bigger than 1-- 00:16:40.360 --> 00:16:43.060 align:middle line:84% because I need to divide by n minus 1-- 00:16:43.060 --> 00:16:45.740 align:middle line:90% 00:16:45.740 --> 00:16:48.800 align:middle line:84% 0, which is bigger than or equal to n, 00:16:48.800 --> 00:16:57.900 align:middle line:84% x sub n is less than or equal to 2 over n minus 1 square root. 00:16:57.900 --> 00:16:59.990 align:middle line:90% OK? 00:16:59.990 --> 00:17:05.150 align:middle line:84% And now this is just 0, so it converges to 0. 00:17:05.150 --> 00:17:08.660 align:middle line:84% This right-hand side is square root of 2 over n minus 1. 00:17:08.660 --> 00:17:11.569 align:middle line:84% Now, the limit as n goes to infinity of 2 over n minus 1 00:17:11.569 --> 00:17:12.710 align:middle line:90% is 0. 00:17:12.710 --> 00:17:14.930 align:middle line:84% Square root of that also converges to 0. 00:17:14.930 --> 00:17:17.880 align:middle line:84% That's a fact we did from the end of last time. 00:17:17.880 --> 00:17:20.810 align:middle line:84% So this whole thing converges to 0. 00:17:20.810 --> 00:17:30.590 align:middle line:84% So by the squeeze theorem, limit as n goes to infinity of x sub 00:17:30.590 --> 00:17:35.830 align:middle line:84% n-- which, remember, this is just n to the 1 over n minus 1, 00:17:35.830 --> 00:17:36.455 align:middle line:90% which implies-- 00:17:36.455 --> 00:17:45.430 align:middle line:90% 00:17:45.430 --> 00:17:45.930 align:middle line:90% OK? 00:17:45.930 --> 00:17:50.410 align:middle line:90% 00:17:50.410 --> 00:17:51.780 align:middle line:90% And that completes the proof. 00:17:51.780 --> 00:18:05.080 align:middle line:90% 00:18:05.080 --> 00:18:08.800 align:middle line:84% All right, so now we are going to study a couple of objects 00:18:08.800 --> 00:18:10.210 align:middle line:90% related to a bounded sequence. 00:18:10.210 --> 00:18:15.946 align:middle line:90% 00:18:15.946 --> 00:18:18.820 align:middle line:84% What's the underlying question we're going to try to answer? 00:18:18.820 --> 00:18:21.940 align:middle line:84% Whenever something's introduced, you 00:18:21.940 --> 00:18:23.920 align:middle line:84% should think of it in terms of, what's 00:18:23.920 --> 00:18:27.620 align:middle line:84% the question that was asked that this is trying to answer? 00:18:27.620 --> 00:18:37.750 align:middle line:84% So we're now moving on to the topic of limsup and liminf 00:18:37.750 --> 00:18:38.545 align:middle line:90% of a sequence. 00:18:38.545 --> 00:18:41.050 align:middle line:90% 00:18:41.050 --> 00:18:42.060 align:middle line:90% So here's the question. 00:18:42.060 --> 00:18:47.260 align:middle line:90% 00:18:47.260 --> 00:18:52.560 align:middle line:84% So we've seen sequences that don't necessarily converge, 00:18:52.560 --> 00:18:57.420 align:middle line:90% like minus 1 to the n, but-- 00:18:57.420 --> 00:18:59.490 align:middle line:84% and we know about subsequences now, 00:18:59.490 --> 00:19:02.490 align:middle line:84% where you just pick entries along the sequence. 00:19:02.490 --> 00:19:07.040 align:middle line:84% You pick one, move to the right, and pick the next one. 00:19:07.040 --> 00:19:08.960 align:middle line:84% Now, if you look at minus 1 to the n, 00:19:08.960 --> 00:19:11.300 align:middle line:84% if I just look at the subsequence consisting 00:19:11.300 --> 00:19:14.390 align:middle line:90% of picking the odd entries-- 00:19:14.390 --> 00:19:16.460 align:middle line:84% the odd numbered entries in the sequence, 00:19:16.460 --> 00:19:19.790 align:middle line:84% then I'd just get minus 1 minus 1 minus 1 00:19:19.790 --> 00:19:21.728 align:middle line:90% for my new subsequence. 00:19:21.728 --> 00:19:22.520 align:middle line:90% And this converges. 00:19:22.520 --> 00:19:23.720 align:middle line:90% It's just constant. 00:19:23.720 --> 00:19:29.840 align:middle line:84% Or if I chose the even ones, I'd get 1, 1, 1, 1, 1. 00:19:29.840 --> 00:19:31.860 align:middle line:90% And that converges. 00:19:31.860 --> 00:19:33.440 align:middle line:84% So this sequence, which is bounded, 00:19:33.440 --> 00:19:36.920 align:middle line:90% has a convergent subsequence. 00:19:36.920 --> 00:19:41.240 align:middle line:84% Now, not all sequences have convergent subsequences. 00:19:41.240 --> 00:19:45.560 align:middle line:84% For example, if you look at the sequence x sub n equals n-- 00:19:45.560 --> 00:19:49.700 align:middle line:84% so x sub n is just 1, 2, 3, 4, 5, 6, and so on-- 00:19:49.700 --> 00:19:51.920 align:middle line:84% that will not have any convergent subsequences, 00:19:51.920 --> 00:19:56.250 align:middle line:84% because any subsequence of that has to be unbounded. 00:19:56.250 --> 00:19:56.750 align:middle line:90% OK? 00:19:56.750 --> 00:19:59.960 align:middle line:90% That's pretty easy to show. 00:19:59.960 --> 00:20:04.880 align:middle line:84% And we know that a convergent sequence is bounded. 00:20:04.880 --> 00:20:05.630 align:middle line:90% OK? 00:20:05.630 --> 00:20:10.830 align:middle line:90% So what am I getting at? 00:20:10.830 --> 00:20:13.490 align:middle line:84% So the question that we're going to ask and try to answer 00:20:13.490 --> 00:20:14.480 align:middle line:90% is the following. 00:20:14.480 --> 00:20:18.830 align:middle line:90% 00:20:18.830 --> 00:20:33.809 align:middle line:84% Does every bounded sequence have a convergent subsequence? 00:20:33.809 --> 00:20:38.800 align:middle line:90% 00:20:38.800 --> 00:20:39.550 align:middle line:90% OK? 00:20:39.550 --> 00:20:41.890 align:middle line:84% And what I was just going on about right 00:20:41.890 --> 00:20:44.635 align:middle line:84% there is that we know this is true, 00:20:44.635 --> 00:20:49.120 align:middle line:84% for example, for the sequence minus 1 to the n. 00:20:49.120 --> 00:20:51.850 align:middle line:84% We also this is true for convergent sequences. 00:20:51.850 --> 00:20:54.700 align:middle line:84% Convergent sequences are bounded, so-- 00:20:54.700 --> 00:20:57.280 align:middle line:84% and they have convergent subsequences. 00:20:57.280 --> 00:20:59.920 align:middle line:84% Just take the subsequence to be the whole sequence 00:20:59.920 --> 00:21:01.970 align:middle line:90% to begin with. 00:21:01.970 --> 00:21:04.280 align:middle line:84% And we have to throw in this bounded part, 00:21:04.280 --> 00:21:06.920 align:middle line:84% because there are sequences that don't have 00:21:06.920 --> 00:21:08.450 align:middle line:90% any convergent subsequences. 00:21:08.450 --> 00:21:10.220 align:middle line:84% Like I just said, x sub n equals n 00:21:10.220 --> 00:21:13.970 align:middle line:84% is an example of an unbounded sequence that doesn't have 00:21:13.970 --> 00:21:18.730 align:middle line:90% any convergent subsequences. 00:21:18.730 --> 00:21:24.230 align:middle line:84% So the answer to this question, as we'll see, 00:21:24.230 --> 00:21:27.880 align:middle line:90% is yes, this is a very-- 00:21:27.880 --> 00:21:30.460 align:middle line:84% really a very powerful statement. 00:21:30.460 --> 00:21:38.890 align:middle line:84% And this is due to Bolzano and Weierstrass. 00:21:38.890 --> 00:21:42.560 align:middle line:90% 00:21:42.560 --> 00:21:44.390 align:middle line:84% And I'll restate it in a little bit 00:21:44.390 --> 00:21:46.430 align:middle line:84% when we get to the statement of that theorem-- 00:21:46.430 --> 00:21:48.260 align:middle line:84% namely, that every bounded sequence does 00:21:48.260 --> 00:21:49.940 align:middle line:90% have a convergent subsequence. 00:21:49.940 --> 00:21:53.870 align:middle line:84% There are several different ways to prove this theorem. 00:21:53.870 --> 00:21:56.450 align:middle line:84% We're going to prove it by introducing limsup and liminf, 00:21:56.450 --> 00:21:59.630 align:middle line:84% because these are also two important objects that 00:21:59.630 --> 00:22:03.450 align:middle line:90% arise in analysis. 00:22:03.450 --> 00:22:05.510 align:middle line:84% So let's get to the definition of these guys. 00:22:05.510 --> 00:22:08.530 align:middle line:90% 00:22:08.530 --> 00:22:22.130 align:middle line:90% So let xn be bounded sequence. 00:22:22.130 --> 00:22:24.820 align:middle line:90% 00:22:24.820 --> 00:22:37.525 align:middle line:84% And we define-- if they exist, there 00:22:37.525 --> 00:22:40.300 align:middle line:84% are going to be certain limits, so it's not clear that they 00:22:40.300 --> 00:22:42.100 align:middle line:90% exist at all to begin with. 00:22:42.100 --> 00:22:44.980 align:middle line:84% But we'll show that they always do exist. 00:22:44.980 --> 00:22:50.730 align:middle line:90% We define limsup x sub n. 00:22:50.730 --> 00:22:53.880 align:middle line:84% And sometimes I'll write n goes to infinity underneath. 00:22:53.880 --> 00:22:57.210 align:middle line:84% Sometimes I'll just write limsup or I'll just 00:22:57.210 --> 00:22:58.230 align:middle line:90% have an n underneath it. 00:22:58.230 --> 00:23:01.470 align:middle line:84% This is supposed to be a number, and this 00:23:01.470 --> 00:23:04.830 align:middle line:84% is equal to the limit as n goes to infinity 00:23:04.830 --> 00:23:11.160 align:middle line:84% of a new sequence obtained from the old sequence x sub n. 00:23:11.160 --> 00:23:13.380 align:middle line:84% What are the entries of this new sequence? 00:23:13.380 --> 00:23:19.620 align:middle line:84% This is sup of x of k, k bigger than or equal to n. 00:23:19.620 --> 00:23:22.160 align:middle line:90% 00:23:22.160 --> 00:23:24.040 align:middle line:90% OK? 00:23:24.040 --> 00:23:30.655 align:middle line:84% And the liminf is similar, except it's now with infs. 00:23:30.655 --> 00:23:42.650 align:middle line:90% 00:23:42.650 --> 00:23:43.430 align:middle line:90% OK? 00:23:43.430 --> 00:23:47.260 align:middle line:84% So for each natural number n, I take 00:23:47.260 --> 00:23:51.530 align:middle line:84% the supremum of the set of elements x sub k for k 00:23:51.530 --> 00:23:52.790 align:middle line:90% bigger than or equal to n. 00:23:52.790 --> 00:23:55.620 align:middle line:84% So this is a bounded set because the sequence is bounded. 00:23:55.620 --> 00:23:57.800 align:middle line:84% So this the supremum is well-defined-- same thing 00:23:57.800 --> 00:23:58.670 align:middle line:90% for the inf. 00:23:58.670 --> 00:23:59.210 align:middle line:90% OK? 00:23:59.210 --> 00:24:02.850 align:middle line:84% So I get a number, a new number for each n. 00:24:02.850 --> 00:24:05.040 align:middle line:90% OK? 00:24:05.040 --> 00:24:08.610 align:middle line:84% And I take the limit of those numbers and I 00:24:08.610 --> 00:24:12.540 align:middle line:84% define one to be the limsup, one to be the liminf, 00:24:12.540 --> 00:24:13.500 align:middle line:90% if they exist-- 00:24:13.500 --> 00:24:15.880 align:middle line:84% because limits don't always exist, 00:24:15.880 --> 00:24:18.960 align:middle line:84% so it's not even clear that these two limits actually 00:24:18.960 --> 00:24:20.130 align:middle line:90% are meaningful. 00:24:20.130 --> 00:24:21.780 align:middle line:84% And the first thing that we'll do 00:24:21.780 --> 00:24:25.425 align:middle line:84% is we'll prove that these limits do actually always exist. 00:24:25.425 --> 00:24:32.430 align:middle line:90% 00:24:32.430 --> 00:24:37.410 align:middle line:84% So rather than keep writing this, I'm going to give these-- 00:24:37.410 --> 00:24:41.180 align:middle line:84% write some symbols for these two things. 00:24:41.180 --> 00:24:43.980 align:middle line:90% So let a sub n to be-- 00:24:43.980 --> 00:24:48.100 align:middle line:90% 00:24:48.100 --> 00:24:50.950 align:middle line:84% so of course, if I forget to say this at least in what I'm 00:24:50.950 --> 00:24:52.990 align:middle line:84% talking about with respect to limsup and liminf, 00:24:52.990 --> 00:24:55.210 align:middle line:84% I'm always talking about a bound sequence. 00:24:55.210 --> 00:24:58.960 align:middle line:84% But let me continue to state that as one of my hypotheses. 00:24:58.960 --> 00:25:08.520 align:middle line:84% Let x of n be a bounded sequence, 00:25:08.520 --> 00:25:21.860 align:middle line:84% and let a sub n be the supremum of the set of all elements 00:25:21.860 --> 00:25:27.350 align:middle line:84% x sub k, where k is bigger than or equal to n, and b 00:25:27.350 --> 00:25:36.160 align:middle line:84% to be the infimum of x sub k, a bigger than or equal to n. 00:25:36.160 --> 00:25:36.660 align:middle line:90% OK? 00:25:36.660 --> 00:25:42.510 align:middle line:90% 00:25:42.510 --> 00:25:46.210 align:middle line:84% So then there's couple of statements. 00:25:46.210 --> 00:25:54.270 align:middle line:84% One is that the sequence a sub n-- 00:25:54.270 --> 00:25:57.180 align:middle line:90% 00:25:57.180 --> 00:26:01.000 align:middle line:84% so let me just put over here what 00:26:01.000 --> 00:26:03.145 align:middle line:84% these things have to do with the limsup and liminf. 00:26:03.145 --> 00:26:05.740 align:middle line:90% 00:26:05.740 --> 00:26:07.990 align:middle line:90% Then the limsup of x sub n-- 00:26:07.990 --> 00:26:12.250 align:middle line:84% this is defined to be the limit as n goes to infinity 00:26:12.250 --> 00:26:13.890 align:middle line:90% of the a sub n's. 00:26:13.890 --> 00:26:21.130 align:middle line:84% And the liminf of x sub n-- this is defined to be the limit as n 00:26:21.130 --> 00:26:26.400 align:middle line:84% goes to infinity of the b sub n's. 00:26:26.400 --> 00:26:28.660 align:middle line:90% OK? 00:26:28.660 --> 00:26:32.200 align:middle line:84% What we're going to show is that the limit of a sub n as n 00:26:32.200 --> 00:26:34.480 align:middle line:84% goes to infinity of a sub n exists, and the limit as n 00:26:34.480 --> 00:26:36.340 align:middle line:84% goes to infinity of b sub n exists. 00:26:36.340 --> 00:26:39.220 align:middle line:84% So how we're going to show that is we're 00:26:39.220 --> 00:26:48.960 align:middle line:84% going to show that a sub n is monotone 00:26:48.960 --> 00:26:55.355 align:middle line:90% decreasing and bounded. 00:26:55.355 --> 00:26:59.150 align:middle line:90% 00:26:59.150 --> 00:27:01.230 align:middle line:84% And so these are the conclusions. 00:27:01.230 --> 00:27:05.780 align:middle line:90% So [INAUDIBLE] should say then. 00:27:05.780 --> 00:27:09.983 align:middle line:90% 00:27:09.983 --> 00:27:14.800 align:middle line:84% The sequence b sub n is monotone increasing and bounded. 00:27:14.800 --> 00:27:26.980 align:middle line:90% 00:27:26.980 --> 00:27:31.240 align:middle line:84% So in particular, since we know that, if we 00:27:31.240 --> 00:27:44.660 align:middle line:84% have a monotone sequence which is bounded, it has to converge, 00:27:44.660 --> 00:27:47.255 align:middle line:84% that means these two limits exist. 00:27:47.255 --> 00:27:50.828 align:middle line:90% 00:27:50.828 --> 00:27:52.370 align:middle line:84% And then the second part of this term 00:27:52.370 --> 00:28:03.510 align:middle line:84% is the simple statement that the liminf of x sub n 00:28:03.510 --> 00:28:09.230 align:middle line:84% is less than or equal to the limsup [INAUDIBLE] x sub n. 00:28:09.230 --> 00:28:09.730 align:middle line:90% OK? 00:28:09.730 --> 00:28:18.500 align:middle line:90% 00:28:18.500 --> 00:28:19.715 align:middle line:90% OK, to prove one-- 00:28:19.715 --> 00:28:23.230 align:middle line:90% 00:28:23.230 --> 00:28:28.700 align:middle line:84% so in fact, before I prove this theorem, 00:28:28.700 --> 00:28:36.160 align:middle line:84% let me prove a small simple theorem first. 00:28:36.160 --> 00:28:37.930 align:middle line:84% Let's put the proof of this theorem 00:28:37.930 --> 00:28:42.010 align:middle line:84% on hold just for a second, and now prove a very simple theorem 00:28:42.010 --> 00:28:52.660 align:middle line:84% that, if A and B are subsets of real numbers, A, B, 00:28:52.660 --> 00:28:58.450 align:middle line:84% both not equal to the empty set, and is A subset of B-- 00:28:58.450 --> 00:29:03.060 align:middle line:90% 00:29:03.060 --> 00:29:05.780 align:middle line:90% so also need to be bounded. 00:29:05.780 --> 00:29:09.380 align:middle line:84% So if we take two non-empty subsets of real numbers, 00:29:09.380 --> 00:29:23.610 align:middle line:84% such that [INAUDIBLE] bounded and A is a subset of B, 00:29:23.610 --> 00:29:30.600 align:middle line:84% and the conclusion is that the inf of B 00:29:30.600 --> 00:29:37.680 align:middle line:84% is less than or equal to the inf of A. 00:29:37.680 --> 00:29:41.680 align:middle line:84% And this is always less than or equal to the sup of A, 00:29:41.680 --> 00:29:48.360 align:middle line:84% and this is less than or equal to the sup of B. 00:29:48.360 --> 00:29:55.270 align:middle line:84% So what this says is that, if I take a subset of B, 00:29:55.270 --> 00:30:00.240 align:middle line:84% then that increases the inf and decreases the sup. 00:30:00.240 --> 00:30:00.850 align:middle line:90% OK? 00:30:00.850 --> 00:30:03.280 align:middle line:84% So the sup of a smaller set is smaller 00:30:03.280 --> 00:30:04.810 align:middle line:90% than the sup of the bigger set. 00:30:04.810 --> 00:30:07.120 align:middle line:84% And that inequality reverses for infs. 00:30:07.120 --> 00:30:09.910 align:middle line:84% The inf of the smaller set is bigger than 00:30:09.910 --> 00:30:12.530 align:middle line:84% or equal to the inf of the bigger set. 00:30:12.530 --> 00:30:13.030 align:middle line:90% All right? 00:30:13.030 --> 00:30:14.950 align:middle line:84% And this just follows immediately 00:30:14.950 --> 00:30:18.460 align:middle line:84% from the definition of sup and inf, 00:30:18.460 --> 00:30:21.560 align:middle line:84% so I'll just prove the sup statement. 00:30:21.560 --> 00:30:40.730 align:middle line:84% So since sup of B is an upper bound for B, 00:30:40.730 --> 00:30:47.240 align:middle line:84% and A is a subset of B, this implies 00:30:47.240 --> 00:31:01.640 align:middle line:84% that sup B is an upper bound for A. Sup B sits above everything 00:31:01.640 --> 00:31:06.620 align:middle line:84% and B. A's a subset of B, so it sits above everything for A. 00:31:06.620 --> 00:31:09.200 align:middle line:84% Now, the supremum of A is supposed to be the least upper 00:31:09.200 --> 00:31:11.750 align:middle line:84% bound, and therefore, if I take any upper bound for A, 00:31:11.750 --> 00:31:15.170 align:middle line:84% that has to be bigger than or equal to the sup of A. 00:31:15.170 --> 00:31:17.330 align:middle line:84% So since sup B is an upper bound for A, 00:31:17.330 --> 00:31:23.187 align:middle line:84% this implies that sup A is less than or equal to sup B-- 00:31:23.187 --> 00:31:25.520 align:middle line:84% and similarly with the infs, so I'm not going to write-- 00:31:25.520 --> 00:31:36.390 align:middle line:90% 00:31:36.390 --> 00:31:37.685 align:middle line:90% so similar for infs. 00:31:37.685 --> 00:31:44.880 align:middle line:90% 00:31:44.880 --> 00:31:52.080 align:middle line:84% OK, so let's go back to the proof of this theorem 00:31:52.080 --> 00:31:54.000 align:middle line:84% here, that if I have a bounded sequence, 00:31:54.000 --> 00:31:56.160 align:middle line:84% then the limsup and the liminf exist. 00:31:56.160 --> 00:32:02.480 align:middle line:84% And we show that by proving that these two sequences have 00:32:02.480 --> 00:32:04.190 align:middle line:90% these monotonicity properties. 00:32:04.190 --> 00:32:11.760 align:middle line:84% So proof now of the theorem we were-- 00:32:11.760 --> 00:32:12.690 align:middle line:90% started off proving-- 00:32:12.690 --> 00:32:16.405 align:middle line:90% 00:32:16.405 --> 00:32:16.905 align:middle line:90% OK. 00:32:16.905 --> 00:32:24.320 align:middle line:90% 00:32:24.320 --> 00:32:27.260 align:middle line:84% Since the set of xk's, with k bigger than 00:32:27.260 --> 00:32:34.220 align:middle line:84% or equal to n plus 1, is a subset of xk's for k 00:32:34.220 --> 00:32:35.870 align:middle line:90% bigger than or equal to n-- 00:32:35.870 --> 00:32:39.650 align:middle line:84% because now I'm-- I have a set where k's starting at n plus 1. 00:32:39.650 --> 00:32:42.260 align:middle line:84% Here's a set with k starting at n, 00:32:42.260 --> 00:32:45.590 align:middle line:84% so this is clearly contained in here. 00:32:45.590 --> 00:32:48.320 align:middle line:84% This implies that a sub n plus 1, 00:32:48.320 --> 00:32:58.270 align:middle line:84% which is the sup of the left hand side, 00:32:58.270 --> 00:33:00.790 align:middle line:84% by this little theorem that I stated here, 00:33:00.790 --> 00:33:07.810 align:middle line:84% is less than or equal to the sup of the bigger set. 00:33:07.810 --> 00:33:10.000 align:middle line:90% And this is just a sub n. 00:33:10.000 --> 00:33:13.200 align:middle line:90% 00:33:13.200 --> 00:33:18.360 align:middle line:84% So we've proven, for all n, a sub n plus 1 is less than 00:33:18.360 --> 00:33:19.380 align:middle line:90% or equal to a sub n. 00:33:19.380 --> 00:33:22.460 align:middle line:90% 00:33:22.460 --> 00:33:23.000 align:middle line:90% OK. 00:33:23.000 --> 00:33:26.990 align:middle line:84% Therefore, this sequence is monotone decreasing. 00:33:26.990 --> 00:33:31.080 align:middle line:90% 00:33:31.080 --> 00:33:31.580 align:middle line:90% OK? 00:33:31.580 --> 00:33:36.830 align:middle line:90% 00:33:36.830 --> 00:33:41.240 align:middle line:84% And so this uses the sup part of this previous theorem, 00:33:41.240 --> 00:33:43.730 align:middle line:84% but if we use the inf part, then we also get 00:33:43.730 --> 00:33:45.110 align:middle line:90% the statement for the b sub n. 00:33:45.110 --> 00:33:49.058 align:middle line:90% 00:33:49.058 --> 00:33:50.600 align:middle line:84% So rather than write out the details, 00:33:50.600 --> 00:33:53.540 align:middle line:84% I'll leave it to you just to flip around the inequalities 00:33:53.540 --> 00:33:55.010 align:middle line:90% for the inf in your notes. 00:33:55.010 --> 00:34:02.810 align:middle line:84% So similarly, for all N, a natural number, b sub n 00:34:02.810 --> 00:34:09.110 align:middle line:84% is bigger than or equal to b sub n plus 1-- 00:34:09.110 --> 00:34:12.790 align:middle line:90% or other way. 00:34:12.790 --> 00:34:17.179 align:middle line:90% 00:34:17.179 --> 00:34:17.679 align:middle line:90% OK? 00:34:17.679 --> 00:34:21.080 align:middle line:90% 00:34:21.080 --> 00:34:25.400 align:middle line:84% Now, so that shows these two sequences are monotone. 00:34:25.400 --> 00:34:27.650 align:middle line:90% Now we show they're bounded. 00:34:27.650 --> 00:34:29.780 align:middle line:84% And this follows simply from the fact 00:34:29.780 --> 00:34:31.310 align:middle line:90% that the x sub n's are bounded. 00:34:31.310 --> 00:34:36.320 align:middle line:84% Since there exist a b bigger than or equal to 0, 00:34:36.320 --> 00:34:44.425 align:middle line:84% such that all N natural numbers, x sub n in absolute value 00:34:44.425 --> 00:34:45.800 align:middle line:84% is less than or equal to b, which 00:34:45.800 --> 00:34:47.659 align:middle line:90% is the same as saying b is-- 00:34:47.659 --> 00:34:54.429 align:middle line:84% x sub n is bounded between minus B and capital B. 00:34:54.429 --> 00:35:04.280 align:middle line:84% So taken as elements of each of these sets, 00:35:04.280 --> 00:35:07.100 align:middle line:84% if you like this x of k, k bigger than or equal to n, 00:35:07.100 --> 00:35:09.740 align:middle line:84% that means minus B is always a lower bound 00:35:09.740 --> 00:35:14.330 align:middle line:84% for these sets and B is always an upper bound for these sets. 00:35:14.330 --> 00:35:19.010 align:middle line:84% So this implies that minus B is always 00:35:19.010 --> 00:35:28.310 align:middle line:84% less than or equal to x sub k, k bigger than or equal to n. 00:35:28.310 --> 00:35:33.680 align:middle line:84% And this always sits below the supremum. 00:35:33.680 --> 00:35:39.530 align:middle line:90% 00:35:39.530 --> 00:35:41.690 align:middle line:84% Since all of these are bounded above by B, 00:35:41.690 --> 00:35:46.820 align:middle line:84% the supremum has to be less than or equal to B, 00:35:46.820 --> 00:35:51.200 align:middle line:84% which I'll state in terms of the a sub n's and b sub n's-- means 00:35:51.200 --> 00:35:53.960 align:middle line:84% minus B is less than or equal to b sub n is less than or equal 00:35:53.960 --> 00:35:56.360 align:middle line:84% to a sub n is less than or equal to b. 00:35:56.360 --> 00:36:01.190 align:middle line:90% So these sequences are bounded. 00:36:01.190 --> 00:36:01.690 align:middle line:90% OK? 00:36:01.690 --> 00:36:06.450 align:middle line:90% 00:36:06.450 --> 00:36:09.300 align:middle line:84% So in fact, these two implies that-- 00:36:09.300 --> 00:36:17.190 align:middle line:90% 00:36:17.190 --> 00:36:19.830 align:middle line:84% all right, so we've shown that these two sequences 00:36:19.830 --> 00:36:20.790 align:middle line:90% that define-- 00:36:20.790 --> 00:36:25.360 align:middle line:84% that we use to define the limsup and the liminf 00:36:25.360 --> 00:36:28.720 align:middle line:84% are monotone and bounded, so therefore, the limit 00:36:28.720 --> 00:36:30.610 align:middle line:90% of these two sequences-- 00:36:30.610 --> 00:36:33.850 align:middle line:84% the limits, which define the limsup and liminf, 00:36:33.850 --> 00:36:35.000 align:middle line:90% actually exist. 00:36:35.000 --> 00:36:38.680 align:middle line:84% So limsup and liminf is always a well-defined object. 00:36:38.680 --> 00:36:41.440 align:middle line:90% Now, this proves one. 00:36:41.440 --> 00:36:43.900 align:middle line:84% To prove two, this follows immediately 00:36:43.900 --> 00:36:47.350 align:middle line:84% from what we've proven right here. 00:36:47.350 --> 00:36:54.090 align:middle line:84% So by this part, we have b sub n is 00:36:54.090 --> 00:36:56.100 align:middle line:90% less than or equal to a sub n. 00:36:56.100 --> 00:36:58.950 align:middle line:84% So for all n, I have these two sequences, 00:36:58.950 --> 00:37:00.760 align:middle line:90% one sitting below the other. 00:37:00.760 --> 00:37:04.560 align:middle line:84% And last time we proved that taking the limit respects 00:37:04.560 --> 00:37:09.660 align:middle line:84% inequality, so limit as n goes to infinity of b sub n-- 00:37:09.660 --> 00:37:11.910 align:middle line:90% which is the liminf-- 00:37:11.910 --> 00:37:16.477 align:middle line:84% sits below limit as n goes to infinity of a sub 00:37:16.477 --> 00:37:19.536 align:middle line:90% n, which is the limsup. 00:37:19.536 --> 00:37:21.060 align:middle line:90% And that completes the proof. 00:37:21.060 --> 00:37:25.800 align:middle line:90% 00:37:25.800 --> 00:37:28.800 align:middle line:84% So I've shown that these two objects we defined-- 00:37:28.800 --> 00:37:30.150 align:middle line:90% the limsup and the liminf-- 00:37:30.150 --> 00:37:33.360 align:middle line:90% 00:37:33.360 --> 00:37:38.130 align:middle line:84% exist, and are well-defined for every bounded sequence. 00:37:38.130 --> 00:37:41.830 align:middle line:90% 00:37:41.830 --> 00:37:43.080 align:middle line:90% So this is kind of-- 00:37:43.080 --> 00:37:48.380 align:middle line:84% can be a little bit of a daunting couple 00:37:48.380 --> 00:37:51.090 align:middle line:84% of objects to come across when you-- 00:37:51.090 --> 00:37:53.460 align:middle line:84% especially in your first analysis class. 00:37:53.460 --> 00:37:55.843 align:middle line:84% So the best thing to do is look at examples. 00:37:55.843 --> 00:37:57.260 align:middle line:84% Whenever you come across something 00:37:57.260 --> 00:37:59.030 align:middle line:84% that you just don't quite understand, 00:37:59.030 --> 00:38:04.990 align:middle line:84% start writing down some actual things. 00:38:04.990 --> 00:38:10.380 align:middle line:84% So for example, let's again look at our favorite example 00:38:10.380 --> 00:38:17.140 align:middle line:84% of a bounded sequence which does not converge, 00:38:17.140 --> 00:38:19.085 align:middle line:90% x sub n equals minus 1 to the n. 00:38:19.085 --> 00:38:22.380 align:middle line:90% 00:38:22.380 --> 00:38:29.310 align:middle line:84% Then, if I am looking at this set x sub n, 00:38:29.310 --> 00:38:35.248 align:middle line:84% n bigger than or equal to k, and writing-- 00:38:35.248 --> 00:38:36.790 align:middle line:84% instead of x sub n, let me just write 00:38:36.790 --> 00:38:39.040 align:middle line:90% what it is, minus 1 to the n. 00:38:39.040 --> 00:38:40.660 align:middle line:90% So what is this set? 00:38:40.660 --> 00:38:44.860 align:middle line:84% This is just a set consisting of two elements, 1 and minus 1. 00:38:44.860 --> 00:38:48.700 align:middle line:90% 00:38:48.700 --> 00:38:50.980 align:middle line:90% Yeah? 00:38:50.980 --> 00:38:54.610 align:middle line:84% And therefore, the sup of this set, 00:38:54.610 --> 00:38:59.830 align:middle line:84% which is the sup of this set, is just 1. 00:38:59.830 --> 00:39:13.890 align:middle line:90% Oops-- so if I take-- 00:39:13.890 --> 00:39:18.260 align:middle line:84% which implies that the limsup of minus 1 00:39:18.260 --> 00:39:23.330 align:middle line:84% to the n, which is the limit as n 00:39:23.330 --> 00:39:32.370 align:middle line:84% goes to infinity of sup minus 1 to the n, n bigger 00:39:32.370 --> 00:39:34.170 align:middle line:84% than or equal to the k-- as we just saw, 00:39:34.170 --> 00:39:37.590 align:middle line:84% this is just the sup of this set consisting of two elements, 1 00:39:37.590 --> 00:39:38.610 align:middle line:90% and minus 1. 00:39:38.610 --> 00:39:44.670 align:middle line:84% This is equal to the limit as n goes to infinity of 1 equals 1. 00:39:44.670 --> 00:39:47.990 align:middle line:84% So the limsup of minus 1 to the n is 1. 00:39:47.990 --> 00:39:52.180 align:middle line:84% Now, if I change all these sups to infs, 00:39:52.180 --> 00:39:54.040 align:middle line:84% then the inf of this set is going 00:39:54.040 --> 00:39:57.250 align:middle line:84% to be the inf of this set, which is just minus 1. 00:39:57.250 --> 00:39:58.645 align:middle line:90% And therefore, we also get-- 00:39:58.645 --> 00:40:07.370 align:middle line:90% 00:40:07.370 --> 00:40:07.870 align:middle line:90% OK? 00:40:07.870 --> 00:40:10.700 align:middle line:90% 00:40:10.700 --> 00:40:13.850 align:middle line:90% So the limsup is 1. 00:40:13.850 --> 00:40:16.672 align:middle line:84% The liminf is minus 1 for this set. 00:40:16.672 --> 00:40:19.130 align:middle line:84% That's just supposed to be a squiggly line, not necessarily 00:40:19.130 --> 00:40:22.160 align:middle line:90% looking like sigma. 00:40:22.160 --> 00:40:24.830 align:middle line:90% 00:40:24.830 --> 00:40:28.520 align:middle line:90% OK, so there's one sequence. 00:40:28.520 --> 00:40:37.060 align:middle line:84% How about our next favorite sequence, x sub n equals 1/n? 00:40:37.060 --> 00:40:40.450 align:middle line:90% 00:40:40.450 --> 00:40:46.390 align:middle line:84% So the set of elements 1/n, such that n 00:40:46.390 --> 00:40:49.180 align:middle line:84% is bigger than or equal-- so 1/k I should write. 00:40:49.180 --> 00:40:50.530 align:middle line:90% Oh, so I was using some-- 00:40:50.530 --> 00:40:54.200 align:middle line:90% 00:40:54.200 --> 00:40:56.900 align:middle line:90% this should have been a k. 00:40:56.900 --> 00:40:59.180 align:middle line:84% Hope didn't make that mistake throughout. 00:40:59.180 --> 00:41:01.414 align:middle line:90% No, I kept writing x sub k. 00:41:01.414 --> 00:41:03.335 align:middle line:90% So that should be x sub k. 00:41:03.335 --> 00:41:10.730 align:middle line:90% 00:41:10.730 --> 00:41:13.110 align:middle line:84% OK, all of that is written correctly. 00:41:13.110 --> 00:41:14.870 align:middle line:84% This should have been minus 1 to the k-- 00:41:14.870 --> 00:41:18.620 align:middle line:90% 00:41:18.620 --> 00:41:21.515 align:middle line:90% minus 1 to the k. 00:41:21.515 --> 00:41:22.390 align:middle line:90% All right, very good. 00:41:22.390 --> 00:41:26.460 align:middle line:84% So we're looking at now the set 1 over k, where n is-- 00:41:26.460 --> 00:41:29.060 align:middle line:90% 00:41:29.060 --> 00:41:32.850 align:middle line:84% where k is bigger than or equal to n. 00:41:32.850 --> 00:41:48.670 align:middle line:84% So this is just a set 1/n, 1/n plus 1, 1/n plus 2, 1/n plus 3, 00:41:48.670 --> 00:41:51.240 align:middle line:90% and so on. 00:41:51.240 --> 00:41:53.830 align:middle line:90% OK? 00:41:53.830 --> 00:41:58.460 align:middle line:84% So as I move to the next entry, things 00:41:58.460 --> 00:41:59.930 align:middle line:90% are getting smaller and smaller. 00:41:59.930 --> 00:42:07.880 align:middle line:84% And in fact, this sequence just here now, written as-- 00:42:07.880 --> 00:42:09.380 align:middle line:90% thinking of this as a new-- 00:42:09.380 --> 00:42:11.990 align:middle line:84% so this is not a sequence-- this is a set. 00:42:11.990 --> 00:42:15.200 align:middle line:84% Taking entries to be these guys, it's 00:42:15.200 --> 00:42:16.940 align:middle line:90% easy to see that converges to 0. 00:42:16.940 --> 00:42:21.410 align:middle line:84% So anyways, let's say I take the supremum 00:42:21.410 --> 00:42:25.790 align:middle line:84% of this set, which is what I need to compute the limsup. 00:42:25.790 --> 00:42:29.180 align:middle line:84% So I'm now taking the supremum of this set. 00:42:29.180 --> 00:42:33.410 align:middle line:84% 1/n plus 1-- that's always smaller than 1/n, and so 00:42:33.410 --> 00:42:35.880 align:middle line:84% is 1/n plus 2, and so on and so on. 00:42:35.880 --> 00:42:39.380 align:middle line:84% And 1/n is an element of the set that's 00:42:39.380 --> 00:42:42.800 align:middle line:84% bigger than or equal to everything else in the set. 00:42:42.800 --> 00:42:45.050 align:middle line:84% And I think there's an exercise in the homework 00:42:45.050 --> 00:42:46.730 align:middle line:84% that-- as you just show, that if you 00:42:46.730 --> 00:42:49.083 align:middle line:84% have a set that contains an element which 00:42:49.083 --> 00:42:50.750 align:middle line:84% is an upper bound for the set, then that 00:42:50.750 --> 00:42:52.340 align:middle line:90% has to be the supremum. 00:42:52.340 --> 00:42:54.770 align:middle line:84% So the supremum of this set is simply 1/n. 00:42:54.770 --> 00:42:57.980 align:middle line:90% 00:42:57.980 --> 00:43:00.640 align:middle line:90% OK? 00:43:00.640 --> 00:43:04.240 align:middle line:84% So this supremum is equal to 1/n. 00:43:04.240 --> 00:43:22.370 align:middle line:84% So as n goes to infinity, the limit of the supremum here, 00:43:22.370 --> 00:43:28.400 align:middle line:84% which equals the limit as n goes to infinity of 1/n, equals 0. 00:43:28.400 --> 00:43:28.900 align:middle line:90% OK? 00:43:28.900 --> 00:43:36.880 align:middle line:90% So the limsup of 1/n equals 0. 00:43:36.880 --> 00:43:40.000 align:middle line:90% 00:43:40.000 --> 00:43:40.500 align:middle line:90% OK? 00:43:40.500 --> 00:43:44.006 align:middle line:90% 00:43:44.006 --> 00:43:46.770 align:middle line:84% But now let's say I look at infs of this set. 00:43:46.770 --> 00:43:49.080 align:middle line:84% So I had to take sups to look at the limsup. 00:43:49.080 --> 00:43:50.610 align:middle line:90% Now let's say I take infs. 00:43:50.610 --> 00:43:53.370 align:middle line:90% OK? 00:43:53.370 --> 00:43:55.080 align:middle line:84% Now, the inf of this set-- these are 00:43:55.080 --> 00:43:58.740 align:middle line:84% elements that are getting closer and closer and closer to 0. 00:43:58.740 --> 00:44:04.680 align:middle line:84% So there's 0, 1/n, and then they just 00:44:04.680 --> 00:44:07.170 align:middle line:84% keep getting smaller and smaller and smaller and smaller, 00:44:07.170 --> 00:44:10.440 align:middle line:90% converging to 0. 00:44:10.440 --> 00:44:13.590 align:middle line:84% And you can, in fact, prove this rigorously if you'd like, 00:44:13.590 --> 00:44:17.460 align:middle line:84% but I think it's easy to at least convince yourself 00:44:17.460 --> 00:44:22.410 align:middle line:84% that infimum of this set is equal to 0. 00:44:22.410 --> 00:44:25.590 align:middle line:84% The smallest thing-- so first off, 0 00:44:25.590 --> 00:44:28.540 align:middle line:90% is a lower bound for this set. 00:44:28.540 --> 00:44:34.100 align:middle line:84% And if I take anything bigger than 0, 00:44:34.100 --> 00:44:38.015 align:middle line:84% that thing cannot be an upper bound simply because of, if you 00:44:38.015 --> 00:44:40.640 align:middle line:84% like the Archimedean property, I can always find something from 00:44:40.640 --> 00:44:44.030 align:middle line:90% the set less than that real-- 00:44:44.030 --> 00:44:45.150 align:middle line:90% positive real number. 00:44:45.150 --> 00:44:48.480 align:middle line:84% So 0 must be the least upper bound. 00:44:48.480 --> 00:44:58.180 align:middle line:84% So in summary, the liminf of 1/n, which is the inf-- 00:44:58.180 --> 00:45:01.330 align:middle line:84% the limit as n goes to infinity of the inf of this set, 00:45:01.330 --> 00:45:05.590 align:middle line:84% is just the limit as n goes to infinity of 0 equals 00:45:05.590 --> 00:45:08.510 align:middle line:90% 0, all right. 00:45:08.510 --> 00:45:11.180 align:middle line:90% So the limsup of 1/n equals 0. 00:45:11.180 --> 00:45:13.340 align:middle line:90% The liminf equals 0 as well. 00:45:13.340 --> 00:45:19.970 align:middle line:90% 00:45:19.970 --> 00:45:20.470 align:middle line:90% OK. 00:45:20.470 --> 00:45:34.650 align:middle line:90% 00:45:34.650 --> 00:45:41.230 align:middle line:84% So let's take a look at these two examples a little bit, 00:45:41.230 --> 00:45:44.040 align:middle line:84% and let me just make a few remarks. 00:45:44.040 --> 00:45:50.360 align:middle line:84% First off, what's to notice about this sequence 00:45:50.360 --> 00:45:54.350 align:middle line:84% is the fact that the limsup is 1. 00:45:54.350 --> 00:45:56.150 align:middle line:90% Liminf is equal to minus 1. 00:45:56.150 --> 00:45:59.780 align:middle line:84% So the limsup does not equal the liminf. 00:45:59.780 --> 00:46:03.680 align:middle line:84% However, in this example, the limsup, which is 0, 00:46:03.680 --> 00:46:06.200 align:middle line:90% equals the liminf, which is 0. 00:46:06.200 --> 00:46:09.770 align:middle line:84% Now, what's the difference between these two sequences? 00:46:09.770 --> 00:46:12.830 align:middle line:84% What's the property that one holds, but the-- one has, 00:46:12.830 --> 00:46:14.180 align:middle line:90% but the other doesn't? 00:46:14.180 --> 00:46:16.895 align:middle line:84% This is a convergent sequence and this one is not. 00:46:16.895 --> 00:46:19.588 align:middle line:90% 00:46:19.588 --> 00:46:21.130 align:middle line:84% And we'll see this is a general fact, 00:46:21.130 --> 00:46:23.110 align:middle line:84% that if we have a convergent sequence, 00:46:23.110 --> 00:46:27.330 align:middle line:84% then the limsup and the liminf equal each other 00:46:27.330 --> 00:46:29.910 align:middle line:84% and are equal to the limit of the original sequence, 00:46:29.910 --> 00:46:33.620 align:middle line:84% because this sequence converges to 0. 00:46:33.620 --> 00:46:35.340 align:middle line:84% But that's not just a one-way street. 00:46:35.340 --> 00:46:39.390 align:middle line:84% It's a two-way street that, in fact, we'll prove, 00:46:39.390 --> 00:46:44.080 align:middle line:84% if the limsup equals the liminf, then the original sequence 00:46:44.080 --> 00:46:46.040 align:middle line:90% converges. 00:46:46.040 --> 00:46:48.730 align:middle line:84% So the sequence converges if and only 00:46:48.730 --> 00:46:51.680 align:middle line:90% if the limsup equals the liminf. 00:46:51.680 --> 00:46:54.440 align:middle line:90% And we saw here that in-- 00:46:54.440 --> 00:46:57.140 align:middle line:84% on display, that the limsup and the liminf 00:46:57.140 --> 00:46:59.640 align:middle line:84% don't equal each other, and the original sequence, 00:46:59.640 --> 00:47:01.370 align:middle line:90% which we know-- 00:47:01.370 --> 00:47:03.065 align:middle line:84% or shown last time-- doesn't converge. 00:47:03.065 --> 00:47:10.730 align:middle line:90% 00:47:10.730 --> 00:47:14.600 align:middle line:84% And so one other thing I'd like to point out is that-- 00:47:14.600 --> 00:47:16.430 align:middle line:90% so the limsup of this guy is 1. 00:47:16.430 --> 00:47:18.560 align:middle line:84% The liminf of this guy is a minus 1. 00:47:18.560 --> 00:47:22.180 align:middle line:90% 00:47:22.180 --> 00:47:24.880 align:middle line:84% Now, we can also find a subsequence which converges 00:47:24.880 --> 00:47:27.220 align:middle line:90% to 1, which is the limsup. 00:47:27.220 --> 00:47:29.620 align:middle line:84% We just take the entries that we choose 00:47:29.620 --> 00:47:34.940 align:middle line:84% to be the even numbered entries, and that just gives-- 00:47:34.940 --> 00:47:36.830 align:middle line:90% produces a sequence 1. 00:47:36.830 --> 00:47:39.770 align:middle line:84% And that sequence converges to 1. 00:47:39.770 --> 00:47:41.880 align:middle line:84% If we take the odd entries of the sequence, 00:47:41.880 --> 00:47:44.690 align:middle line:84% that subsequence is just minus 1 and converges to minus 1, 00:47:44.690 --> 00:47:45.950 align:middle line:90% which is the liminf. 00:47:45.950 --> 00:47:47.780 align:middle line:84% And that's also a general fact will prove, 00:47:47.780 --> 00:47:50.390 align:middle line:84% that for any bounded sequence, there 00:47:50.390 --> 00:47:54.050 align:middle line:84% exists some sequences converging to the limsup and the liminf. 00:47:54.050 --> 00:47:57.480 align:middle line:84% And that will give us the proof of this Bolzano-Weierstrass 00:47:57.480 --> 00:47:57.980 align:middle line:90% theorem. 00:47:57.980 --> 00:48:01.600 align:middle line:90% 00:48:01.600 --> 00:48:02.550 align:middle line:90% All right. 00:48:02.550 --> 00:48:05.610 align:middle line:84% And I think that's all of the remarks I want to say. 00:48:05.610 --> 00:48:08.300 align:middle line:90% 00:48:08.300 --> 00:48:10.550 align:middle line:84% And let me make one other important comment. 00:48:10.550 --> 00:48:15.600 align:middle line:84% So this sequence, which we use to define the limsup and also 00:48:15.600 --> 00:48:16.100 align:middle line:90% the liminf-- 00:48:16.100 --> 00:48:19.160 align:middle line:90% 00:48:19.160 --> 00:48:21.770 align:middle line:84% in three out of these four cases, 00:48:21.770 --> 00:48:24.080 align:middle line:90% they were actual subsequences. 00:48:24.080 --> 00:48:26.750 align:middle line:84% So the sup of this set equals 1, which 00:48:26.750 --> 00:48:30.230 align:middle line:84% I can consider as a subsequence of the original guy-- 00:48:30.230 --> 00:48:33.230 align:middle line:84% and the same thing for the liminf. 00:48:33.230 --> 00:48:36.590 align:middle line:84% The inf of this set was minus 1, which 00:48:36.590 --> 00:48:39.320 align:middle line:84% I can consider as a subsequence of minus 1 to the n. 00:48:39.320 --> 00:48:41.830 align:middle line:90% 00:48:41.830 --> 00:48:46.650 align:middle line:84% And for the sup of this set, I got 1/n, which I can-- 00:48:46.650 --> 00:48:50.520 align:middle line:84% this is just equal to the original sequence. 00:48:50.520 --> 00:48:52.620 align:middle line:84% Definitely, I can consider it as a subsequence 00:48:52.620 --> 00:48:54.120 align:middle line:90% of the original sequence. 00:48:54.120 --> 00:48:58.010 align:middle line:84% But if I take the infimum of this set, which 00:48:58.010 --> 00:49:00.950 align:middle line:84% I need to define the liminf, I got 0 00:49:00.950 --> 00:49:03.260 align:middle line:84% for every entry, which is not a subsequence 00:49:03.260 --> 00:49:04.910 align:middle line:90% of this original sequence. 00:49:04.910 --> 00:49:05.480 align:middle line:90% OK? 00:49:05.480 --> 00:49:10.430 align:middle line:84% So all of that is to say that the sequences I get 00:49:10.430 --> 00:49:13.670 align:middle line:84% through this process to define the limsup and liminf 00:49:13.670 --> 00:49:18.120 align:middle line:84% are not necessarily subsequences of the original sequence. 00:49:18.120 --> 00:49:18.620 align:middle line:90% OK? 00:49:18.620 --> 00:49:21.700 align:middle line:90% 00:49:21.700 --> 00:49:24.730 align:middle line:84% What I just said a minute ago about there actually 00:49:24.730 --> 00:49:26.860 align:middle line:84% being some sequences which do converge 00:49:26.860 --> 00:49:28.180 align:middle line:90% to the limsup and liminf-- 00:49:28.180 --> 00:49:31.105 align:middle line:84% this is a non-trivial fact, which we're going to prove. 00:49:31.105 --> 00:49:37.740 align:middle line:90% 00:49:37.740 --> 00:49:38.240 align:middle line:90% OK. 00:49:38.240 --> 00:49:42.320 align:middle line:90% 00:49:42.320 --> 00:49:45.170 align:middle line:84% So theorem-- and this will give us the Bolzano-Weierstrass 00:49:45.170 --> 00:49:47.960 align:middle line:84% essentially immediately right after it-- 00:49:47.960 --> 00:49:53.660 align:middle line:90% 00:49:53.660 --> 00:49:58.265 align:middle line:90% let xn be a bounded sequence. 00:49:58.265 --> 00:50:04.410 align:middle line:90% 00:50:04.410 --> 00:50:17.430 align:middle line:84% Then there exists subsequences xn sub k and xm sub k-- 00:50:17.430 --> 00:50:22.890 align:middle line:84% so they don't necessarily have to be the same subsequence-- 00:50:22.890 --> 00:50:28.890 align:middle line:84% such that limit as k goes to infinity of x sub n sub k 00:50:28.890 --> 00:50:32.250 align:middle line:90% equals the limsup-- 00:50:32.250 --> 00:50:35.160 align:middle line:84% so it's a convergent subsequence-- 00:50:35.160 --> 00:50:40.140 align:middle line:84% and the limit as k goes to infinity of xm sub k 00:50:40.140 --> 00:50:41.050 align:middle line:90% produces the liminf. 00:50:41.050 --> 00:50:46.670 align:middle line:90% 00:50:46.670 --> 00:50:47.170 align:middle line:90% OK. 00:50:47.170 --> 00:50:50.040 align:middle line:90% 00:50:50.040 --> 00:50:57.000 align:middle line:84% And so before I prove this, this immediately gives 00:50:57.000 --> 00:51:05.210 align:middle line:84% the Bolzano-Weierstrass theorem, which 00:51:05.210 --> 00:51:20.220 align:middle line:84% is that every bounded sequence has a convergence subsequence. 00:51:20.220 --> 00:51:30.740 align:middle line:90% 00:51:30.740 --> 00:51:32.600 align:middle line:90% OK. 00:51:32.600 --> 00:51:34.100 align:middle line:84% So again, this follows immediately 00:51:34.100 --> 00:51:35.725 align:middle line:84% from the previous theorem, because if I 00:51:35.725 --> 00:51:38.780 align:middle line:84% take a bounded sequence, by the previous theorem, 00:51:38.780 --> 00:51:40.700 align:middle line:84% I can find a subsequence which converges 00:51:40.700 --> 00:51:43.560 align:middle line:84% to the limsup, which always exists for a bounded sequence. 00:51:43.560 --> 00:51:44.060 align:middle line:90% OK? 00:51:44.060 --> 00:51:46.760 align:middle line:90% And then that's [INAUDIBLE]. 00:51:46.760 --> 00:51:48.260 align:middle line:84% In fact, we have something stronger, 00:51:48.260 --> 00:51:52.730 align:middle line:84% in that we have at least two subsequences which 00:51:52.730 --> 00:51:54.860 align:middle line:84% converge to these two numbers, which may or may not 00:51:54.860 --> 00:51:56.270 align:middle line:90% be the same. 00:51:56.270 --> 00:51:56.770 align:middle line:90% OK? 00:51:56.770 --> 00:52:05.290 align:middle line:90% 00:52:05.290 --> 00:52:13.770 align:middle line:84% So the reason this is so powerful and so strong 00:52:13.770 --> 00:52:15.900 align:middle line:90% is that it-- 00:52:15.900 --> 00:52:19.530 align:middle line:84% to get your hands on something, it 00:52:19.530 --> 00:52:21.540 align:middle line:84% doesn't require you to show something 00:52:21.540 --> 00:52:24.450 align:middle line:84% as strong as showing there is a sequence converging to that. 00:52:24.450 --> 00:52:30.350 align:middle line:84% So quite often, you can think in terms of variational problems, 00:52:30.350 --> 00:52:35.780 align:middle line:84% where you want to show that a minimum of something 00:52:35.780 --> 00:52:40.010 align:middle line:84% always exists or a maximum of something always exists. 00:52:40.010 --> 00:52:46.850 align:middle line:84% Well, what you can try and do is take a sequence of guys 00:52:46.850 --> 00:52:50.150 align:middle line:84% that you stick into your-- so this is a general nonsense 00:52:50.150 --> 00:52:54.170 align:middle line:84% that you stick into your machine or function that spits out 00:52:54.170 --> 00:52:54.845 align:middle line:90% output. 00:52:54.845 --> 00:52:56.720 align:middle line:84% And these outputs are approaching the maximum 00:52:56.720 --> 00:52:58.720 align:middle line:90% or approaching the minimum. 00:52:58.720 --> 00:53:01.210 align:middle line:84% And what you'd like to say is that there does, in fact, 00:53:01.210 --> 00:53:06.880 align:middle line:84% exist an element that you can stick into your machine 00:53:06.880 --> 00:53:08.965 align:middle line:84% and produce the maximum amount of output. 00:53:08.965 --> 00:53:12.110 align:middle line:90% 00:53:12.110 --> 00:53:15.600 align:middle line:84% Now, maybe it's not clear how to do that. 00:53:15.600 --> 00:53:17.960 align:middle line:84% So first, take a sequence approaching-- 00:53:17.960 --> 00:53:21.530 align:middle line:84% so that the values are approaching that-- 00:53:21.530 --> 00:53:24.110 align:middle line:84% the outputs are approaching the maximum. 00:53:24.110 --> 00:53:28.880 align:middle line:84% Maybe you could show that the inputs converge to something, 00:53:28.880 --> 00:53:31.940 align:middle line:84% but that's typically really hard. 00:53:31.940 --> 00:53:34.680 align:middle line:84% But you don't have to work that hard is what this theorem says. 00:53:34.680 --> 00:53:38.570 align:middle line:84% It says that what you really need to do, 00:53:38.570 --> 00:53:43.280 align:middle line:84% and which is much more straightforward, or simpler, 00:53:43.280 --> 00:53:46.970 align:middle line:84% or impossible really, is to show that that sequence of inputs 00:53:46.970 --> 00:53:50.030 align:middle line:84% that you put into your machine to get the outputs 00:53:50.030 --> 00:53:51.930 align:middle line:90% is a bounded sequence. 00:53:51.930 --> 00:53:52.490 align:middle line:90% OK? 00:53:52.490 --> 00:53:55.100 align:middle line:84% Then you could pass to a subsequence, which actually 00:53:55.100 --> 00:53:57.500 align:middle line:84% does converge to something by this theorem, 00:53:57.500 --> 00:54:03.430 align:middle line:84% and proceed in that way by showing that you do 00:54:03.430 --> 00:54:05.130 align:middle line:90% have some minimum or maximum-- 00:54:05.130 --> 00:54:09.170 align:middle line:90% 00:54:09.170 --> 00:54:11.930 align:middle line:84% some input that produces a maximum output 00:54:11.930 --> 00:54:12.680 align:middle line:90% or minimum output. 00:54:12.680 --> 00:54:15.380 align:middle line:90% 00:54:15.380 --> 00:54:18.610 align:middle line:84% So that's a bit small bit of rambling about why this theorem 00:54:18.610 --> 00:54:23.508 align:middle line:84% is so useful is that, again, it's-- 00:54:23.508 --> 00:54:25.550 align:middle line:84% to get your hands on something that typically you 00:54:25.550 --> 00:54:29.210 align:middle line:84% want to study, it's very difficult 00:54:29.210 --> 00:54:33.170 align:middle line:84% to show convergence to that thing you want to study, 00:54:33.170 --> 00:54:36.980 align:middle line:84% or that thing exists, because there's a sequence 00:54:36.980 --> 00:54:39.980 align:middle line:84% that you come up with ad hoc actually converges 00:54:39.980 --> 00:54:40.790 align:middle line:90% to that thing. 00:54:40.790 --> 00:54:44.420 align:middle line:84% But this theorem says you don't need to work that hard 00:54:44.420 --> 00:54:45.635 align:middle line:90% or try to do the impossible. 00:54:45.635 --> 00:54:49.620 align:middle line:84% And typically, it's much easier just 00:54:49.620 --> 00:54:54.620 align:middle line:84% to proceed by showing your sequence of inputs is bounded. 00:54:54.620 --> 00:54:58.760 align:middle line:84% OK, so that's enough of that bit of rambling about why 00:54:58.760 --> 00:55:00.170 align:middle line:90% this theorem is so useful. 00:55:00.170 --> 00:55:03.410 align:middle line:90% 00:55:03.410 --> 00:55:07.880 align:middle line:84% It's also useful in the study of PDEs, which is what I study. 00:55:07.880 --> 00:55:11.000 align:middle line:90% So I have a soft spot for it. 00:55:11.000 --> 00:55:16.250 align:middle line:84% Actually, one of its generalizations-- 00:55:16.250 --> 00:55:18.710 align:middle line:84% all right, so let's prove this theorem 00:55:18.710 --> 00:55:23.060 align:middle line:84% that there exists some sequences which converge to the limsup 00:55:23.060 --> 00:55:24.650 align:middle line:90% and liminf. 00:55:24.650 --> 00:55:28.340 align:middle line:84% So as before, I'm going to use-- rather than keep writing 00:55:28.340 --> 00:55:32.287 align:middle line:90% the supremum of this set and-- 00:55:32.287 --> 00:55:34.370 align:middle line:84% actually, I'm not even going to do this statement. 00:55:34.370 --> 00:55:36.000 align:middle line:84% I'm going to leave this as an exercise. 00:55:36.000 --> 00:55:39.290 align:middle line:90% I'm going to do this statement. 00:55:39.290 --> 00:55:42.095 align:middle line:84% I'm going to use this notation a sub n, as before-- 00:55:42.095 --> 00:55:44.600 align:middle line:90% 00:55:44.600 --> 00:55:54.875 align:middle line:84% a sub n b sup of x sub k's, k bigger than or equal to n. 00:55:54.875 --> 00:55:57.000 align:middle line:84% So I want to show that there's something converging 00:55:57.000 --> 00:56:00.140 align:middle line:84% to the limit as n goes to infinity of the a sub n's. 00:56:00.140 --> 00:56:02.223 align:middle line:90% 00:56:02.223 --> 00:56:03.640 align:middle line:84% And so what I'm going to do is I'm 00:56:03.640 --> 00:56:07.720 align:middle line:84% going to try up a subsequence of elements between a sub n 00:56:07.720 --> 00:56:12.460 align:middle line:84% and a sub n minus something which is converging to 0, 00:56:12.460 --> 00:56:14.320 align:middle line:90% and not quite a sub n. 00:56:14.320 --> 00:56:16.720 align:middle line:84% So it'll be along a subsequence as well. 00:56:16.720 --> 00:56:22.680 align:middle line:90% 00:56:22.680 --> 00:56:25.500 align:middle line:84% So we know that there exists an n sub 1 00:56:25.500 --> 00:56:28.560 align:middle line:84% bigger than or equal to 1 simply by how 00:56:28.560 --> 00:56:33.270 align:middle line:84% this is defined as a supremum, and by the exercise 00:56:33.270 --> 00:56:38.970 align:middle line:84% from assignment 3 that there exists an n sub 1 00:56:38.970 --> 00:56:44.370 align:middle line:84% bigger than or equal to 1 such that a sub 1 minus 1 00:56:44.370 --> 00:56:53.060 align:middle line:84% is less than or equal to is less than x sub 1 is less than 00:56:53.060 --> 00:56:55.370 align:middle line:90% or equal to a sub 1. 00:56:55.370 --> 00:56:55.870 align:middle line:90% OK? 00:56:55.870 --> 00:56:59.210 align:middle line:90% 00:56:59.210 --> 00:57:01.190 align:middle line:90% All right. 00:57:01.190 --> 00:57:08.210 align:middle line:84% a sub 1 minus 1 is not an upper bound for the set a sub 1, 00:57:08.210 --> 00:57:12.620 align:middle line:84% which is x sub k's, where k's bigger than or equal to 1. 00:57:12.620 --> 00:57:14.600 align:middle line:84% Therefore, I should be able to find 00:57:14.600 --> 00:57:17.510 align:middle line:84% an element from this set strictly bigger than that. 00:57:17.510 --> 00:57:19.010 align:middle line:84% And of course, it's always less than 00:57:19.010 --> 00:57:21.570 align:middle line:84% or equal to the supremum, which is a sub 1. 00:57:21.570 --> 00:57:22.070 align:middle line:90% OK. 00:57:22.070 --> 00:57:27.010 align:middle line:90% 00:57:27.010 --> 00:57:42.780 align:middle line:84% So now, since a sub n sub 1 plus 1-- so a sub n sub 1 plus 1, 00:57:42.780 --> 00:57:45.060 align:middle line:90% not 1 plus 1-- 00:57:45.060 --> 00:57:50.310 align:middle line:84% since this thing equals the supremum of xk, such-- 00:57:50.310 --> 00:57:56.250 align:middle line:84% k is bigger than or equal to of 1 plus 1, 00:57:56.250 --> 00:58:01.855 align:middle line:90% there exists an a sub n sub 2-- 00:58:01.855 --> 00:58:04.390 align:middle line:90% 00:58:04.390 --> 00:58:11.360 align:middle line:84% at least there exists an n sub 2 bigger than n sub 1-- in fact, 00:58:11.360 --> 00:58:15.280 align:middle line:84% it has to be bigger than or equal to n sub 1 plus 1-- 00:58:15.280 --> 00:58:24.940 align:middle line:84% such that a sub n plus 1 minus 1/2 is less than a sub n sub 2 00:58:24.940 --> 00:58:28.970 align:middle line:84% is less than or equal to a sub n plus 1. 00:58:28.970 --> 00:58:29.470 align:middle line:90% OK? 00:58:29.470 --> 00:58:32.270 align:middle line:90% 00:58:32.270 --> 00:58:34.910 align:middle line:84% So why the n sub 1 plus 1 is because I 00:58:34.910 --> 00:58:40.580 align:middle line:84% wanted to obtain a new entry from the sequence that 00:58:40.580 --> 00:58:44.850 align:middle line:84% comes from farther than the index n sub 1. 00:58:44.850 --> 00:58:45.350 align:middle line:90% OK? 00:58:45.350 --> 00:58:46.790 align:middle line:84% I'm trying to build a subsequence. 00:58:46.790 --> 00:58:50.600 align:middle line:84% And like I said, the idea is I want to somehow sandwich-- 00:58:50.600 --> 00:58:53.630 align:middle line:90% this should be x sub n-- 00:58:53.630 --> 00:58:56.780 align:middle line:84% I want to build up a subsequence which 00:58:56.780 --> 00:59:01.040 align:middle line:84% is sandwiched between things converging to the limsup, 00:59:01.040 --> 00:59:03.760 align:middle line:90% and use the squeeze theorem. 00:59:03.760 --> 00:59:04.260 align:middle line:90% All right? 00:59:04.260 --> 00:59:09.180 align:middle line:90% 00:59:09.180 --> 00:59:12.540 align:middle line:84% OK, so since this is the supremum of this set 00:59:12.540 --> 00:59:14.520 align:middle line:84% that-- and because this thing is not 00:59:14.520 --> 00:59:17.070 align:middle line:84% an upper bound for this set, there 00:59:17.070 --> 00:59:21.690 align:middle line:84% exists something from the set-- so some element k bigger than 00:59:21.690 --> 00:59:24.200 align:middle line:84% or equal to n sub 1 plus 1, which I'm going to call n sub 00:59:24.200 --> 00:59:26.700 align:middle line:84% 2, so that x sub n sub 2 is bigger 00:59:26.700 --> 00:59:34.210 align:middle line:84% than a sub n plus 1 and a sub n sub 1 plus 1. 00:59:34.210 --> 00:59:35.740 align:middle line:90% And then I just keep doing this. 00:59:35.740 --> 00:59:39.310 align:middle line:90% 00:59:39.310 --> 00:59:48.290 align:middle line:84% Since a sub 2 plus 1 equals the supremum of the x of k, 00:59:48.290 --> 00:59:54.970 align:middle line:84% such that k is bigger than or equal to n sub 2 plus 1, 00:59:54.970 --> 01:00:00.310 align:middle line:84% there exists an n sub 3 bigger than n sub 2, 01:00:00.310 --> 01:00:07.600 align:middle line:84% such that a sub n sub 2 plus 1, minus now a third, 01:00:07.600 --> 01:00:12.280 align:middle line:84% is less than x sub n sub 3 is less than or equal to a sub n 01:00:12.280 --> 01:00:14.050 align:middle line:90% sub 2 plus 1. 01:00:14.050 --> 01:00:24.330 align:middle line:90% 01:00:24.330 --> 01:00:24.830 align:middle line:90% OK. 01:00:24.830 --> 01:00:32.350 align:middle line:90% 01:00:32.350 --> 01:00:35.320 align:middle line:84% But now we're essentially home free. 01:00:35.320 --> 01:00:36.835 align:middle line:84% We'll just continue in this manner. 01:00:36.835 --> 01:00:41.030 align:middle line:90% 01:00:41.030 --> 01:00:43.480 align:middle line:90% Let me write down. 01:00:43.480 --> 01:00:52.190 align:middle line:84% Now, strictly speaking, I need to state the construction 01:00:52.190 --> 01:00:57.200 align:middle line:84% of this sequence as an inductive argument, 01:00:57.200 --> 01:01:01.440 align:middle line:84% but for the purposes of this class, 01:01:01.440 --> 01:01:02.980 align:middle line:90% I'm not going to do that. 01:01:02.980 --> 01:01:08.820 align:middle line:84% I'm just going to say-- continuing in this manner. 01:01:08.820 --> 01:01:17.370 align:middle line:84% And that will be what I say for this part. 01:01:17.370 --> 01:01:34.390 align:middle line:84% So continuing in this manner, we obtain a sequence 01:01:34.390 --> 01:01:40.245 align:middle line:90% of integers, natural numbers-- 01:01:40.245 --> 01:01:45.680 align:middle line:90% 01:01:45.680 --> 01:01:50.840 align:middle line:84% n sub 1 less than n sub 2 less than n sub 3, so on-- 01:01:50.840 --> 01:01:52.190 align:middle line:90% such that what holds-- 01:01:52.190 --> 01:01:58.330 align:middle line:90% 01:01:58.330 --> 01:02:03.220 align:middle line:84% such that, for all k [? N, ?] natural number, 01:02:03.220 --> 01:02:16.620 align:middle line:84% a sub n sub k minus 1 plus 1 minus 1/k 01:02:16.620 --> 01:02:23.370 align:middle line:84% is less than or equal to x n sub k less than 01:02:23.370 --> 01:02:29.980 align:middle line:84% or equal to a sub n sub k minus 1 plus 1. 01:02:29.980 --> 01:02:30.480 align:middle line:90% OK? 01:02:30.480 --> 01:02:33.430 align:middle line:90% 01:02:33.430 --> 01:02:38.035 align:middle line:84% And here, if you like, we didn't define what n sub 0 is, so-- 01:02:38.035 --> 01:02:49.760 align:middle line:90% 01:02:49.760 --> 01:02:52.800 align:middle line:84% so really, we're only interested in the n sub 1, n sub 2. 01:02:52.800 --> 01:02:55.530 align:middle line:84% But for the sake of this whole thing making 01:02:55.530 --> 01:03:01.060 align:middle line:84% sense for all integers k, with n sub 0 being defined to be 0-- 01:03:01.060 --> 01:03:01.560 align:middle line:90% OK? 01:03:01.560 --> 01:03:07.770 align:middle line:84% That's just the first case that we dealt with. 01:03:07.770 --> 01:03:11.030 align:middle line:84% So we obtain this subsequence x sub n sub 01:03:11.030 --> 01:03:19.460 align:middle line:84% k that's sandwiched in between this subsequence of a's and 01:03:19.460 --> 01:03:23.640 align:middle line:84% minus 1 over k, and this sequence 01:03:23.640 --> 01:03:27.270 align:middle line:90% of a sub n sub k minus 1. 01:03:27.270 --> 01:03:35.370 align:middle line:84% So since n sub 1 is less than n sub 2 and so on, 01:03:35.370 --> 01:03:36.435 align:middle line:90% this implies that-- 01:03:36.435 --> 01:03:46.040 align:middle line:90% 01:03:46.040 --> 01:03:49.770 align:middle line:84% write it this way-- n plus 1 is less than [INAUDIBLE] 2 plus 1 01:03:49.770 --> 01:03:53.060 align:middle line:90% is less than n sub 3 plus 1. 01:03:53.060 --> 01:03:55.760 align:middle line:90% So this is a subsequence-- 01:03:55.760 --> 01:03:58.970 align:middle line:90% 01:03:58.970 --> 01:04:15.570 align:middle line:84% a sub n sub k minus 1 plus 1 is a subsequence of a sub n's. 01:04:15.570 --> 01:04:18.090 align:middle line:84% Now, what do we know" We know the a sub n's converge 01:04:18.090 --> 01:04:20.640 align:middle line:90% to the limsup. 01:04:20.640 --> 01:04:23.100 align:middle line:84% And we proved last time that every subsequence 01:04:23.100 --> 01:04:26.100 align:middle line:84% of a convergent sequence converges to the same thing. 01:04:26.100 --> 01:04:31.800 align:middle line:90% 01:04:31.800 --> 01:04:35.680 align:middle line:84% That's the limit as k goes to infinity of a sub n sub 01:04:35.680 --> 01:04:43.590 align:middle line:84% k minus 1 plus 1 equals the limit of the original sequence, 01:04:43.590 --> 01:04:45.750 align:middle line:84% which is, by definition, the limsup. 01:04:45.750 --> 01:04:48.500 align:middle line:90% 01:04:48.500 --> 01:04:49.000 align:middle line:90% OK? 01:04:49.000 --> 01:04:51.880 align:middle line:90% 01:04:51.880 --> 01:04:56.360 align:middle line:90% So now I have this subsequence. 01:04:56.360 --> 01:05:00.560 align:middle line:84% So now I have this subsequence of x's sandwiched 01:05:00.560 --> 01:05:02.940 align:middle line:90% in between two sequences-- 01:05:02.940 --> 01:05:05.240 align:middle line:84% this guy on the left, this guy on the right. 01:05:05.240 --> 01:05:07.850 align:middle line:84% This guy on the right converges to the limsup. 01:05:07.850 --> 01:05:11.330 align:middle line:84% This guy on the left converges to the limsup minus 0, 01:05:11.330 --> 01:05:13.100 align:middle line:90% because 1/k converges to 0. 01:05:13.100 --> 01:05:17.420 align:middle line:90% 01:05:17.420 --> 01:05:25.070 align:middle line:84% So by the squeeze theorem, we get that the limit as k 01:05:25.070 --> 01:05:29.180 align:middle line:84% goes to infinity of x sub n sub k equals the limit as k 01:05:29.180 --> 01:05:33.747 align:middle line:84% goes to infinity of this and this whole thing, 01:05:33.747 --> 01:05:34.580 align:middle line:90% which is the limsup. 01:05:34.580 --> 01:05:41.790 align:middle line:90% 01:05:41.790 --> 01:05:42.290 align:middle line:90% OK. 01:05:42.290 --> 01:05:44.910 align:middle line:90% 01:05:44.910 --> 01:05:50.580 align:middle line:84% And again, so I'll leave it to you to do the liminf part. 01:05:50.580 --> 01:05:56.140 align:middle line:84% But the point is that now, basically, this 1/k 01:05:56.140 --> 01:05:58.570 align:middle line:84% gets moved to over here, and this becomes a sub n sub 01:05:58.570 --> 01:06:01.360 align:middle line:90% k minus 1 plus 1 plus 1/k. 01:06:01.360 --> 01:06:04.820 align:middle line:84% And now I just have this on sitting below this guy. 01:06:04.820 --> 01:06:05.320 align:middle line:90% All right? 01:06:05.320 --> 01:06:09.460 align:middle line:84% That's really the only change for the infs 01:06:09.460 --> 01:06:11.395 align:middle line:84% to get a subsequence converging to the liminf. 01:06:11.395 --> 01:06:14.720 align:middle line:90% 01:06:14.720 --> 01:06:16.660 align:middle line:90% OK? 01:06:16.660 --> 01:06:22.110 align:middle line:84% Now, we've shown that there exists a subsequence converging 01:06:22.110 --> 01:06:29.970 align:middle line:84% to the limsup and the liminf, which 01:06:29.970 --> 01:06:31.760 align:middle line:84% gives us the Bolzano-Weierstrass theorem. 01:06:31.760 --> 01:06:36.392 align:middle line:90% 01:06:36.392 --> 01:06:38.100 align:middle line:84% And now let me come back to the statement 01:06:38.100 --> 01:06:42.360 align:middle line:84% I made about these two sequences here-- 01:06:42.360 --> 01:06:46.380 align:middle line:84% namely, that the limsup equals the liminf if and only 01:06:46.380 --> 01:06:49.530 align:middle line:84% if the original sequence converges. 01:06:49.530 --> 01:06:50.730 align:middle line:90% So let's prove that now. 01:06:50.730 --> 01:06:59.830 align:middle line:90% 01:06:59.830 --> 01:07:04.840 align:middle line:84% Let x sub n be a bounded sequence. 01:07:04.840 --> 01:07:09.090 align:middle line:90% 01:07:09.090 --> 01:07:21.490 align:middle line:84% Then xn converges if and only if limsup equals the liminf. 01:07:21.490 --> 01:07:26.120 align:middle line:90% 01:07:26.120 --> 01:07:27.770 align:middle line:90% And there's one more part. 01:07:27.770 --> 01:07:30.910 align:middle line:90% 01:07:30.910 --> 01:07:39.040 align:middle line:84% Moreover, if xn converges, then all these limits agree-- 01:07:39.040 --> 01:07:43.930 align:middle line:84% equals the limsup, equals liminf. 01:07:43.930 --> 01:07:47.960 align:middle line:90% 01:07:47.960 --> 01:07:48.460 align:middle line:90% OK? 01:07:48.460 --> 01:07:52.620 align:middle line:90% 01:07:52.620 --> 01:07:56.040 align:middle line:84% So the sequence converges if and only 01:07:56.040 --> 01:07:58.110 align:middle line:84% if the limsup equals the liminf-- 01:07:58.110 --> 01:08:03.480 align:middle line:84% and in the case that we do have this limit of the sequences 01:08:03.480 --> 01:08:05.745 align:middle line:84% given by this common value of the limsup and liminf. 01:08:05.745 --> 01:08:14.240 align:middle line:90% 01:08:14.240 --> 01:08:22.859 align:middle line:84% OK, so let's go in one direction. 01:08:22.859 --> 01:08:26.040 align:middle line:84% To do this direction, we'll use the squeeze theorem. 01:08:26.040 --> 01:08:40.290 align:middle line:84% So suppose L equals limsup x sub n equals liminf of x sub n. 01:08:40.290 --> 01:08:42.990 align:middle line:84% We're assuming that these two things equal each other. 01:08:42.990 --> 01:08:44.880 align:middle line:84% They're given by a common value L. 01:08:44.880 --> 01:08:46.740 align:middle line:84% And what we're going to end up showing 01:08:46.740 --> 01:08:48.540 align:middle line:84% is that this sequence converges to L. 01:08:48.540 --> 01:08:52.859 align:middle line:84% So that actually gives the second part of the statement 01:08:52.859 --> 01:08:54.130 align:middle line:90% of this theorem here. 01:08:54.130 --> 01:09:00.814 align:middle line:84% So suppose L is this common number, limsup and liminf. 01:09:00.814 --> 01:09:05.450 align:middle line:90% 01:09:05.450 --> 01:09:07.670 align:middle line:84% Then, for all N, [? a ?] natural number, 01:09:07.670 --> 01:09:11.600 align:middle line:84% we have that the inf of the x sub k's, 01:09:11.600 --> 01:09:14.270 align:middle line:84% which is for k bigger than or equal to n-- 01:09:14.270 --> 01:09:17.370 align:middle line:90% 01:09:17.370 --> 01:09:22.080 align:middle line:84% so x sub n is in this set, so it's certainly bigger than 01:09:22.080 --> 01:09:24.689 align:middle line:90% or equal to this inf. 01:09:24.689 --> 01:09:30.050 align:middle line:84% And it's less than or equal to the sup of that set, 01:09:30.050 --> 01:09:34.220 align:middle line:84% again, because x sub n sits in this set. 01:09:34.220 --> 01:09:37.060 align:middle line:90% All right? 01:09:37.060 --> 01:09:40.500 align:middle line:84% But as n goes to infinity, this sequence 01:09:40.500 --> 01:09:42.850 align:middle line:84% of numbers converges to the liminf, 01:09:42.850 --> 01:09:46.740 align:middle line:84% which is L. As n goes to infinity, 01:09:46.740 --> 01:09:49.350 align:middle line:84% this sequence of numbers converges to the limsup, which 01:09:49.350 --> 01:09:52.930 align:middle line:84% is L. So by the squeeze theorem, the thing in between, 01:09:52.930 --> 01:10:14.080 align:middle line:84% which is x sub n, converges to L. 01:10:14.080 --> 01:10:15.820 align:middle line:90% So for the second part, it's-- 01:10:15.820 --> 01:10:19.130 align:middle line:90% 01:10:19.130 --> 01:10:20.825 align:middle line:90% follows from what we've proven-- 01:10:20.825 --> 01:10:24.930 align:middle line:90% 01:10:24.930 --> 01:10:30.170 align:middle line:84% what we proved to obtain the Bolzano-Weierstrass 01:10:30.170 --> 01:10:33.650 align:middle line:84% and what we know about convergent sequences 01:10:33.650 --> 01:10:35.000 align:middle line:90% and their subsequences. 01:10:35.000 --> 01:10:37.724 align:middle line:90% 01:10:37.724 --> 01:10:39.710 align:middle line:84% So this is for this direction-- namely, 01:10:39.710 --> 01:10:43.070 align:middle line:84% that convergence implies the limsup equals the liminf. 01:10:43.070 --> 01:10:49.630 align:middle line:90% 01:10:49.630 --> 01:10:55.150 align:middle line:84% Let L be the limit as n goes to infinity of x sub n. 01:10:55.150 --> 01:10:58.870 align:middle line:84% So now we're assuming that the sequence converges to something 01:10:58.870 --> 01:11:01.135 align:middle line:84% which we call L-- doesn't have to be the same L from-- 01:11:01.135 --> 01:11:03.750 align:middle line:90% 01:11:03.750 --> 01:11:07.350 align:middle line:84% so it's just L I'm just using for this limit. 01:11:07.350 --> 01:11:10.430 align:middle line:90% 01:11:10.430 --> 01:11:25.180 align:middle line:84% By previous theorem, there exists a subsequence x sub n 01:11:25.180 --> 01:11:30.010 align:middle line:84% sub k, such that the limit as k goes 01:11:30.010 --> 01:11:37.330 align:middle line:84% to infinity of x sub n sub k gives me limsup of x sub n. 01:11:37.330 --> 01:11:40.240 align:middle line:90% 01:11:40.240 --> 01:11:46.450 align:middle line:84% But this is a subsequence of a convergent sequence, 01:11:46.450 --> 01:11:50.560 align:middle line:84% and a subsequence of a convergent sequence 01:11:50.560 --> 01:11:53.210 align:middle line:84% is convergent, and converges to the same thing. 01:11:53.210 --> 01:12:02.380 align:middle line:84% So this thing on the left is just equal to L. OK? 01:12:02.380 --> 01:12:10.710 align:middle line:90% 01:12:10.710 --> 01:12:18.450 align:middle line:84% Similarly, there exists a subsequence x sub n sub k, 01:12:18.450 --> 01:12:22.440 align:middle line:84% such that the limit as k goes to infinity of x sub n sub 01:12:22.440 --> 01:12:28.770 align:middle line:90% k equals liminf of x sub n. 01:12:28.770 --> 01:12:34.930 align:middle line:84% And again, this is a subsequence of a convergent sequence, 01:12:34.930 --> 01:12:37.530 align:middle line:84% so it's convergent and convergence of the same thing, 01:12:37.530 --> 01:12:47.530 align:middle line:84% L. So that implies that this thing is equal to L. 01:12:47.530 --> 01:12:49.703 align:middle line:90% All right. 01:12:49.703 --> 01:12:52.120 align:middle line:84% And therefore, the limsup and the liminf equal each other, 01:12:52.120 --> 01:12:55.340 align:middle line:84% and they also equal the limit of the original sequence. 01:12:55.340 --> 01:13:07.400 align:middle line:84% So that is the end of the proof of this theorem that the limsup 01:13:07.400 --> 01:13:08.510 align:middle line:90% and liminf-- 01:13:08.510 --> 01:13:12.650 align:middle line:84% when they coincide and tell you that the original sequence 01:13:12.650 --> 01:13:13.320 align:middle line:90% converges. 01:13:13.320 --> 01:13:14.695 align:middle line:84% So that's another way of thinking 01:13:14.695 --> 01:13:18.170 align:middle line:84% about limsups and liminfs is that they also 01:13:18.170 --> 01:13:22.850 align:middle line:84% somehow measure just how divergent your sequence is, 01:13:22.850 --> 01:13:24.530 align:middle line:84% or at least the difference between them. 01:13:24.530 --> 01:13:27.290 align:middle line:84% If that difference is 0, then your original sequence 01:13:27.290 --> 01:13:28.380 align:middle line:90% is convergent. 01:13:28.380 --> 01:13:28.880 align:middle line:90% OK? 01:13:28.880 --> 01:13:32.640 align:middle line:90% 01:13:32.640 --> 01:13:35.630 align:middle line:84% All right, so I think we'll stop there. 01:13:35.630 --> 01:13:42.241 align:middle line:90%