WEBVTT 00:00:00.000 --> 00:00:13.050 align:middle line:84% [SQUEAKING] [RUSTLING] [CLICKING] 00:00:13.050 --> 00:00:16.980 align:middle line:84% CASEY RODRIGUEZ: OK, so last time, we 00:00:16.980 --> 00:00:22.380 align:middle line:84% were talking about sequences, and I introduced the notion 00:00:22.380 --> 00:00:25.540 align:middle line:90% of a limit of a sequence. 00:00:25.540 --> 00:00:29.580 align:middle line:84% So we say that x n converges to x if for all epsilon positive 00:00:29.580 --> 00:00:31.260 align:middle line:84% there exists an M, a natural number, 00:00:31.260 --> 00:00:33.960 align:middle line:84% such that for all little n bigger than or equal to capital 00:00:33.960 --> 00:00:37.720 align:middle line:84% M we have x sub n minus x is less than epsilon. 00:00:37.720 --> 00:00:43.550 align:middle line:84% So what this definition says is that given 00:00:43.550 --> 00:00:46.580 align:middle line:84% any little bit of tolerance, as long as I go far enough 00:00:46.580 --> 00:00:49.160 align:middle line:84% out in the sequence, the entries are getting 00:00:49.160 --> 00:00:51.620 align:middle line:90% within that tolerance to x. 00:00:51.620 --> 00:00:54.680 align:middle line:84% And when x then converges to x, we'll 00:00:54.680 --> 00:00:58.310 align:middle line:84% write x is equal to the limit as n goes to infinity of x sub n, 00:00:58.310 --> 00:01:04.560 align:middle line:84% or using this notation, x sub n arrow x. 00:01:04.560 --> 00:01:08.530 align:middle line:84% And so I've said before that if you ever 00:01:08.530 --> 00:01:13.630 align:middle line:84% have a reasonably complicated or interesting definition, 00:01:13.630 --> 00:01:17.270 align:middle line:84% you should try to come up with examples and negate it. 00:01:17.270 --> 00:01:24.250 align:middle line:84% So the end of last time, we saw a few examples of-- 00:01:24.250 --> 00:01:27.130 align:middle line:84% at least one example of-- a convergent sequence. 00:01:27.130 --> 00:01:29.830 align:middle line:84% Let's negate this definition because then we'll 00:01:29.830 --> 00:01:31.960 align:middle line:84% also come up with an example of a sequence which 00:01:31.960 --> 00:01:33.320 align:middle line:90% does not converge. 00:01:33.320 --> 00:01:38.560 align:middle line:84% And so the negation of this definition 00:01:38.560 --> 00:01:49.040 align:middle line:84% is that x n does not converge to x. 00:01:49.040 --> 00:01:49.980 align:middle line:90% What does this mean? 00:01:49.980 --> 00:01:51.950 align:middle line:84% So this is the actual negation part. 00:01:51.950 --> 00:01:57.980 align:middle line:84% If there exists an epsilon positive-- so 00:01:57.980 --> 00:02:01.430 align:middle line:84% whenever we negate a "for all," it becomes "there exists," 00:02:01.430 --> 00:02:03.350 align:middle line:84% and whenever we negate a "there exists," 00:02:03.350 --> 00:02:04.880 align:middle line:90% it becomes a "for all." 00:02:04.880 --> 00:02:10.520 align:middle line:84% So there exists some bad epsilon 0 positive such 00:02:10.520 --> 00:02:18.740 align:middle line:84% that for all natural numbers N, there exists an n bigger 00:02:18.740 --> 00:02:25.280 align:middle line:84% than or equal to M such that x n minus 00:02:25.280 --> 00:02:32.420 align:middle line:84% x is bigger than or equal to this bad epsilon 0 in distance. 00:02:32.420 --> 00:02:35.030 align:middle line:84% So we have a picture that goes along with convergence. 00:02:35.030 --> 00:02:38.330 align:middle line:84% We can have a picture that goes along with a sequence not 00:02:38.330 --> 00:02:39.140 align:middle line:90% converging to x. 00:02:39.140 --> 00:02:46.520 align:middle line:84% So this means if I go out within this bad epsilon 0, 00:02:46.520 --> 00:02:50.780 align:middle line:84% and if I go arbitrarily out in the sequence, 00:02:50.780 --> 00:02:53.990 align:middle line:84% I can always find an element x sub n which 00:02:53.990 --> 00:02:58.595 align:middle line:90% is not in that interval. 00:02:58.595 --> 00:03:01.330 align:middle line:90% 00:03:01.330 --> 00:03:04.030 align:middle line:84% And we use this negation to prove a certain sequence 00:03:04.030 --> 00:03:07.120 align:middle line:90% does not converge to x. 00:03:07.120 --> 00:03:09.880 align:middle line:84% But let's go back do another example 00:03:09.880 --> 00:03:13.465 align:middle line:84% of a sequence which does converge, so-- 00:03:13.465 --> 00:03:25.310 align:middle line:90% 00:03:25.310 --> 00:03:25.910 align:middle line:90% equals 0. 00:03:25.910 --> 00:03:28.780 align:middle line:90% 00:03:28.780 --> 00:03:30.910 align:middle line:90% So again, how does the proof go? 00:03:30.910 --> 00:03:34.510 align:middle line:84% You have to verify the definition of the limit. 00:03:34.510 --> 00:03:38.770 align:middle line:84% I mean, there's nothing else that we have to use right now. 00:03:38.770 --> 00:03:42.262 align:middle line:90% So let epsilon be positive. 00:03:42.262 --> 00:03:43.720 align:middle line:84% And now, off to the side over here, 00:03:43.720 --> 00:03:48.100 align:middle line:84% I'm going to do a little bit of work 00:03:48.100 --> 00:03:51.640 align:middle line:84% to show you how exactly in practice one 00:03:51.640 --> 00:03:55.180 align:middle line:84% would find such a capital M or choose a capital M to get 00:03:55.180 --> 00:03:59.470 align:middle line:84% that 1 over n squared plus 30n plus 1 would be less 00:03:59.470 --> 00:04:04.380 align:middle line:84% than epsilon for n bigger than or equal to capital M. 00:04:04.380 --> 00:04:08.640 align:middle line:84% So typically how this works is, unlike in calculus 00:04:08.640 --> 00:04:12.030 align:middle line:84% where you have some inequality that you want to get 00:04:12.030 --> 00:04:16.649 align:middle line:84% and then you solve for n, in analysis what's 00:04:16.649 --> 00:04:19.500 align:middle line:84% great is that you can replace that thing 00:04:19.500 --> 00:04:22.050 align:middle line:84% you want to make small by something simpler, 00:04:22.050 --> 00:04:23.490 align:middle line:90% and then solve that inequality. 00:04:23.490 --> 00:04:25.020 align:middle line:90% What do I mean? 00:04:25.020 --> 00:04:33.450 align:middle line:84% So what I want to do, I want to find M so 00:04:33.450 --> 00:04:38.250 align:middle line:84% that if n it is bigger than or equal to M, then-- 00:04:38.250 --> 00:04:43.750 align:middle line:84% so I'm saying the limit is 0, so I want to show 1 over n 00:04:43.750 --> 00:04:50.690 align:middle line:84% squared plus 30n plus 1 is less than epsilon. 00:04:50.690 --> 00:04:53.990 align:middle line:84% Now, I could try to solve this inequality for n, 00:04:53.990 --> 00:05:00.020 align:middle line:84% but this is a quadratic function and imagine I put 20 there, 00:05:00.020 --> 00:05:04.940 align:middle line:84% so I can't exactly solve exactly for n to guarantee this. 00:05:04.940 --> 00:05:09.670 align:middle line:84% But what's great about analysis is that I 00:05:09.670 --> 00:05:13.630 align:middle line:90% don't have to work that hard. 00:05:13.630 --> 00:05:15.760 align:middle line:84% I could start with this thing which 00:05:15.760 --> 00:05:20.380 align:middle line:84% I want to bound by epsilon, replace it by something bigger, 00:05:20.380 --> 00:05:23.260 align:middle line:84% and then find capital M so that that bigger 00:05:23.260 --> 00:05:24.560 align:middle line:90% thing is less than epsilon. 00:05:24.560 --> 00:05:25.460 align:middle line:90% So what do I mean? 00:05:25.460 --> 00:05:27.130 align:middle line:90% So let's start with this thing-- 00:05:27.130 --> 00:05:33.600 align:middle line:84% 1 over n squared plus 30n plus 1. 00:05:33.600 --> 00:05:36.020 align:middle line:84% So this one is only making things bigger. 00:05:36.020 --> 00:05:37.395 align:middle line:84% I'm going to do this very slowly. 00:05:37.395 --> 00:05:42.520 align:middle line:90% 00:05:42.520 --> 00:05:46.600 align:middle line:84% So this is less than or equal to this 00:05:46.600 --> 00:05:51.880 align:middle line:84% because for this, I've just added 1 to the denominator. 00:05:51.880 --> 00:05:54.462 align:middle line:90% And 1 over n squared plus 30n-- 00:05:54.462 --> 00:05:56.170 align:middle line:84% this certainly looks a little bit simpler 00:05:56.170 --> 00:05:58.280 align:middle line:84% to solve for n less than epsilon. 00:05:58.280 --> 00:06:01.060 align:middle line:84% But I can make it even simpler by dropping the n squared, 00:06:01.060 --> 00:06:02.740 align:middle line:84% because n squared is positive, so that's 00:06:02.740 --> 00:06:03.990 align:middle line:90% only making the bottom bigger. 00:06:03.990 --> 00:06:07.460 align:middle line:90% 00:06:07.460 --> 00:06:10.700 align:middle line:84% So I've 1 over 30n, and this is certainly 00:06:10.700 --> 00:06:14.240 align:middle line:90% less than or equal to 1 over n. 00:06:14.240 --> 00:06:19.190 align:middle line:84% So what does this computation show? 00:06:19.190 --> 00:06:24.680 align:middle line:84% This shows that if 1 over n is less than epsilon, 00:06:24.680 --> 00:06:30.500 align:middle line:84% then this implies that 1 over n squared plus 30n plus 1 is 00:06:30.500 --> 00:06:31.250 align:middle line:90% less than epsilon. 00:06:31.250 --> 00:06:35.380 align:middle line:90% 00:06:35.380 --> 00:06:36.940 align:middle line:84% If this thing is less than epsilon, 00:06:36.940 --> 00:06:38.830 align:middle line:84% then by this string of inequalities, 00:06:38.830 --> 00:06:40.430 align:middle line:84% this thing will be less than epsilon. 00:06:40.430 --> 00:06:45.917 align:middle line:84% So we choose capital M so that we get this. 00:06:45.917 --> 00:06:47.500 align:middle line:84% Now, when we actually write the proof, 00:06:47.500 --> 00:06:48.792 align:middle line:90% it'll look something like this. 00:06:48.792 --> 00:06:51.460 align:middle line:84% But the actual style of the proof 00:06:51.460 --> 00:06:54.522 align:middle line:90% is a little bit different. 00:06:54.522 --> 00:06:56.230 align:middle line:84% And if you didn't see this, it would just 00:06:56.230 --> 00:07:00.575 align:middle line:84% seem like I pulled this capital M out of nowhere, 00:07:00.575 --> 00:07:02.450 align:middle line:84% but this is the thinking that goes behind it. 00:07:02.450 --> 00:07:04.340 align:middle line:90% So let epsilon be positive. 00:07:04.340 --> 00:07:09.730 align:middle line:84% Now I'll tell you how to choose capital M. Choose 00:07:09.730 --> 00:07:14.410 align:middle line:84% M a natural number so that-- so we want it 00:07:14.410 --> 00:07:16.960 align:middle line:84% for all n bigger than or equal to capital M, 1 over n 00:07:16.960 --> 00:07:18.410 align:middle line:90% is less than epsilon. 00:07:18.410 --> 00:07:21.400 align:middle line:84% So this will certainly be true if n 00:07:21.400 --> 00:07:23.740 align:middle line:84% is bigger than or equal to capital M 00:07:23.740 --> 00:07:25.810 align:middle line:84% so that 1 over capital M is less than epsilon. 00:07:25.810 --> 00:07:29.985 align:middle line:90% 00:07:29.985 --> 00:07:31.485 align:middle line:84% And the reason we can choose capital 00:07:31.485 --> 00:07:33.910 align:middle line:84% M like this-- again, where is this coming from? 00:07:33.910 --> 00:07:37.515 align:middle line:84% It's coming from the Archimedean property of the real numbers. 00:07:37.515 --> 00:07:40.710 align:middle line:90% 00:07:40.710 --> 00:07:43.260 align:middle line:84% So we've chosen capital M. Now we 00:07:43.260 --> 00:07:44.925 align:middle line:84% need to show that this capital M works. 00:07:44.925 --> 00:07:48.030 align:middle line:90% 00:07:48.030 --> 00:07:53.190 align:middle line:84% Then for all n bigger than or equal to capital M, 00:07:53.190 --> 00:07:59.190 align:middle line:84% if I look at 1 over n squared plus 30n plus 1 00:07:59.190 --> 00:08:03.090 align:middle line:84% minus 0, my proposed limit, this is equal to 1 00:08:03.090 --> 00:08:07.890 align:middle line:90% over n squared plus 1. 00:08:07.890 --> 00:08:09.390 align:middle line:84% And this is less than or equal to, 00:08:09.390 --> 00:08:12.990 align:middle line:84% if I drop n squared and 1, 30n, which 00:08:12.990 --> 00:08:16.110 align:middle line:84% is less than or equal to 1 over n, which is less than 00:08:16.110 --> 00:08:18.840 align:middle line:84% or equal to 1 over capital M, which is less than epsilon. 00:08:18.840 --> 00:08:37.030 align:middle line:90% 00:08:37.030 --> 00:08:42.970 align:middle line:90% So something of a non-example-- 00:08:42.970 --> 00:08:49.960 align:middle line:84% let's show that the sequence minus 1 to the n DNC 00:08:49.960 --> 00:08:51.310 align:middle line:90% does not converge-- 00:08:51.310 --> 00:08:53.170 align:middle line:84% not Democratic National Committee-- 00:08:53.170 --> 00:08:56.130 align:middle line:90% 00:08:56.130 --> 00:08:57.570 align:middle line:90% does not converge. 00:08:57.570 --> 00:09:02.000 align:middle line:84% So this is just a sequence minus 1, 00:09:02.000 --> 00:09:06.870 align:middle line:84% 1, minus 1, 1, minus 1, and so on. 00:09:06.870 --> 00:09:09.180 align:middle line:84% So we have to show it does not converge, 00:09:09.180 --> 00:09:11.580 align:middle line:84% meaning for every x in R, we'll show 00:09:11.580 --> 00:09:15.180 align:middle line:84% that x n does not converge to x using the negation 00:09:15.180 --> 00:09:17.310 align:middle line:90% of the definition-- 00:09:17.310 --> 00:09:20.595 align:middle line:90% so proof. 00:09:20.595 --> 00:09:34.640 align:middle line:90% 00:09:34.640 --> 00:09:36.545 align:middle line:84% Let x be in R. And what do we want to show? 00:09:36.545 --> 00:09:41.200 align:middle line:90% 00:09:41.200 --> 00:09:53.020 align:middle line:84% Minus 1 to the n-th does not converge to x. 00:09:53.020 --> 00:09:55.700 align:middle line:90% 00:09:55.700 --> 00:09:57.120 align:middle line:90% And we'll use this definition. 00:09:57.120 --> 00:10:02.660 align:middle line:84% So this definition means we need to find a bad epsilon 0. 00:10:02.660 --> 00:10:06.780 align:middle line:90% 00:10:06.780 --> 00:10:11.470 align:middle line:84% And here's the kind of thinking that goes along with this-- 00:10:11.470 --> 00:10:13.930 align:middle line:90% there's 1, there's minus 1. 00:10:13.930 --> 00:10:17.650 align:middle line:84% This sequence just keeps alternating, so intuitively 00:10:17.650 --> 00:10:21.280 align:middle line:84% there's no way that every element of the sequence 00:10:21.280 --> 00:10:22.210 align:middle line:90% can be getting-- 00:10:22.210 --> 00:10:24.190 align:middle line:84% or every entry in the sequence can be getting-- 00:10:24.190 --> 00:10:26.020 align:middle line:90% close to some single x. 00:10:26.020 --> 00:10:27.380 align:middle line:90% So what's the idea? 00:10:27.380 --> 00:10:31.780 align:middle line:84% The idea is that the distance between these two guys 00:10:31.780 --> 00:10:32.620 align:middle line:90% is always 2. 00:10:32.620 --> 00:10:35.210 align:middle line:90% 00:10:35.210 --> 00:10:44.220 align:middle line:84% So somehow, the distance between every entry, 00:10:44.220 --> 00:10:48.330 align:middle line:84% or basically, some entry has to be within distance-- 00:10:48.330 --> 00:10:52.830 align:middle line:84% if this is within distance 1, say, to a given x, 00:10:52.830 --> 00:10:58.200 align:middle line:84% then this will be greater than or equal to 1 in distance to x. 00:10:58.200 --> 00:11:00.090 align:middle line:84% So you can imagine x is over here. 00:11:00.090 --> 00:11:03.060 align:middle line:84% Minus 1 is within distance 1 to x, 00:11:03.060 --> 00:11:09.120 align:middle line:84% but then 1 would be greater than distance 1 to x. 00:11:09.120 --> 00:11:13.290 align:middle line:84% So our bad epsilon 0 was going to be 1, as we'll see. 00:11:13.290 --> 00:11:17.010 align:middle line:90% 00:11:17.010 --> 00:11:18.300 align:middle line:90% Let epsilon 0 equals 1. 00:11:18.300 --> 00:11:19.630 align:middle line:90% That's our bad epsilon. 00:11:19.630 --> 00:11:21.300 align:middle line:84% So now we have to show for all capital 00:11:21.300 --> 00:11:23.190 align:middle line:84% M in the natural numbers, there exists a little n 00:11:23.190 --> 00:11:24.565 align:middle line:84% bigger than or equal to capital M 00:11:24.565 --> 00:11:28.210 align:middle line:90% so that we have that inequality. 00:11:28.210 --> 00:11:31.575 align:middle line:90% So let M be a natural number. 00:11:31.575 --> 00:11:42.020 align:middle line:90% 00:11:42.020 --> 00:11:43.445 align:middle line:90% Then what do we see? 00:11:43.445 --> 00:11:46.330 align:middle line:90% 00:11:46.330 --> 00:11:49.300 align:middle line:84% So then it says, they're exists an n greater than 00:11:49.300 --> 00:11:54.180 align:middle line:84% or equal to capital M. And I will 00:11:54.180 --> 00:11:57.330 align:middle line:84% show that n is, in fact, either capital M or capital 00:11:57.330 --> 00:12:08.730 align:middle line:84% M plus 1, then 2, which is the distance between minus 1 00:12:08.730 --> 00:12:14.770 align:middle line:84% to the M and minus 1 to the M plus 1 is 00:12:14.770 --> 00:12:17.440 align:middle line:84% less than or equal to-- now using the triangle inequality, 00:12:17.440 --> 00:12:21.010 align:middle line:84% I add and subtract x and then use the triangle inequality. 00:12:21.010 --> 00:12:31.320 align:middle line:90% 00:12:31.320 --> 00:12:37.220 align:middle line:84% So I have the sum of two numbers is bigger than or equal to 2. 00:12:37.220 --> 00:12:39.800 align:middle line:84% And the only way that can happen is if one of these numbers 00:12:39.800 --> 00:12:41.270 align:middle line:90% is bigger than or equal to 1. 00:12:41.270 --> 00:12:43.400 align:middle line:84% If both of these numbers are less than 1, 00:12:43.400 --> 00:12:46.580 align:middle line:84% then the sum is less than 2, and I would have 2 is less than 2. 00:12:46.580 --> 00:12:47.790 align:middle line:90% That's not possible. 00:12:47.790 --> 00:12:50.780 align:middle line:84% So one of these numbers has to be bigger than or equal to 1. 00:12:50.780 --> 00:13:09.140 align:middle line:90% 00:13:09.140 --> 00:13:12.140 align:middle line:84% And therefore, I could take n equal to capital M 00:13:12.140 --> 00:13:13.940 align:middle line:84% if this is bigger than or equal to 1, 00:13:13.940 --> 00:13:16.280 align:middle line:84% or if this is bigger than or equal to 1, 00:13:16.280 --> 00:13:18.590 align:middle line:84% I take little n to be capital M plus 1. 00:13:18.590 --> 00:13:21.030 align:middle line:90% So that's the end of that. 00:13:21.030 --> 00:13:28.080 align:middle line:90% 00:13:28.080 --> 00:13:33.870 align:middle line:84% So that's good for examples for now. 00:13:33.870 --> 00:13:37.560 align:middle line:84% Let's prove a general theorem about convergent sequences-- 00:13:37.560 --> 00:13:40.230 align:middle line:84% namely, that if I have a convergent sequence, then 00:13:40.230 --> 00:13:40.860 align:middle line:90% it's bounded. 00:13:40.860 --> 00:13:57.690 align:middle line:90% 00:13:57.690 --> 00:14:04.230 align:middle line:84% And let me just give you a quick reminder 00:14:04.230 --> 00:14:10.890 align:middle line:84% about what being bounded means for a sequence, i.e., 00:14:10.890 --> 00:14:16.590 align:middle line:84% there exists some number such that for all n, 00:14:16.590 --> 00:14:22.430 align:middle line:84% x n is less than or equal to capital B. Now, 00:14:22.430 --> 00:14:30.510 align:middle line:84% as beginning students of math, what types of questions 00:14:30.510 --> 00:14:32.130 align:middle line:84% should we start asking ourselves? 00:14:32.130 --> 00:14:37.220 align:middle line:90% 00:14:37.220 --> 00:14:40.510 align:middle line:84% So one of the types, or a type of question 00:14:40.510 --> 00:14:44.050 align:middle line:84% that we should ask when we come across the theorem that 00:14:44.050 --> 00:14:47.830 align:middle line:84% is a one-way street, meaning if something 00:14:47.830 --> 00:14:50.560 align:middle line:84% happens then something else happens, 00:14:50.560 --> 00:14:51.940 align:middle line:90% is does the converse hold? 00:14:51.940 --> 00:14:55.450 align:middle line:84% Namely, if I assume x sub n is a bounded sequence, 00:14:55.450 --> 00:14:58.390 align:middle line:84% does this imply that x n is convergent? 00:14:58.390 --> 00:15:01.090 align:middle line:90% 00:15:01.090 --> 00:15:03.670 align:middle line:84% So if you ever hear the words, "does the converse hold," 00:15:03.670 --> 00:15:05.500 align:middle line:90% that's what they mean. 00:15:05.500 --> 00:15:07.980 align:middle line:84% Now, for this statement, this is false. 00:15:07.980 --> 00:15:10.900 align:middle line:84% If x n is bounded implies x n is convergent, 00:15:10.900 --> 00:15:12.990 align:middle line:84% that's false because we have an example right 00:15:12.990 --> 00:15:14.700 align:middle line:90% there of a bounded sequence. 00:15:14.700 --> 00:15:16.560 align:middle line:84% This sequence here, minus 1 to the n, 00:15:16.560 --> 00:15:22.590 align:middle line:84% is bounded since the sequence in absolute value 00:15:22.590 --> 00:15:26.100 align:middle line:84% equals 1 for all n, so we could take B equals 1. 00:15:26.100 --> 00:15:28.050 align:middle line:84% So minus 1 to the n is a bounded sequence, 00:15:28.050 --> 00:15:29.110 align:middle line:90% which is not convergent. 00:15:29.110 --> 00:15:31.790 align:middle line:90% So the converse does not hold. 00:15:31.790 --> 00:15:35.250 align:middle line:90% So let's prove the theorem. 00:15:35.250 --> 00:15:39.620 align:middle line:84% And let me draw a little picture to go along 00:15:39.620 --> 00:15:43.360 align:middle line:90% with it of what's going on. 00:15:43.360 --> 00:15:47.440 align:middle line:90% So here is 0, let's imagine. 00:15:47.440 --> 00:15:52.660 align:middle line:84% Let me make this picture a little bit bigger 00:15:52.660 --> 00:15:53.950 align:middle line:90% before I go into the proof. 00:15:53.950 --> 00:15:59.910 align:middle line:90% 00:15:59.910 --> 00:16:01.635 align:middle line:90% So let x be the limit. 00:16:01.635 --> 00:16:04.690 align:middle line:90% 00:16:04.690 --> 00:16:06.000 align:middle line:90% So this is not the proof yet. 00:16:06.000 --> 00:16:11.200 align:middle line:90% This is just more discussion. 00:16:11.200 --> 00:16:14.610 align:middle line:84% So if this x sub n is converging to x, 00:16:14.610 --> 00:16:21.780 align:middle line:84% then let's say I go out distance 1 to x. 00:16:21.780 --> 00:16:24.030 align:middle line:84% Then all of the x sub n's eventually 00:16:24.030 --> 00:16:27.210 align:middle line:90% have to land in this strip here. 00:16:27.210 --> 00:16:30.880 align:middle line:90% 00:16:30.880 --> 00:16:39.340 align:middle line:84% So x sub n, n bigger than or equal to some capital M. 00:16:39.340 --> 00:16:41.830 align:middle line:84% So I know that they're all within distance 1 to x. 00:16:41.830 --> 00:16:45.730 align:middle line:84% So in absolute value, they'd be bounded by, 00:16:45.730 --> 00:16:47.860 align:middle line:84% let's say for this picture, it'd be x plus 1. 00:16:47.860 --> 00:16:51.520 align:middle line:84% But if this was over there, capital, 00:16:51.520 --> 00:16:54.820 align:middle line:84% it would be the absolute value of x plus 1. 00:16:54.820 --> 00:16:57.610 align:middle line:84% So that takes care of all x sub n's with n bigger than 00:16:57.610 --> 00:17:03.930 align:middle line:84% or equal to M. And then there's only finitely many guys 00:17:03.930 --> 00:17:05.280 align:middle line:90% to deal with. 00:17:05.280 --> 00:17:10.079 align:middle line:84% Maybe there's a bigger one out here, x sub n minus 1 00:17:10.079 --> 00:17:14.099 align:middle line:84% to deal with, and that's how we define our number, 00:17:14.099 --> 00:17:18.569 align:middle line:84% is by the absolute value of x plus 1 00:17:18.569 --> 00:17:21.835 align:middle line:84% plus the absolute value of these finitely many. 00:17:21.835 --> 00:17:23.460 align:middle line:84% And then that would be something that's 00:17:23.460 --> 00:17:25.710 align:middle line:84% bigger than or equal to every element in the sequence. 00:17:25.710 --> 00:17:28.298 align:middle line:90% 00:17:28.298 --> 00:17:29.840 align:middle line:84% That's the picture that goes with it. 00:17:29.840 --> 00:17:33.830 align:middle line:90% 00:17:33.830 --> 00:17:35.630 align:middle line:84% How do we take that picture to proof? 00:17:35.630 --> 00:17:40.240 align:middle line:90% 00:17:40.240 --> 00:17:50.500 align:middle line:84% So suppose x n converges to x-- that's our assumption, 00:17:50.500 --> 00:17:55.030 align:middle line:84% and there exists a natural number M such 00:17:55.030 --> 00:18:02.980 align:middle line:84% that for all n bigger than or equal to capital m, x sub 00:18:02.980 --> 00:18:07.770 align:middle line:84% n minus x is within distance 1, or x sub n is within distance 1 00:18:07.770 --> 00:18:14.410 align:middle line:84% to x, then for all n bigger than or equal to capital M, 00:18:14.410 --> 00:18:20.160 align:middle line:84% if we look at the absolute value of x sub n, 00:18:20.160 --> 00:18:23.970 align:middle line:84% add and subtract x, and now use the triangle inequality, 00:18:23.970 --> 00:18:26.790 align:middle line:84% this is less than or equal to absolute value of x sub 00:18:26.790 --> 00:18:33.060 align:middle line:84% n minus x plus the absolute value of x equals 1 plus x. 00:18:33.060 --> 00:18:37.870 align:middle line:84% So for all entries in the sequence past this point 00:18:37.870 --> 00:18:42.230 align:middle line:84% capital M, they're bounded in absolute value by 1 plus x. 00:18:42.230 --> 00:18:43.105 align:middle line:90% That's just a number. 00:18:43.105 --> 00:18:46.300 align:middle line:90% 00:18:46.300 --> 00:18:52.660 align:middle line:84% So then, we just have to take care of the 1 00:18:52.660 --> 00:18:54.760 align:middle line:90% up to capital M minus 1 guys. 00:18:54.760 --> 00:18:56.260 align:middle line:84% We just have to find a number bigger 00:18:56.260 --> 00:18:59.440 align:middle line:84% than those guys and this number, and we'll 00:18:59.440 --> 00:19:02.410 align:middle line:84% have found a B. So then let's take 00:19:02.410 --> 00:19:11.670 align:middle line:84% B to be x sub 1 plus x sub 2 plus x sub capital 00:19:11.670 --> 00:19:18.870 align:middle line:90% M minus 1 plus this. 00:19:18.870 --> 00:19:29.530 align:middle line:90% 00:19:29.530 --> 00:19:38.190 align:middle line:84% Then for all n a natural number, the absolute value of x of n 00:19:38.190 --> 00:19:40.590 align:middle line:84% is less than or equal to B. Because if n 00:19:40.590 --> 00:19:44.460 align:middle line:84% is less than capital M, then its absolute value 00:19:44.460 --> 00:19:45.820 align:middle line:90% is bounded by one of these. 00:19:45.820 --> 00:19:47.910 align:middle line:84% Its absolute value is equal to 1 over these, 00:19:47.910 --> 00:19:50.910 align:middle line:84% and therefore, less than or equal to one of these 00:19:50.910 --> 00:19:54.020 align:middle line:90% plus some non-negative numbers. 00:19:54.020 --> 00:19:56.840 align:middle line:84% And if n is bigger than or equal to capital M, 00:19:56.840 --> 00:20:00.230 align:middle line:84% then we have this bound that we use. 00:20:00.230 --> 00:20:01.970 align:middle line:84% And this number is certainly less than 00:20:01.970 --> 00:20:04.670 align:middle line:84% or equal to capital B, again, because it's 00:20:04.670 --> 00:20:06.800 align:middle line:84% this number plus a sum of non-negative numbers. 00:20:06.800 --> 00:20:24.230 align:middle line:90% 00:20:24.230 --> 00:20:29.870 align:middle line:84% Now in general, there's no easily checkable criterion 00:20:29.870 --> 00:20:32.720 align:middle line:90% for a sequence to converge. 00:20:32.720 --> 00:20:34.940 align:middle line:84% All we can do is verify the definition. 00:20:34.940 --> 00:20:38.690 align:middle line:90% 00:20:38.690 --> 00:20:41.500 align:middle line:84% But there are some sequences which you can easily 00:20:41.500 --> 00:20:45.040 align:middle line:84% check or figure out if they converge just 00:20:45.040 --> 00:20:48.190 align:middle line:90% by telling if they're bounded. 00:20:48.190 --> 00:20:50.400 align:middle line:84% Those are what are called "monotone sequences." 00:20:50.400 --> 00:20:55.710 align:middle line:90% 00:20:55.710 --> 00:20:57.750 align:middle line:84% So this is, in some sense, the class 00:20:57.750 --> 00:21:00.030 align:middle line:84% of sequences that does kind of satisfy 00:21:00.030 --> 00:21:04.440 align:middle line:84% the converse of that theorem I stated a minute ago. 00:21:04.440 --> 00:21:22.730 align:middle line:84% So sequence x sub n is monotone increasing 00:21:22.730 --> 00:21:32.520 align:middle line:84% if for all natural numbers little n, x sub n is less than 00:21:32.520 --> 00:21:34.350 align:middle line:90% or equal to x sub n plus 1. 00:21:34.350 --> 00:21:38.880 align:middle line:84% So this means that x sub 1 is less 00:21:38.880 --> 00:21:43.020 align:middle line:84% than or equal to x sub 2 is less than or equal to x to 3, 00:21:43.020 --> 00:21:43.740 align:middle line:90% and so on. 00:21:43.740 --> 00:21:54.580 align:middle line:90% 00:21:54.580 --> 00:22:03.050 align:middle line:84% Sequence is monotone decreasing if we 00:22:03.050 --> 00:22:04.500 align:middle line:90% have the other inequality. 00:22:04.500 --> 00:22:09.080 align:middle line:84% So things are getting smaller, so x sub n 00:22:09.080 --> 00:22:11.285 align:middle line:84% is bigger than or equal to x sub n plus 1. 00:22:11.285 --> 00:22:32.240 align:middle line:90% 00:22:32.240 --> 00:22:39.830 align:middle line:84% And if we have a sequence which is monotone increasing 00:22:39.830 --> 00:22:41.510 align:middle line:90% or monotone decreasing-- 00:22:41.510 --> 00:22:44.850 align:middle line:90% 00:22:44.850 --> 00:22:47.190 align:middle line:90% so it's one or the other-- 00:22:47.190 --> 00:22:55.440 align:middle line:84% we say x n is monotone or monotonic. 00:22:55.440 --> 00:23:00.140 align:middle line:90% 00:23:00.140 --> 00:23:06.260 align:middle line:84% So it's simple enough to come up with examples of sequences 00:23:06.260 --> 00:23:09.620 align:middle line:84% which are monotone increasing, monotone 00:23:09.620 --> 00:23:11.490 align:middle line:90% decreasing, or neither. 00:23:11.490 --> 00:23:16.940 align:middle line:90% So 1 over n-- 00:23:16.940 --> 00:23:22.760 align:middle line:90% this is just 1, 1/2, 1/3, 1/4-- 00:23:22.760 --> 00:23:27.770 align:middle line:90% this is monotone decreasing. 00:23:27.770 --> 00:23:32.780 align:middle line:84% And if I take minus the sequence, 00:23:32.780 --> 00:23:36.260 align:middle line:84% then that reverses basically all of the inequalities, 00:23:36.260 --> 00:23:38.130 align:middle line:84% and I get a monotone increasing one. 00:23:38.130 --> 00:23:42.080 align:middle line:84% So then this is minus 1, minus 1/2, minus 1/3, 00:23:42.080 --> 00:23:46.790 align:middle line:90% minus 1/4, monotone increasing. 00:23:46.790 --> 00:23:48.470 align:middle line:84% And then something that's not is-- we've 00:23:48.470 --> 00:23:57.310 align:middle line:84% already come across that-- minus 1 to the n, and so on. 00:23:57.310 --> 00:24:04.060 align:middle line:90% 00:24:04.060 --> 00:24:07.540 align:middle line:84% And so the theorem about monotone sequences 00:24:07.540 --> 00:24:12.820 align:middle line:84% is that there is a simpler criterion than the definition 00:24:12.820 --> 00:24:15.400 align:middle line:84% for determining when they're convergent, 00:24:15.400 --> 00:24:19.000 align:middle line:84% or if you like, they satisfy the converse of this theorem up 00:24:19.000 --> 00:24:20.180 align:middle line:90% here. 00:24:20.180 --> 00:24:38.660 align:middle line:84% So a monotonic sequence is convergent if and only 00:24:38.660 --> 00:24:44.850 align:middle line:90% if it is bounded. 00:24:44.850 --> 00:24:47.420 align:middle line:84% So we proved a minute ago that convergent sequences 00:24:47.420 --> 00:24:50.975 align:middle line:84% are bounded, so for monotonic sequences, the converse holds. 00:24:50.975 --> 00:24:54.510 align:middle line:90% 00:24:54.510 --> 00:24:55.760 align:middle line:90% So let's do the proof of this. 00:24:55.760 --> 00:25:00.200 align:middle line:84% I'm going to do the proof for monotone increasing sequences, 00:25:00.200 --> 00:25:02.720 align:middle line:84% and as an exercise, I'll leave it to you 00:25:02.720 --> 00:25:08.090 align:middle line:84% to do the proof for monotone decreasing sequences. 00:25:08.090 --> 00:25:19.160 align:middle line:84% So suppose x sub n is a monotone increasing sequence. 00:25:19.160 --> 00:25:23.080 align:middle line:90% 00:25:23.080 --> 00:25:28.350 align:middle line:84% So there's two directions to prove convergence if and only 00:25:28.350 --> 00:25:29.560 align:middle line:90% if bounded. 00:25:29.560 --> 00:25:33.270 align:middle line:84% So we have one direction, meaning 00:25:33.270 --> 00:25:35.160 align:middle line:90% convergence implies bounded. 00:25:35.160 --> 00:25:40.570 align:middle line:84% This is just the previous theorem, 00:25:40.570 --> 00:25:43.150 align:middle line:84% where we proved that every convergent sequence is bounded, 00:25:43.150 --> 00:25:46.220 align:middle line:90% not just monotonic sequences. 00:25:46.220 --> 00:25:50.020 align:middle line:84% So the meat is in proving the converse direction. 00:25:50.020 --> 00:25:53.050 align:middle line:84% And in fact, we'll be able to pick out what 00:25:53.050 --> 00:25:54.685 align:middle line:90% the limit is of this sequence. 00:25:54.685 --> 00:25:57.200 align:middle line:90% 00:25:57.200 --> 00:26:08.610 align:middle line:90% So suppose x sub n is bounded. 00:26:08.610 --> 00:26:14.060 align:middle line:90% 00:26:14.060 --> 00:26:16.470 align:middle line:84% Then if I look at the set of entries-- 00:26:16.470 --> 00:26:22.130 align:middle line:84% so not the sequence but if I look at the set of values 00:26:22.130 --> 00:26:27.110 align:middle line:84% that this sequence takes-- so x sub n is just a natural number, 00:26:27.110 --> 00:26:30.725 align:middle line:84% this is now a subset of the real numbers. 00:26:30.725 --> 00:26:34.880 align:middle line:90% 00:26:34.880 --> 00:26:51.060 align:middle line:84% This is a bounded set, meaning it's bounded above and below 00:26:51.060 --> 00:26:53.760 align:middle line:84% because there exists some capital B so that x sub 00:26:53.760 --> 00:26:57.390 align:middle line:84% n is, in absolute value, less than 00:26:57.390 --> 00:26:58.920 align:middle line:90% or equal to capital B for all n. 00:26:58.920 --> 00:27:06.600 align:middle line:84% So that means x sub n is between B and minus B. 00:27:06.600 --> 00:27:10.360 align:middle line:84% All right, so what's the picture again that goes with this? 00:27:10.360 --> 00:27:16.000 align:middle line:84% So we have these x sub n's, x sub 1, x sub 2, x sub 3. 00:27:16.000 --> 00:27:21.100 align:middle line:84% We know that they cannot go past a certain B, 00:27:21.100 --> 00:27:28.990 align:middle line:84% and they're steadily increasing, but they can't keep strictly-- 00:27:28.990 --> 00:27:31.060 align:middle line:84% they can't increase without bound. 00:27:31.060 --> 00:27:34.160 align:middle line:90% 00:27:34.160 --> 00:27:42.160 align:middle line:84% Not only that, they're bounded by some number 00:27:42.160 --> 00:27:46.630 align:middle line:84% x, which I'll define to be the supremum of this set. 00:27:46.630 --> 00:27:49.510 align:middle line:84% And these guys are just getting, in essence-- 00:27:49.510 --> 00:27:52.570 align:middle line:84% so this is a picture, just me trying 00:27:52.570 --> 00:27:54.250 align:middle line:84% to explain to you what's going on. 00:27:54.250 --> 00:28:02.620 align:middle line:84% If x is the sup of this set of entries, then 00:28:02.620 --> 00:28:05.550 align:middle line:90% what do I know about x sub n? 00:28:05.550 --> 00:28:09.880 align:middle line:84% I mean x is that if I go a little bit to the left of x-- 00:28:09.880 --> 00:28:13.190 align:middle line:90% so let me draw this again. 00:28:13.190 --> 00:28:16.900 align:middle line:90% So let me back up a minute here. 00:28:16.900 --> 00:28:21.070 align:middle line:90% 00:28:21.070 --> 00:28:33.790 align:middle line:84% So since this set is bounded in R, it has a supremum in R. 00:28:33.790 --> 00:28:36.710 align:middle line:84% And what I claim is that this supremum is, in fact, 00:28:36.710 --> 00:28:38.590 align:middle line:90% the limit of this sequence. 00:28:38.590 --> 00:28:41.170 align:middle line:90% This is a supremum of a set. 00:28:41.170 --> 00:28:43.720 align:middle line:84% I'm saying it's the limit of the sequence. 00:28:43.720 --> 00:28:47.210 align:middle line:90% So what's the thinking here? 00:28:47.210 --> 00:28:50.200 align:middle line:84% So x is the supremum of all the entries in x sub n, 00:28:50.200 --> 00:28:51.700 align:middle line:84% so nothing's ever going to be bigger 00:28:51.700 --> 00:28:56.300 align:middle line:84% than some tolerance x plus epsilon. 00:28:56.300 --> 00:28:59.960 align:middle line:84% So we just need to worry about what's to the left of it. 00:28:59.960 --> 00:29:01.460 align:middle line:84% And we need to find a capital number 00:29:01.460 --> 00:29:06.560 align:middle line:84% M so that all the x sub n's are in this interval 00:29:06.560 --> 00:29:23.735 align:middle line:84% here, because we're trying to show that the x sub 00:29:23.735 --> 00:29:26.680 align:middle line:84% n's converge to x, the supremum of this guy. 00:29:26.680 --> 00:29:31.030 align:middle line:84% So let's walk through this by this picture. 00:29:31.030 --> 00:29:33.370 align:middle line:90% So x is the supremum. 00:29:33.370 --> 00:29:37.000 align:middle line:84% Nothing's bigger than x, so all of the x sub n's 00:29:37.000 --> 00:29:39.140 align:middle line:84% are to the left of x plus epsilon. 00:29:39.140 --> 00:29:42.080 align:middle line:84% So now we just need to worry about x minus epsilon. 00:29:42.080 --> 00:29:46.120 align:middle line:84% Now, since x is the supremum of this set, x minus epsilon 00:29:46.120 --> 00:29:50.440 align:middle line:84% can't be an upper bound for the set of entries of x sub n. 00:29:50.440 --> 00:29:56.830 align:middle line:84% So that means there has to exist some x sub M so that it's 00:29:56.830 --> 00:30:00.130 align:middle line:90% bigger than x minus epsilon. 00:30:00.130 --> 00:30:02.770 align:middle line:84% But now, this is where we use the fact that this is 00:30:02.770 --> 00:30:06.100 align:middle line:90% a monotone increasing sequence. 00:30:06.100 --> 00:30:09.130 align:middle line:84% Because then, if n is bigger than or equal to capital M, 00:30:09.130 --> 00:30:13.360 align:middle line:84% then x sub n is strictly to the right of x sub M, 00:30:13.360 --> 00:30:20.060 align:middle line:84% or it's equal to x sub M and still less than or equal 00:30:20.060 --> 00:30:23.370 align:middle line:84% to x because x is the supremum of all the entries. 00:30:23.370 --> 00:30:25.820 align:middle line:84% So then, for n bigger than or equal to capital M, 00:30:25.820 --> 00:30:27.830 align:middle line:84% they all have to lie in this interval. 00:30:27.830 --> 00:30:30.080 align:middle line:84% And that's why the sequence converges 00:30:30.080 --> 00:30:32.210 align:middle line:90% to the supremum of this set. 00:30:32.210 --> 00:30:35.840 align:middle line:84% So we're using, in a crucial way, 00:30:35.840 --> 00:30:41.570 align:middle line:84% the fact that this sequence is monotonic increasing. 00:30:41.570 --> 00:30:42.750 align:middle line:90% I'll prove this claim. 00:30:42.750 --> 00:30:45.840 align:middle line:90% Let epsilon be positive. 00:30:45.840 --> 00:30:57.110 align:middle line:84% Then since x minus epsilon is not 00:30:57.110 --> 00:31:04.700 align:middle line:84% an upper bound for this set, x sub n, 00:31:04.700 --> 00:31:08.870 align:middle line:84% there exists an M0, natural number, 00:31:08.870 --> 00:31:13.970 align:middle line:84% so that x sub n minus epsilon is less than x 00:31:13.970 --> 00:31:17.825 align:middle line:84% sub M0 is less than or equal to x. 00:31:17.825 --> 00:31:20.380 align:middle line:90% 00:31:20.380 --> 00:31:24.640 align:middle line:84% And we'll choose M to be this M0. 00:31:24.640 --> 00:31:28.760 align:middle line:90% 00:31:28.760 --> 00:31:32.260 align:middle line:84% Then for all n bigger than or equal to M, 00:31:32.260 --> 00:31:38.290 align:middle line:84% we have x sub n minus epsilon, which is less than x sub M0. 00:31:38.290 --> 00:31:40.810 align:middle line:84% And because n is bigger than or equal to capital M, 00:31:40.810 --> 00:31:43.510 align:middle line:84% and this is a monotone increasing sequence, 00:31:43.510 --> 00:31:46.420 align:middle line:84% x sub M0 is less than or equal to x sub n. 00:31:46.420 --> 00:31:52.480 align:middle line:84% And because x is a supremum of the set 00:31:52.480 --> 00:31:54.220 align:middle line:84% of all entries of this sequence, this 00:31:54.220 --> 00:31:58.570 align:middle line:84% is less than or equal to x, which is less than x 00:31:58.570 --> 00:32:00.520 align:middle line:90% plus epsilon. 00:32:00.520 --> 00:32:02.110 align:middle line:84% Or to summarize-- so this should not 00:32:02.110 --> 00:32:04.720 align:middle line:84% be x sub n, that should be x minus epsilon. 00:32:04.720 --> 00:32:10.240 align:middle line:84% Or x minus epsilon is less than x sub n is less than x 00:32:10.240 --> 00:32:10.960 align:middle line:90% plus epsilon. 00:32:10.960 --> 00:32:13.720 align:middle line:84% And that's the same as showing the absolute value of x sub 00:32:13.720 --> 00:32:15.400 align:middle line:90% n minus x is less than epsilon. 00:32:15.400 --> 00:32:19.530 align:middle line:90% 00:32:19.530 --> 00:32:22.170 align:middle line:84% Now, what's the change for monotone decreasing 00:32:22.170 --> 00:32:24.960 align:middle line:84% is that we'll be able to identify 00:32:24.960 --> 00:32:28.140 align:middle line:84% the limit of the sequence as being 00:32:28.140 --> 00:32:32.990 align:middle line:84% the imf, the infimum of this set. 00:32:32.990 --> 00:32:34.930 align:middle line:84% But I'll leave that to you as an exercise. 00:32:34.930 --> 00:32:57.860 align:middle line:90% 00:32:57.860 --> 00:33:02.410 align:middle line:84% So let's use this real quick to prove the limit of a little bit 00:33:02.410 --> 00:33:07.030 align:middle line:84% more interesting sequence than 1 over this one, 00:33:07.030 --> 00:33:11.110 align:middle line:84% which is reasonably interesting, I guess. 00:33:11.110 --> 00:33:26.490 align:middle line:84% But typically in math, we just don't prove theorems 00:33:26.490 --> 00:33:28.710 align:middle line:84% for the sake of proving theorems. 00:33:28.710 --> 00:33:35.170 align:middle line:84% Typically, there's some concrete reason we do things. 00:33:35.170 --> 00:33:36.720 align:middle line:84% There's some concrete sequence we're 00:33:36.720 --> 00:33:40.890 align:middle line:84% trying to prove converges, or has a certain property, 00:33:40.890 --> 00:33:43.450 align:middle line:84% or doesn't have a certain property. 00:33:43.450 --> 00:33:53.200 align:middle line:84% And so the basic one is if you like a geometric sequence. 00:33:53.200 --> 00:34:04.290 align:middle line:84% So if c is a positive number between 0 and 1, 00:34:04.290 --> 00:34:11.639 align:middle line:84% then we'll prove that this sequence c to the n 00:34:11.639 --> 00:34:13.530 align:middle line:84% is convergent and it converges to 0. 00:34:13.530 --> 00:34:17.969 align:middle line:90% 00:34:17.969 --> 00:34:26.090 align:middle line:84% And we'll prove if c is bigger than 1, 00:34:26.090 --> 00:34:31.974 align:middle line:84% then the sequence c to the n is unbounded. 00:34:31.974 --> 00:34:33.349 align:middle line:84% In particular, it can't converge. 00:34:33.349 --> 00:34:40.650 align:middle line:90% 00:34:40.650 --> 00:34:44.280 align:middle line:90% So let's go in reverse order. 00:34:44.280 --> 00:34:47.409 align:middle line:84% Let's prove that if c is bigger than 1, 00:34:47.409 --> 00:34:49.239 align:middle line:90% then this thing is unbounded. 00:34:49.239 --> 00:34:51.690 align:middle line:84% It doesn't require 1, but it's shorter than 1, 00:34:51.690 --> 00:34:53.560 align:middle line:90% and we can go ahead and do it. 00:34:53.560 --> 00:34:56.100 align:middle line:84% And we'll use this fact that we proved, 00:34:56.100 --> 00:35:02.920 align:middle line:84% I think, in the first or second lecture using induction. 00:35:02.920 --> 00:35:06.660 align:middle line:84% So what does it mean to show something is unbounded? 00:35:06.660 --> 00:35:09.420 align:middle line:84% Again, we get to use our negation skills. 00:35:09.420 --> 00:35:11.835 align:middle line:84% It means that-- what do we want to show-- 00:35:11.835 --> 00:35:15.290 align:middle line:90% 00:35:15.290 --> 00:35:20.300 align:middle line:84% for all B bigger than or equal to 0, 00:35:20.300 --> 00:35:26.120 align:middle line:84% there exists an n, a natural number, so that c to the n 00:35:26.120 --> 00:35:43.980 align:middle line:84% is bigger than B. Now, how are we going to find this n? 00:35:43.980 --> 00:35:45.585 align:middle line:84% This seems like a complicated thing. 00:35:45.585 --> 00:35:49.410 align:middle line:90% 00:35:49.410 --> 00:35:52.790 align:middle line:84% So let's replace it by something simpler. 00:35:52.790 --> 00:35:58.130 align:middle line:84% Again, this is analysis, which means we get to use our wits, 00:35:58.130 --> 00:36:00.520 align:middle line:84% and try to replace complicated things by simpler things, 00:36:00.520 --> 00:36:01.895 align:middle line:84% and work with the simpler things. 00:36:01.895 --> 00:36:04.460 align:middle line:90% 00:36:04.460 --> 00:36:06.040 align:middle line:90% So let me do a little bit of-- 00:36:06.040 --> 00:36:08.860 align:middle line:84% again, this is off to the side, how 00:36:08.860 --> 00:36:11.650 align:middle line:90% would one think through this. 00:36:11.650 --> 00:36:13.360 align:middle line:84% If you look at c to the n, remember we 00:36:13.360 --> 00:36:16.570 align:middle line:84% have this inequality from infinite time 00:36:16.570 --> 00:36:24.760 align:middle line:84% ago that as long as x is bigger than or equal to minus 1, 00:36:24.760 --> 00:36:27.850 align:middle line:84% then I get 1 plus x to the n is bigger than 1 00:36:27.850 --> 00:36:32.850 align:middle line:84% plus n times x for all natural numbers n. 00:36:32.850 --> 00:36:38.450 align:middle line:90% 00:36:38.450 --> 00:36:47.050 align:middle line:84% So that means c to the n, which is equal to 1 plus c minus 1 00:36:47.050 --> 00:36:48.850 align:middle line:90% to the n-- so this is my x-- 00:36:48.850 --> 00:36:52.870 align:middle line:84% is bigger than or equal to 1 plus n times 00:36:52.870 --> 00:37:00.440 align:middle line:84% c minus 1, which is bigger than or equal to n times c minus 1. 00:37:00.440 --> 00:37:01.070 align:middle line:90% You see? 00:37:01.070 --> 00:37:08.530 align:middle line:84% So if I want to make this big, it suffices to make this big. 00:37:08.530 --> 00:37:10.780 align:middle line:90% So that's what I'll do. 00:37:10.780 --> 00:37:21.880 align:middle line:84% So let n be a natural number such that n is 00:37:21.880 --> 00:37:25.120 align:middle line:90% bigger than B over c minus 1. 00:37:25.120 --> 00:37:35.920 align:middle line:90% 00:37:35.920 --> 00:37:38.650 align:middle line:84% Then now this we did off to the side we'll just 00:37:38.650 --> 00:37:39.970 align:middle line:90% put in the proof now. 00:37:39.970 --> 00:37:42.510 align:middle line:90% 00:37:42.510 --> 00:37:47.850 align:middle line:84% And c to the n equals 1 plus 1 minus c to the n 00:37:47.850 --> 00:37:51.060 align:middle line:84% is bigger than or equal to 1 plus n times 00:37:51.060 --> 00:37:54.530 align:middle line:84% c minus 1, which is bigger than or equal to n times 00:37:54.530 --> 00:38:00.570 align:middle line:84% c minus 1, which is bigger than B over c minus 1 times c 00:38:00.570 --> 00:38:10.930 align:middle line:84% minus 1 equals B. So now let's prove the first claim, 00:38:10.930 --> 00:38:13.870 align:middle line:84% that if c is between 0 and 1, then the limit as n 00:38:13.870 --> 00:38:17.710 align:middle line:84% goes to infinity of c to the n equals 0. 00:38:17.710 --> 00:38:20.470 align:middle line:84% So first, I want to prove the following claim-- 00:38:20.470 --> 00:38:24.040 align:middle line:84% that we, in fact, have a decreasing sequence, which 00:38:24.040 --> 00:38:28.600 align:middle line:90% is bounded below by 0. 00:38:28.600 --> 00:38:39.210 align:middle line:84% So claim for all n, 0 is less than c to the n 00:38:39.210 --> 00:38:45.510 align:middle line:84% is less than, I should say, c to the n plus 1 is less than c 00:38:45.510 --> 00:38:46.020 align:middle line:90% to the n. 00:38:46.020 --> 00:38:53.720 align:middle line:90% 00:38:53.720 --> 00:38:57.990 align:middle line:84% So the proof of this claim is a very simple induction argument, 00:38:57.990 --> 00:38:59.910 align:middle line:90% so we'll do this by induction. 00:38:59.910 --> 00:39:06.020 align:middle line:84% So we have the base case n equals 1. 00:39:06.020 --> 00:39:17.410 align:middle line:84% So we are assuming that c is between 0 and 1. 00:39:17.410 --> 00:39:19.690 align:middle line:84% And if I multiply through by c, I 00:39:19.690 --> 00:39:24.210 align:middle line:84% conclude that 0 is less than c squared is less than c. 00:39:24.210 --> 00:39:26.790 align:middle line:90% And that's the n equals 1 case. 00:39:26.790 --> 00:39:32.785 align:middle line:84% And the inductive step is essentially the same proof. 00:39:32.785 --> 00:39:35.500 align:middle line:90% 00:39:35.500 --> 00:39:42.450 align:middle line:84% Suppose 0 is less than c to the m plus 1 is less than c 00:39:42.450 --> 00:39:43.090 align:middle line:90% to the m. 00:39:43.090 --> 00:39:46.170 align:middle line:90% 00:39:46.170 --> 00:39:51.280 align:middle line:84% Then multiplying through by c, we get that 0 is less than c 00:39:51.280 --> 00:39:55.020 align:middle line:84% to the m plus 2 is less than c to the m 00:39:55.020 --> 00:39:58.880 align:middle line:84% plus 1, which is n equals m plus 1. 00:39:58.880 --> 00:40:01.790 align:middle line:84% Here, we're using this fact here-- 00:40:01.790 --> 00:40:05.400 align:middle line:84% that c is positive so it doesn't flip the inequalities, 00:40:05.400 --> 00:40:10.010 align:middle line:84% so I can multiply it through and preserve the inequalities. 00:40:10.010 --> 00:40:20.740 align:middle line:84% So this shows that this sequence is monotone decreasing 00:40:20.740 --> 00:40:25.540 align:middle line:84% and it's bounded below by 0, and in fact bounded, 00:40:25.540 --> 00:40:40.530 align:middle line:84% because c to the n and absolute value, these are all positive, 00:40:40.530 --> 00:40:42.120 align:middle line:90% is equal to c to the n. 00:40:42.120 --> 00:40:46.800 align:middle line:84% And c to the n is less than c to the n minus 1, 00:40:46.800 --> 00:40:50.640 align:middle line:84% which is less than c to the n minus 2, so on, and so on, 00:40:50.640 --> 00:40:52.800 align:middle line:90% which is less than c. 00:40:52.800 --> 00:40:56.340 align:middle line:90% So it's bounded. 00:40:56.340 --> 00:41:02.340 align:middle line:84% I guess I could have built that into this inequality 00:41:02.340 --> 00:41:06.620 align:middle line:84% and proved that as well, but that's OK. 00:41:06.620 --> 00:41:08.750 align:middle line:90% So it has a limit. 00:41:08.750 --> 00:41:25.090 align:middle line:84% So by the previous theorem, has a limit, and I'll call it L. 00:41:25.090 --> 00:41:27.235 align:middle line:84% And now what I want to show is that L is 0. 00:41:27.235 --> 00:41:34.350 align:middle line:90% 00:41:34.350 --> 00:41:39.900 align:middle line:84% And how we'll do that is one of these analysis tricks, where-- 00:41:39.900 --> 00:41:42.750 align:middle line:84% not really tricks-- but rather than show with L 00:41:42.750 --> 00:41:44.970 align:middle line:84% is directly equal to 0, we'll show 00:41:44.970 --> 00:41:47.760 align:middle line:84% that the absolute value of L is less than epsilon 00:41:47.760 --> 00:41:49.500 align:middle line:84% for every epsilon positive, and therefore 00:41:49.500 --> 00:41:51.500 align:middle line:84% it has to be 0 because it's just a fixed number. 00:41:51.500 --> 00:41:58.010 align:middle line:90% 00:41:58.010 --> 00:41:59.170 align:middle line:90% So let epsilon be positive. 00:41:59.170 --> 00:42:01.753 align:middle line:84% Again, we're going to show that the absolute value of L, which 00:42:01.753 --> 00:42:04.790 align:middle line:84% is just a fixed number, is less than epsilon. 00:42:04.790 --> 00:42:07.300 align:middle line:84% And therefore, capital L has to be 0. 00:42:07.300 --> 00:42:12.430 align:middle line:90% 00:42:12.430 --> 00:42:18.490 align:middle line:84% Then there exists, since this sequence converges, an M, 00:42:18.490 --> 00:42:23.680 align:middle line:84% a natural number, such that for all n bigger 00:42:23.680 --> 00:42:27.790 align:middle line:84% than or equal to capital M, c to the n minus L 00:42:27.790 --> 00:42:35.870 align:middle line:84% is less than 1 minus c times epsilon over 2. 00:42:35.870 --> 00:42:42.480 align:middle line:84% Now, maybe you're wondering why didn't I just use epsilon here? 00:42:42.480 --> 00:42:45.560 align:middle line:84% Well, in the end, it's just going 00:42:45.560 --> 00:42:49.378 align:middle line:84% to come out to this being less than epsilon in absolute value. 00:42:49.378 --> 00:42:51.920 align:middle line:84% Otherwise, I would have come out with less than epsilon times 00:42:51.920 --> 00:42:54.110 align:middle line:84% some number, and I did away with that number 00:42:54.110 --> 00:42:57.210 align:middle line:84% by choosing a different number here. 00:42:57.210 --> 00:43:04.820 align:middle line:84% So now we compute that if I look at 1 minus c 00:43:04.820 --> 00:43:07.100 align:middle line:84% times the absolute value of L, this 00:43:07.100 --> 00:43:12.520 align:middle line:84% is equal to L minus c times L. And this 00:43:12.520 --> 00:43:21.500 align:middle line:84% is less than or equal to L minus c to the capital M plus-- 00:43:21.500 --> 00:43:24.890 align:middle line:90% 00:43:24.890 --> 00:43:29.990 align:middle line:84% let's put a plus 1, plus c to the M plus 1 minus c to the L. 00:43:29.990 --> 00:43:32.420 align:middle line:84% And by the triangle inequality, this 00:43:32.420 --> 00:43:37.880 align:middle line:84% is less than or equal to L minus c to the M plus 1 plus-- 00:43:37.880 --> 00:43:40.730 align:middle line:84% now c to the M plus 1 minus c to the L, 00:43:40.730 --> 00:43:47.860 align:middle line:84% so I can pull out a c which is positive, so I can get L. 00:43:47.860 --> 00:43:52.390 align:middle line:84% Now, M plus 1 is bigger than M. So it 00:43:52.390 --> 00:43:57.820 align:middle line:84% satisfies this inequality, and so does this one. 00:43:57.820 --> 00:44:01.570 align:middle line:84% And therefore, this is less than epsilon over 2 times 00:44:01.570 --> 00:44:08.200 align:middle line:84% 1 minus c plus c times epsilon over 2 times 1 minus c. 00:44:08.200 --> 00:44:11.140 align:middle line:84% And C is less than 1, so this whole thing 00:44:11.140 --> 00:44:12.970 align:middle line:84% is less than epsilon over 2 times 00:44:12.970 --> 00:44:15.610 align:middle line:84% 1 minus c plus another epsilon over 2. 00:44:15.610 --> 00:44:20.850 align:middle line:84% So I get epsilon over 2 times 1 minus c. 00:44:20.850 --> 00:44:26.660 align:middle line:90% 00:44:26.660 --> 00:44:34.960 align:middle line:84% And thus, the absolute value of L is less than epsilon. 00:44:34.960 --> 00:44:37.350 align:middle line:84% And since epsilon was arbitrary, that implies 00:44:37.350 --> 00:44:42.270 align:middle line:84% the absolute value of L is equal to 0, i.e., L is equal to 0. 00:44:42.270 --> 00:44:51.560 align:middle line:90% 00:44:51.560 --> 00:44:56.000 align:middle line:84% So that's a very concrete application 00:44:56.000 --> 00:45:00.980 align:middle line:84% of some of these theorems we've been proving, which is really-- 00:45:00.980 --> 00:45:03.050 align:middle line:84% I think the only real reason one proves 00:45:03.050 --> 00:45:10.213 align:middle line:84% theorems is you typically have a concrete example of something 00:45:10.213 --> 00:45:11.505 align:middle line:90% in mind that you want to study. 00:45:11.505 --> 00:45:14.300 align:middle line:90% 00:45:14.300 --> 00:45:16.360 align:middle line:84% But in order to study it, it often 00:45:16.360 --> 00:45:22.935 align:middle line:84% requires some general machinery, i.e., theorems. 00:45:22.935 --> 00:45:35.680 align:middle line:90% 00:45:35.680 --> 00:45:38.830 align:middle line:84% So now we'll talk a little bit about sequences 00:45:38.830 --> 00:45:42.550 align:middle line:90% obtained from other sequences. 00:45:42.550 --> 00:45:48.940 align:middle line:84% So these are called subsequences, 00:45:48.940 --> 00:45:56.117 align:middle line:84% or sequences obtained from a single sequence. 00:45:56.117 --> 00:45:57.200 align:middle line:90% So what do I mean by this? 00:45:57.200 --> 00:45:59.390 align:middle line:84% Let me give you the precise definition. 00:45:59.390 --> 00:46:07.200 align:middle line:90% 00:46:07.200 --> 00:46:16.380 align:middle line:84% So we started off with a sequence and an increasing 00:46:16.380 --> 00:46:23.250 align:middle line:90% sequence of integers, n sub k. 00:46:23.250 --> 00:46:31.910 align:middle line:90% 00:46:31.910 --> 00:46:35.820 align:middle line:84% So let me just say what this means rather than write out 00:46:35.820 --> 00:46:38.110 align:middle line:84% the word and then say what it means. 00:46:38.110 --> 00:46:51.930 align:middle line:84% So this is the sequence of natural numbers, which 00:46:51.930 --> 00:46:54.330 align:middle line:84% are strictly increasing. n sub 1 is less 00:46:54.330 --> 00:46:58.140 align:middle line:84% than n sub 2 is less than n sub 3, and so on. 00:46:58.140 --> 00:47:05.620 align:middle line:90% 00:47:05.620 --> 00:47:14.130 align:middle line:84% The new sequence, x sub n sub k-- so now the index is not n, 00:47:14.130 --> 00:47:15.350 align:middle line:90% but the index is k-- 00:47:15.350 --> 00:47:20.600 align:middle line:90% 00:47:20.600 --> 00:47:33.470 align:middle line:84% is called a subsequence of the original sequence x sub n. 00:47:33.470 --> 00:47:37.400 align:middle line:90% 00:47:37.400 --> 00:47:42.320 align:middle line:84% So how should you view subsequences of a sequence? 00:47:42.320 --> 00:47:47.780 align:middle line:84% You should think that I line up all the entries of x, 00:47:47.780 --> 00:47:51.830 align:middle line:84% of my original sequence x sub n, and then 00:47:51.830 --> 00:47:57.770 align:middle line:84% I just start picking entries out of the sequence. 00:47:57.770 --> 00:48:01.250 align:middle line:84% But every time I make a choice, I have to move to the right 00:48:01.250 --> 00:48:03.410 align:middle line:90% and make another choice. 00:48:03.410 --> 00:48:06.980 align:middle line:84% That's what I was just saying there, expresses this condition 00:48:06.980 --> 00:48:10.740 align:middle line:84% that this sequence of natural numbers is increasing. 00:48:10.740 --> 00:48:13.430 align:middle line:84% So I pick an entry in the sequence. 00:48:13.430 --> 00:48:16.700 align:middle line:84% That's going to be my first guy, the first element 00:48:16.700 --> 00:48:18.200 align:middle line:90% of my new subsequence. 00:48:18.200 --> 00:48:22.640 align:middle line:84% And then I move to the right, and maybe I pick the next one, 00:48:22.640 --> 00:48:24.320 align:middle line:90% maybe I pick one three down. 00:48:24.320 --> 00:48:25.500 align:middle line:90% I pick that one. 00:48:25.500 --> 00:48:30.400 align:middle line:84% And then I move to the right of that one and pick a new one. 00:48:30.400 --> 00:48:33.310 align:middle line:84% And then I move to the right of that one and pick another one. 00:48:33.310 --> 00:48:36.280 align:middle line:84% So if you want to generate a subsequence 00:48:36.280 --> 00:48:40.480 align:middle line:84% from an original sequence, how do you think about this? 00:48:40.480 --> 00:48:43.065 align:middle line:84% Again, line them all up, start picking entries, 00:48:43.065 --> 00:48:44.440 align:middle line:84% but every time you pick an entry, 00:48:44.440 --> 00:48:47.875 align:middle line:84% you have to move to the right in order to pick your next entry. 00:48:47.875 --> 00:48:52.600 align:middle line:90% 00:48:52.600 --> 00:48:56.180 align:middle line:84% So let me give you some examples and non-examples. 00:48:56.180 --> 00:49:06.730 align:middle line:84% So for example, if our original sequence is 1, 2, 3, 4, 5, 6, 00:49:06.730 --> 00:49:12.880 align:middle line:84% and so on, what would be examples of some subsequences? 00:49:12.880 --> 00:49:19.160 align:middle line:84% The odd numbers 1, 3, 5, 7, 9 and so on. 00:49:19.160 --> 00:49:21.560 align:middle line:84% You see how I'm taking the original sequence, 00:49:21.560 --> 00:49:25.563 align:middle line:90% so this is x sub n. 00:49:25.563 --> 00:49:27.730 align:middle line:84% I'm just picking entries from the original sequence, 00:49:27.730 --> 00:49:29.938 align:middle line:84% but every time I pick an entry, I move to the right 00:49:29.938 --> 00:49:30.730 align:middle line:90% and pick a new one. 00:49:30.730 --> 00:49:33.070 align:middle line:84% So I picked 1, now I get to choose anything 00:49:33.070 --> 00:49:34.310 align:middle line:90% to the right of that. 00:49:34.310 --> 00:49:38.060 align:middle line:84% I pick 3, now I can choose anything to the right of that. 00:49:38.060 --> 00:49:40.480 align:middle line:84% So in the language of this definition, 00:49:40.480 --> 00:49:44.380 align:middle line:84% the increasing sequence of integers 00:49:44.380 --> 00:49:48.250 align:middle line:84% is just the odd integers, so 2k minus 1. 00:49:48.250 --> 00:49:52.090 align:middle line:84% So here, x sub n is equal to just n. 00:49:52.090 --> 00:49:57.420 align:middle line:90% 00:49:57.420 --> 00:50:00.810 align:middle line:84% I could pick another subsequence would be 00:50:00.810 --> 00:50:02.880 align:middle line:90% the sequence of even numbers-- 00:50:02.880 --> 00:50:07.620 align:middle line:90% 2, 4, 6, 8, 10. 00:50:07.620 --> 00:50:10.050 align:middle line:84% This is a subsequence of this original sequence. 00:50:10.050 --> 00:50:13.590 align:middle line:84% Here, the increasing sequence of integers is 2k. 00:50:13.590 --> 00:50:20.160 align:middle line:90% 00:50:20.160 --> 00:50:23.250 align:middle line:84% I could pick the subsequence could be 00:50:23.250 --> 00:50:26.040 align:middle line:90% the sequence of prime numbers-- 00:50:26.040 --> 00:50:30.540 align:middle line:90% 2, 3, 5, 7, 11, 13. 00:50:30.540 --> 00:50:34.000 align:middle line:90% 00:50:34.000 --> 00:50:38.510 align:middle line:84% And the increasing-- so I don't have 00:50:38.510 --> 00:50:40.730 align:middle line:90% a general formula for that. 00:50:40.730 --> 00:50:43.220 align:middle line:84% Good luck finding one, but we'll just 00:50:43.220 --> 00:50:49.125 align:middle line:90% write this as k-th prime number. 00:50:49.125 --> 00:50:57.790 align:middle line:90% 00:50:57.790 --> 00:51:00.410 align:middle line:84% Now, what would not be an example? 00:51:00.410 --> 00:51:02.890 align:middle line:84% So these are examples of subsequences. 00:51:02.890 --> 00:51:05.650 align:middle line:90% 00:51:05.650 --> 00:51:07.780 align:middle line:84% Not examples of subsequences would 00:51:07.780 --> 00:51:12.280 align:middle line:84% be, for example, the sequence 1, 1, 1, 1, 1. 00:51:12.280 --> 00:51:16.120 align:middle line:84% This is not a subsequence of this original sequence. 00:51:16.120 --> 00:51:18.530 align:middle line:84% Remember, I have a sequence of natural numbers 00:51:18.530 --> 00:51:20.020 align:middle line:90% which is increasing. 00:51:20.020 --> 00:51:27.070 align:middle line:84% And x sub n sub k means that this new sequence is 00:51:27.070 --> 00:51:30.160 align:middle line:84% obtained from the old one by picking entries-- 00:51:30.160 --> 00:51:32.227 align:middle line:84% by picking an entry, moving to the right, 00:51:32.227 --> 00:51:34.060 align:middle line:84% picking the next entry, moving to the right, 00:51:34.060 --> 00:51:35.710 align:middle line:90% and picking the next entry. 00:51:35.710 --> 00:51:39.055 align:middle line:84% Here, in order to get this sequence from this one, 00:51:39.055 --> 00:51:42.640 align:middle line:84% I just stay on the first entry and keep picking it. 00:51:42.640 --> 00:51:46.585 align:middle line:84% But that's not how a subsequence is defined. 00:51:46.585 --> 00:51:49.430 align:middle line:90% 00:51:49.430 --> 00:51:56.500 align:middle line:84% So here, if you like, this would mean n sub k is equal to 1. 00:51:56.500 --> 00:51:58.663 align:middle line:84% And this is not a strictly increasing sequence 00:51:58.663 --> 00:51:59.455 align:middle line:90% of natural numbers. 00:51:59.455 --> 00:52:03.040 align:middle line:90% 00:52:03.040 --> 00:52:09.830 align:middle line:84% Or 1, 1 3, 3, 5, 5, and so on-- that's also not. 00:52:09.830 --> 00:52:20.050 align:middle line:84% So these are not subsequences of the original sequence 00:52:20.050 --> 00:52:20.800 align:middle line:90% given here. 00:52:20.800 --> 00:52:27.527 align:middle line:90% 00:52:27.527 --> 00:52:28.485 align:middle line:90% So I hope that's clear. 00:52:28.485 --> 00:52:34.000 align:middle line:90% 00:52:34.000 --> 00:52:40.040 align:middle line:90% Now, I want to emphasize this-- 00:52:40.040 --> 00:52:46.430 align:middle line:84% I'm not saying that the number you pick each time 00:52:46.430 --> 00:52:49.910 align:middle line:84% has to be different from the number you picked before, 00:52:49.910 --> 00:52:52.220 align:middle line:90% meaning the value. 00:52:52.220 --> 00:52:56.750 align:middle line:84% You couldn't have 1, 1, 1, 1 as a subsequence of this guy 00:52:56.750 --> 00:52:59.240 align:middle line:84% because 1 is only in the first entry. 00:52:59.240 --> 00:53:01.400 align:middle line:84% You can't keep picking one entry. 00:53:01.400 --> 00:53:03.110 align:middle line:84% You have to pick an entry and move 00:53:03.110 --> 00:53:06.080 align:middle line:84% to the next entry or the one after that 00:53:06.080 --> 00:53:09.800 align:middle line:84% and pick that entry to obtain your sequence. 00:53:09.800 --> 00:53:11.570 align:middle line:84% But that doesn't necessarily mean 00:53:11.570 --> 00:53:16.130 align:middle line:84% that those actual numbers in those entries 00:53:16.130 --> 00:53:17.250 align:middle line:90% have to be different. 00:53:17.250 --> 00:53:24.110 align:middle line:84% So for example, if I look at the sequence minus 1, 00:53:24.110 --> 00:53:31.270 align:middle line:84% 1, 1, minus 1, 1, one meaning x sub n is equal to minus 1 00:53:31.270 --> 00:53:36.850 align:middle line:84% to the n-th then a perfectly good subsequence is given 00:53:36.850 --> 00:53:42.070 align:middle line:84% by minus 1, minus 1, minus 1, minus 1, 00:53:42.070 --> 00:53:45.130 align:middle line:84% where it looks like I'm just picking one of the entries, 00:53:45.130 --> 00:53:46.120 align:middle line:90% but really I'm not. 00:53:46.120 --> 00:53:49.030 align:middle line:84% Here, the increasing sequence-- so 00:53:49.030 --> 00:53:51.460 align:middle line:84% what I'm doing here to get this new sequence, 00:53:51.460 --> 00:53:54.160 align:middle line:84% I picked the first entry, then I picked the third, 00:53:54.160 --> 00:53:56.410 align:middle line:84% then I picked the fifth, then I picked the seventh. 00:53:56.410 --> 00:54:01.510 align:middle line:90% So n sub k equals 2k minus 1. 00:54:01.510 --> 00:54:04.600 align:middle line:84% So all the actual numbers appearing in the sequence 00:54:04.600 --> 00:54:09.970 align:middle line:84% are the same, but I'm actually picking different entries 00:54:09.970 --> 00:54:14.220 align:middle line:84% from the original sequence, where they appear. 00:54:14.220 --> 00:54:15.630 align:middle line:90% So this is also fine. 00:54:15.630 --> 00:54:18.425 align:middle line:90% So is 1, 1, 1, 1, 1. 00:54:18.425 --> 00:54:22.080 align:middle line:90% 00:54:22.080 --> 00:54:24.430 align:middle line:90% This is n k equals 2 times k. 00:54:24.430 --> 00:54:27.430 align:middle line:84% So these are all good subsequences. 00:54:27.430 --> 00:54:34.190 align:middle line:84% So these were subsequences of the original sequence. 00:54:34.190 --> 00:54:37.070 align:middle line:84% These are subsequences of this sequence. 00:54:37.070 --> 00:55:09.120 align:middle line:90% 00:55:09.120 --> 00:55:14.700 align:middle line:84% So from a given sequence, we can obtain new sequences 00:55:14.700 --> 00:55:19.610 align:middle line:84% from this original sequence in this way. 00:55:19.610 --> 00:55:22.060 align:middle line:84% So a natural question is to ask, how 00:55:22.060 --> 00:55:33.350 align:middle line:84% does going from a sequence to a subsequence 00:55:33.350 --> 00:55:35.570 align:middle line:90% behave with respect to limits? 00:55:35.570 --> 00:55:38.330 align:middle line:84% If the original sequence converges to a limit, 00:55:38.330 --> 00:55:41.060 align:middle line:84% does a subsequence or does every subsequence 00:55:41.060 --> 00:55:43.440 align:middle line:90% converge to that limit as well? 00:55:43.440 --> 00:55:45.920 align:middle line:84% And this shouldn't come as a surprise that it does. 00:55:45.920 --> 00:55:50.648 align:middle line:84% Because all of the elements of the sequence are getting close, 00:55:50.648 --> 00:55:52.440 align:middle line:84% all the entries of the sequence are getting 00:55:52.440 --> 00:55:53.940 align:middle line:90% close to a given number. 00:55:53.940 --> 00:55:57.510 align:middle line:84% And a subsequence is just, if you like, picking certain ones 00:55:57.510 --> 00:55:58.630 align:middle line:90% along the way. 00:55:58.630 --> 00:56:01.620 align:middle line:84% So certainly if I'm in a subsequence, 00:56:01.620 --> 00:56:04.470 align:middle line:84% then as long as I'm far enough out in the subsequence, 00:56:04.470 --> 00:56:06.540 align:middle line:84% I'll be close to that limit as well. 00:56:06.540 --> 00:56:09.540 align:middle line:84% So if you didn't get all that rambling, 00:56:09.540 --> 00:56:13.410 align:middle line:84% I'll go ahead and state the theorem and prove it. 00:56:13.410 --> 00:56:29.270 align:middle line:90% 00:56:29.270 --> 00:56:33.575 align:middle line:84% x n is a sequence which converges to x. 00:56:33.575 --> 00:56:43.150 align:middle line:90% 00:56:43.150 --> 00:56:58.520 align:middle line:84% And x sub n sub k is a subsequence of x sub n. 00:56:58.520 --> 00:57:02.850 align:middle line:84% Then the subsequence converges to x, as well. 00:57:02.850 --> 00:57:07.200 align:middle line:90% 00:57:07.200 --> 00:57:12.510 align:middle line:84% Limit as k goes to infinity of x sub n sub k equals x. 00:57:12.510 --> 00:57:34.250 align:middle line:90% 00:57:34.250 --> 00:57:38.690 align:middle line:84% So let me just start off by making a simple observation 00:57:38.690 --> 00:57:41.990 align:middle line:84% based on the fact that this increasing sequence-- 00:57:41.990 --> 00:57:43.718 align:middle line:84% that the n sub k's are an increasing 00:57:43.718 --> 00:57:44.885 align:middle line:90% sequence of natural numbers. 00:57:44.885 --> 00:57:47.420 align:middle line:90% 00:57:47.420 --> 00:57:55.440 align:middle line:84% Since n sub 1 is bigger than or equal to 1 is less than n 00:57:55.440 --> 00:58:00.000 align:middle line:84% sub 2 is less than n sub 3 and so on, 00:58:00.000 --> 00:58:05.370 align:middle line:84% this implies that for all k, a natural number, 00:58:05.370 --> 00:58:08.190 align:middle line:84% n sub k is bigger than or equal to k. 00:58:08.190 --> 00:58:13.920 align:middle line:90% 00:58:13.920 --> 00:58:16.590 align:middle line:84% So this is not so hard to believe because n sub 1 has 00:58:16.590 --> 00:58:18.150 align:middle line:90% to be at least bigger than 1. 00:58:18.150 --> 00:58:20.880 align:middle line:84% And n sub 2 has to be at least bigger than n sub 1, 00:58:20.880 --> 00:58:24.270 align:middle line:84% so it's either n sub 1 plus 1 or bigger. 00:58:24.270 --> 00:58:28.057 align:middle line:84% And therefore, by a simple induction argument, 00:58:28.057 --> 00:58:29.640 align:middle line:84% which I'll leave to you, you can prove 00:58:29.640 --> 00:58:31.125 align:middle line:84% that for all k, a natural number, 00:58:31.125 --> 00:58:35.705 align:middle line:84% n sub k is bigger than or equal to k, induct on k. 00:58:35.705 --> 00:58:41.340 align:middle line:90% 00:58:41.340 --> 00:58:44.940 align:middle line:84% So we want to prove that this sequence converges to x. 00:58:44.940 --> 00:58:47.490 align:middle line:84% And we're going to do that using the only means we have. 00:58:47.490 --> 00:58:51.780 align:middle line:84% So this is just some arbitrary subsequence of x sub n. 00:58:51.780 --> 00:58:53.475 align:middle line:84% All we have is the definition to use. 00:58:53.475 --> 00:58:56.220 align:middle line:90% 00:58:56.220 --> 00:58:57.650 align:middle line:84% So we're going to now prove this. 00:58:57.650 --> 00:58:59.330 align:middle line:84% So we have to show for all epsilon 00:58:59.330 --> 00:59:02.450 align:middle line:84% positive, blah, blah, blah, so the first thing we have to do 00:59:02.450 --> 00:59:04.590 align:middle line:90% is let epsilon be positive. 00:59:04.590 --> 00:59:07.130 align:middle line:84% Now we'll use the fact that x sub n converges to x. 00:59:07.130 --> 00:59:10.655 align:middle line:90% 00:59:10.655 --> 00:59:14.510 align:middle line:90% So x sub n converges to x. 00:59:14.510 --> 00:59:19.790 align:middle line:84% There exists a natural number M0 such that 00:59:19.790 --> 00:59:30.360 align:middle line:84% for all n bigger than or equal to M0, x 00:59:30.360 --> 00:59:35.720 align:middle line:84% sub n minus x absolute value is less than epsilon. 00:59:35.720 --> 00:59:38.411 align:middle line:84% This M sub 0 we'll choose for our subsequence. 00:59:38.411 --> 00:59:41.060 align:middle line:90% 00:59:41.060 --> 00:59:45.700 align:middle line:84% Choose M to equal M sub 0, so now we 00:59:45.700 --> 00:59:48.250 align:middle line:84% need to show that this capital M works for our subsequence, 00:59:48.250 --> 00:59:51.070 align:middle line:84% meaning if k is bigger than or equal to capital M, 00:59:51.070 --> 00:59:56.350 align:middle line:84% then x sub n sub k minus x is an absolute value less 00:59:56.350 --> 00:59:57.170 align:middle line:90% than epsilon. 00:59:57.170 --> 01:00:01.840 align:middle line:84% But this just follows from this fact. 01:00:01.840 --> 01:00:06.790 align:middle line:84% And if k is bigger than or equal to M, this implies, 01:00:06.790 --> 01:00:11.860 align:middle line:84% by this inequality, n sub k is bigger than 01:00:11.860 --> 01:00:15.190 align:middle line:84% or equal to M, which remember, is equal to M sub 0. 01:00:15.190 --> 01:00:18.370 align:middle line:84% So n sub k is some integer, so natural number 01:00:18.370 --> 01:00:21.490 align:middle line:90% bigger than or equal to M sub 0. 01:00:21.490 --> 01:00:23.710 align:middle line:84% So that implies by this inequality 01:00:23.710 --> 01:00:28.090 align:middle line:84% that x sub n sub k minus x in absolute value 01:00:28.090 --> 01:00:30.105 align:middle line:90% is less than epsilon. 01:00:30.105 --> 01:00:38.451 align:middle line:90%