WEBVTT 00:00:00.000 --> 00:00:02.485 align:middle line:90% [SQUEAKING] 00:00:02.485 --> 00:00:04.473 align:middle line:90% [RUSTLING] 00:00:04.473 --> 00:00:07.952 align:middle line:90% [CLICKING] 00:00:07.952 --> 00:00:15.852 align:middle line:90% 00:00:15.852 --> 00:00:18.060 align:middle line:84% CASEY RODRIGUES: So I'm going to prove a few theorems 00:00:18.060 --> 00:00:22.200 align:middle line:84% about limits, which will allow us to compute limits, 00:00:22.200 --> 00:00:29.920 align:middle line:84% or at least we can use to prove that other non-trivial limits 00:00:29.920 --> 00:00:32.950 align:middle line:84% exist using these theorems, rather than using 00:00:32.950 --> 00:00:34.850 align:middle line:90% the definition directly. 00:00:34.850 --> 00:00:43.720 align:middle line:84% So this first theorem is the easiest theorem in the world 00:00:43.720 --> 00:00:47.440 align:middle line:84% because it's simply just restating the definition 00:00:47.440 --> 00:00:49.670 align:middle line:90% of convergence of a sequence. 00:00:49.670 --> 00:00:57.640 align:middle line:84% So I'm going to state it as follows, so pretty short, 00:00:57.640 --> 00:01:02.710 align:middle line:84% that if I have a sequence x sub n, 00:01:02.710 --> 00:01:12.790 align:middle line:84% then it converges to x if and only if the sequence obtained 00:01:12.790 --> 00:01:19.320 align:middle line:84% by taking the absolute value of x of n minus x equals 0, 00:01:19.320 --> 00:01:22.020 align:middle line:84% or the limit of that sequence is 0. 00:01:22.020 --> 00:01:24.630 align:middle line:90% So what is the proof? 00:01:24.630 --> 00:01:27.730 align:middle line:84% It follows just immediately from the definition. 00:01:27.730 --> 00:01:29.850 align:middle line:84% So I'm not even going to write anything. 00:01:29.850 --> 00:01:30.870 align:middle line:90% I'll leave it to you. 00:01:30.870 --> 00:01:34.590 align:middle line:90% 00:01:34.590 --> 00:01:40.650 align:middle line:84% But the proof follows from the definition and the simple fact 00:01:40.650 --> 00:01:45.900 align:middle line:84% that x sub n minus x in absolute value 00:01:45.900 --> 00:01:51.710 align:middle line:84% is equal to the absolute value of the absolute value of x sub 00:01:51.710 --> 00:01:54.940 align:middle line:90% n minus x minus 0, OK. 00:01:54.940 --> 00:01:59.590 align:middle line:84% So the definition says for all epsilon positive, 00:01:59.590 --> 00:02:02.380 align:middle line:84% you have to find an M so that this is less than epsilon 00:02:02.380 --> 00:02:04.570 align:middle line:84% for all m bigger than or equal to capital M. 00:02:04.570 --> 00:02:08.080 align:middle line:84% But if you found such a capital M for this to be less 00:02:08.080 --> 00:02:09.729 align:middle line:84% than epsilon, then this will be less 00:02:09.729 --> 00:02:13.690 align:middle line:84% than epsilon, which is saying that this limit equals 0. 00:02:13.690 --> 00:02:17.030 align:middle line:84% And then going the other direction, it's the same thing. 00:02:17.030 --> 00:02:19.060 align:middle line:84% So this is just following directly 00:02:19.060 --> 00:02:24.460 align:middle line:84% from the definition and this simple fact. 00:02:24.460 --> 00:02:28.150 align:middle line:84% OK, that was a very silly fact about limits, 00:02:28.150 --> 00:02:31.510 align:middle line:84% but a very useful one in conjunction 00:02:31.510 --> 00:02:37.742 align:middle line:84% with the next theorem, which is not so trivial, 00:02:37.742 --> 00:02:38.950 align:middle line:90% which is the squeeze theorem. 00:02:38.950 --> 00:02:45.750 align:middle line:90% 00:02:45.750 --> 00:02:47.460 align:middle line:90% And it says the following. 00:02:47.460 --> 00:03:03.630 align:middle line:84% So let an, bn, and xn be sequences such 00:03:03.630 --> 00:03:08.650 align:middle line:84% that the following holds for all n 00:03:08.650 --> 00:03:12.370 align:middle line:84% natural numbers, a sub n is less than 00:03:12.370 --> 00:03:17.140 align:middle line:84% or equal x sub n is less than or equal to b sub n. 00:03:17.140 --> 00:03:29.790 align:middle line:84% And a sub n and b sub n are convergent sequences. 00:03:29.790 --> 00:03:32.660 align:middle line:90% 00:03:32.660 --> 00:03:39.590 align:middle line:84% And their limits equal each other. 00:03:39.590 --> 00:03:41.720 align:middle line:84% And they're given by some number, call it x. 00:03:41.720 --> 00:03:44.820 align:middle line:90% 00:03:44.820 --> 00:03:55.700 align:middle line:84% Then the conclusion is that the limit as n 00:03:55.700 --> 00:03:57.800 align:middle line:84% goes to infinity of x sub n equals x. 00:03:57.800 --> 00:04:01.100 align:middle line:84% So when I write something like this, you should also kind of-- 00:04:01.100 --> 00:04:05.090 align:middle line:84% there's half a sentence before that that 00:04:05.090 --> 00:04:08.810 align:middle line:84% goes with this saying, x of n is a conversion sequence. 00:04:08.810 --> 00:04:10.880 align:middle line:90% And its limit is equal to x. 00:04:10.880 --> 00:04:14.180 align:middle line:84% So it's two statements in one when I write that. 00:04:14.180 --> 00:04:18.410 align:middle line:90% 00:04:18.410 --> 00:04:25.610 align:middle line:84% So if you just draw a picture, the squeezed theorem 00:04:25.610 --> 00:04:27.110 align:middle line:90% shouldn't be too surprising. 00:04:27.110 --> 00:04:33.870 align:middle line:90% 00:04:33.870 --> 00:04:38.010 align:middle line:90% So this is a little discussion. 00:04:38.010 --> 00:04:44.370 align:middle line:84% So here's x, the common limit of a sub n and b sub n. 00:04:44.370 --> 00:04:49.770 align:middle line:84% And so we can imagine that we're trying to show the limit as n 00:04:49.770 --> 00:04:51.720 align:middle line:84% goes to infinity of x sub n equals x. 00:04:51.720 --> 00:04:55.770 align:middle line:84% So that means we have to find for every epsilon a capital M 00:04:55.770 --> 00:04:59.940 align:middle line:84% so that x sub n is between x plus epsilon 00:04:59.940 --> 00:05:02.880 align:middle line:90% and x minus epsilon. 00:05:02.880 --> 00:05:07.370 align:middle line:84% So if I go out a little bit, I would 00:05:07.370 --> 00:05:08.880 align:middle line:84% hope I can find a natural number, 00:05:08.880 --> 00:05:15.350 align:middle line:84% a capital M so that x sub n minus x-- 00:05:15.350 --> 00:05:18.230 align:middle line:84% or x sub n is in this little interval. 00:05:18.230 --> 00:05:22.595 align:middle line:84% Now, if I'm assuming that a sub n and x sub n, if a sub n 00:05:22.595 --> 00:05:26.120 align:middle line:84% and b sub n are squeezing x sub n, in other words x 00:05:26.120 --> 00:05:32.450 align:middle line:84% sub n is between the two, and b sub n is converging to x, then 00:05:32.450 --> 00:05:38.450 align:middle line:84% for n bigger than or equal to some integer M0, 00:05:38.450 --> 00:05:41.603 align:middle line:84% all of the b sub n's are in this interval. 00:05:41.603 --> 00:05:42.770 align:middle line:90% OK, maybe they're not there. 00:05:42.770 --> 00:05:45.200 align:middle line:90% Maybe they could be over here. 00:05:45.200 --> 00:05:47.660 align:middle line:84% But the way I drew it is just to the right of x. 00:05:47.660 --> 00:05:52.850 align:middle line:84% And, likewise, since a sub n is converging to x, 00:05:52.850 --> 00:05:57.470 align:middle line:84% there exists some other integer M sub 1 00:05:57.470 --> 00:06:02.240 align:middle line:84% so that if I look at a sub n, it's also in this interval. 00:06:02.240 --> 00:06:03.230 align:middle line:90% Maybe it's over here. 00:06:03.230 --> 00:06:06.890 align:middle line:84% Maybe it's-- well, it can't be to the right of b sub n 00:06:06.890 --> 00:06:11.480 align:middle line:84% because that inequality up there strictly implies that a sub n 00:06:11.480 --> 00:06:14.480 align:middle line:84% is less than or equal to b sub n. 00:06:14.480 --> 00:06:18.100 align:middle line:90% But it's in this interval. 00:06:18.100 --> 00:06:21.580 align:middle line:84% So then what would that say if I look 00:06:21.580 --> 00:06:29.550 align:middle line:84% at n bigger than or equal to n plus M1, M0 plus M1, then n 00:06:29.550 --> 00:06:32.940 align:middle line:90% is bigger than M0, this guy. 00:06:32.940 --> 00:06:36.060 align:middle line:84% And n is bigger than both and M1, this guy. 00:06:36.060 --> 00:06:37.560 align:middle line:84% And therefore if I look at x sub n, 00:06:37.560 --> 00:06:40.212 align:middle line:84% it's going to be between these two. 00:06:40.212 --> 00:06:42.420 align:middle line:84% And in particular, it's going to be in this interval. 00:06:42.420 --> 00:06:45.030 align:middle line:90% 00:06:45.030 --> 00:06:47.010 align:middle line:84% So that's the proof in a picture. 00:06:47.010 --> 00:06:51.810 align:middle line:84% And now our goal is just to write it down. 00:06:51.810 --> 00:06:54.070 align:middle line:84% So that's the picture of the proof. 00:06:54.070 --> 00:06:57.210 align:middle line:84% But if you're actually trying to guess why this would be true? 00:06:57.210 --> 00:06:58.840 align:middle line:84% I mean, the b sub n's, you can imagine, 00:06:58.840 --> 00:07:01.470 align:middle line:90% are getting very close to x. 00:07:01.470 --> 00:07:04.365 align:middle line:84% The a sub n's are also getting very close to x. 00:07:04.365 --> 00:07:06.240 align:middle line:84% x sub n is in between the two, so it's 00:07:06.240 --> 00:07:09.540 align:middle line:84% getting squeezed to x, and thus the name. 00:07:09.540 --> 00:07:24.730 align:middle line:90% 00:07:24.730 --> 00:07:30.340 align:middle line:84% OK, so now we just need to turn this into written word. 00:07:30.340 --> 00:07:31.450 align:middle line:90% So we need to show-- 00:07:31.450 --> 00:07:35.440 align:middle line:84% we're going to show that x sub n converges to x. 00:07:35.440 --> 00:07:39.520 align:middle line:84% So we're-- all we have is an epsilon delta proof. 00:07:39.520 --> 00:07:41.410 align:middle line:90% Or we could use that theorem. 00:07:41.410 --> 00:07:44.740 align:middle line:84% But let's go with an epsilon delta proof. 00:07:44.740 --> 00:07:49.750 align:middle line:84% I mean, not epsilon delta, epsilon M. Epsilon delta proofs 00:07:49.750 --> 00:07:52.090 align:middle line:90% will come later. 00:07:52.090 --> 00:07:55.130 align:middle line:90% So let epsilon be positive. 00:07:55.130 --> 00:08:01.210 align:middle line:84% And since b sub n converges to x, there 00:08:01.210 --> 00:08:07.060 align:middle line:84% exist to M0 in natural numbers such that for all n bigger 00:08:07.060 --> 00:08:18.040 align:middle line:84% than or equal to n sub 0, b sub n minus x is less than epsilon 00:08:18.040 --> 00:08:29.748 align:middle line:84% in absolute value, which is the same as saying-- well, 00:08:29.748 --> 00:08:30.790 align:middle line:90% I mean it's not the same. 00:08:30.790 --> 00:08:35.320 align:middle line:84% But it implies that b sub n is less than x plus epsilon. 00:08:35.320 --> 00:08:41.620 align:middle line:90% 00:08:41.620 --> 00:08:45.220 align:middle line:84% Since a sub n's converge to x as well, 00:08:45.220 --> 00:08:49.720 align:middle line:84% there exists a M sub 1, natural number, 00:08:49.720 --> 00:08:57.310 align:middle line:84% such that for all n bigger than or equal M1, 00:08:57.310 --> 00:08:59.380 align:middle line:84% a sub n minus x in absolute value 00:08:59.380 --> 00:09:02.830 align:middle line:84% is less than epsilon, which implies 00:09:02.830 --> 00:09:06.520 align:middle line:84% a sub n is between x minus epsilon and x plus epsilon, 00:09:06.520 --> 00:09:09.460 align:middle line:84% but I'm only going to use one of those inequalities. 00:09:09.460 --> 00:09:19.760 align:middle line:90% 00:09:19.760 --> 00:09:22.640 align:middle line:84% Now I'm going to choose the capital M for my sequence 00:09:22.640 --> 00:09:25.640 align:middle line:84% x sub n, I'm trying to show convergence to x. 00:09:25.640 --> 00:09:30.590 align:middle line:84% So choose m to be m sub-zero plus M sub 1. 00:09:30.590 --> 00:09:33.770 align:middle line:84% I mean, you could have chosen it to be the maximum of the two. 00:09:33.770 --> 00:09:35.525 align:middle line:84% But this works just fine as well. 00:09:35.525 --> 00:09:38.390 align:middle line:90% 00:09:38.390 --> 00:09:45.110 align:middle line:84% Then if n is bigger than or equal to m, 00:09:45.110 --> 00:09:49.690 align:middle line:84% this implies n is bigger than or equal to M0 00:09:49.690 --> 00:09:53.180 align:middle line:84% and M is bigger than or equal to M1, which 00:09:53.180 --> 00:09:55.880 align:middle line:84% implies that both of these inequalities 00:09:55.880 --> 00:09:59.950 align:middle line:90% here are valid for this n. 00:09:59.950 --> 00:10:04.480 align:middle line:84% So then x minus epsilon is less than a sub n, which 00:10:04.480 --> 00:10:13.970 align:middle line:84% by assumption, is less than or equal to x sub n, which 00:10:13.970 --> 00:10:18.110 align:middle line:84% is less than or equal to b sub n, which is less than x 00:10:18.110 --> 00:10:20.160 align:middle line:90% plus epsilon. 00:10:20.160 --> 00:10:21.590 align:middle line:84% Now, these string of inequalities, 00:10:21.590 --> 00:10:23.660 align:middle line:84% therefore, tell us that x minus epsilon 00:10:23.660 --> 00:10:27.650 align:middle line:84% is less than x sub n is less than x 00:10:27.650 --> 00:10:34.990 align:middle line:84% plus epsilon, which is equivalent to saying 00:10:34.990 --> 00:10:40.520 align:middle line:84% the absolute value of x, so then minus x is less than epsilon. 00:10:40.520 --> 00:10:45.115 align:middle line:84% And therefore x sub n converges to x. 00:10:45.115 --> 00:10:52.620 align:middle line:90% 00:10:52.620 --> 00:11:07.090 align:middle line:84% So these two facts together give us a very robust and short way 00:11:07.090 --> 00:11:10.180 align:middle line:90% to prove limits of sequences. 00:11:10.180 --> 00:11:15.170 align:middle line:90% 00:11:15.170 --> 00:11:20.090 align:middle line:84% So, for example, let me give you a simple one. 00:11:20.090 --> 00:11:37.420 align:middle line:90% 00:11:37.420 --> 00:11:40.560 align:middle line:84% Let's show the limit as n goes to infinity of n squared over n 00:11:40.560 --> 00:11:46.800 align:middle line:90% squared plus n plus 1 equals 1. 00:11:46.800 --> 00:11:49.800 align:middle line:84% So, I mean, this is a very simple limit 00:11:49.800 --> 00:11:51.220 align:middle line:90% to use these theorems on. 00:11:51.220 --> 00:11:53.970 align:middle line:84% But in practice, you don't always 00:11:53.970 --> 00:11:55.830 align:middle line:84% have just a simple expression like this. 00:11:55.830 --> 00:12:02.772 align:middle line:90% 00:12:02.772 --> 00:12:04.730 align:middle line:84% And we'll use these two theorems in conjunction 00:12:04.730 --> 00:12:06.647 align:middle line:84% to prove some other theorems here in a minute. 00:12:06.647 --> 00:12:09.270 align:middle line:84% But let's just see it in action once. 00:12:09.270 --> 00:12:11.980 align:middle line:90% Let me look at-- 00:12:11.980 --> 00:12:16.510 align:middle line:84% so this sequence converges to 1 if and only 00:12:16.510 --> 00:12:19.690 align:middle line:84% if the absolute value of the difference converges to 0. 00:12:19.690 --> 00:12:23.080 align:middle line:84% So let's look at the absolute value of the difference. 00:12:23.080 --> 00:12:28.030 align:middle line:90% 00:12:28.030 --> 00:12:30.460 align:middle line:84% This is equal to-- now just doing the algebra-- 00:12:30.460 --> 00:12:31.510 align:middle line:90% this is equal to-- 00:12:31.510 --> 00:12:35.740 align:middle line:90% 00:12:35.740 --> 00:12:43.190 align:middle line:84% and taking absolute values gives me n plus 1 00:12:43.190 --> 00:12:48.600 align:middle line:90% over n squared plus n plus 1. 00:12:48.600 --> 00:12:50.520 align:middle line:84% And this is less than or equal to. 00:12:50.520 --> 00:12:53.430 align:middle line:84% This 1 is making things bigger, so I can drop it. 00:12:53.430 --> 00:12:56.700 align:middle line:84% And this is less than or equal to n plus 1 00:12:56.700 --> 00:12:59.340 align:middle line:90% over n squared plus n. 00:12:59.340 --> 00:13:03.570 align:middle line:84% Now, n squared plus n I can factor into n times n plus 1. 00:13:03.570 --> 00:13:07.990 align:middle line:84% So then that cancels with this n plus 1 on top. 00:13:07.990 --> 00:13:09.360 align:middle line:90% And I just get 1 over n. 00:13:09.360 --> 00:13:11.970 align:middle line:90% 00:13:11.970 --> 00:13:18.540 align:middle line:84% So 0 is less than or equal to n squared over n squared 00:13:18.540 --> 00:13:25.900 align:middle line:84% plus n plus 1 minus 1, which is less than or equal to 1 over n. 00:13:25.900 --> 00:13:29.410 align:middle line:84% Now, 1 over n we've shown using epsilon delta proof-- 00:13:29.410 --> 00:13:31.040 align:middle line:90% I mean, epsilon M proof. 00:13:31.040 --> 00:13:34.840 align:middle line:84% We've shown 1 over n converges to 0. 00:13:34.840 --> 00:13:42.070 align:middle line:84% So since 0 converges to 0, the left side, 00:13:42.070 --> 00:13:46.310 align:middle line:84% and 1 over n converges to 0, the right side, 00:13:46.310 --> 00:13:50.330 align:middle line:84% that implies by the squeeze theorem, 00:13:50.330 --> 00:13:54.640 align:middle line:84% n squared over n squared plus n plus 1 minus 1 converges 00:13:54.640 --> 00:14:01.370 align:middle line:90% to 0 by the squeeze theorem. 00:14:01.370 --> 00:14:05.720 align:middle line:84% Which implies that n squared over n 00:14:05.720 --> 00:14:09.980 align:middle line:84% squared plus n plus 1 converges to 1 by that first theorem. 00:14:09.980 --> 00:14:18.250 align:middle line:90% 00:14:18.250 --> 00:14:24.450 align:middle line:84% OK now you can imagine that instead of having this 00:14:24.450 --> 00:14:32.630 align:middle line:84% at your disposal, I asked you to do an epsilon 00:14:32.630 --> 00:14:41.720 align:middle line:84% proof, an epsilon M proof of this statement, 00:14:41.720 --> 00:14:44.060 align:middle line:84% then you would have taken this and played with it 00:14:44.060 --> 00:14:48.290 align:middle line:84% just like we did here and gotten to 1 over n. 00:14:48.290 --> 00:14:50.278 align:middle line:84% And therefore if this is less than epsilon, 00:14:50.278 --> 00:14:52.070 align:middle line:84% that would imply this is less than epsilon. 00:14:52.070 --> 00:14:55.650 align:middle line:90% 00:14:55.650 --> 00:14:57.770 align:middle line:84% So you would choose capital M to be-- 00:14:57.770 --> 00:15:01.170 align:middle line:84% so that 1 over capital M is less than epsilon. 00:15:01.170 --> 00:15:06.110 align:middle line:84% But using these two theorems saves us a little work 00:15:06.110 --> 00:15:06.965 align:middle line:90% and a little time. 00:15:06.965 --> 00:15:12.610 align:middle line:90% 00:15:12.610 --> 00:15:19.220 align:middle line:84% OK, now, so at the end of the lecture last time, 00:15:19.220 --> 00:15:21.950 align:middle line:84% we discussed the notion of subsequences. 00:15:21.950 --> 00:15:27.710 align:middle line:84% And we showed that limiting-- limits and subsequences 00:15:27.710 --> 00:15:33.350 align:middle line:84% interact nicely, meaning if I have a convergent sequence, 00:15:33.350 --> 00:15:35.590 align:middle line:84% then every subsequence converges to the same thing. 00:15:35.590 --> 00:15:38.380 align:middle line:90% 00:15:38.380 --> 00:15:42.630 align:middle line:84% So now a natural question is, how 00:15:42.630 --> 00:15:46.500 align:middle line:84% do limits interact with the order of the real numbers? 00:15:46.500 --> 00:16:07.350 align:middle line:90% 00:16:07.350 --> 00:16:10.740 align:middle line:84% R has these two fundamental properties about it, 00:16:10.740 --> 00:16:13.560 align:middle line:90% that it's-- 00:16:13.560 --> 00:16:16.060 align:middle line:84% so first off, that it has the least upper bound property, 00:16:16.060 --> 00:16:20.555 align:middle line:84% and also that it's an ordered field. 00:16:20.555 --> 00:16:21.930 align:middle line:84% So first natural question is, how 00:16:21.930 --> 00:16:25.560 align:middle line:84% does this definition of limit convergence interact 00:16:25.560 --> 00:16:29.050 align:middle line:90% with the order? 00:16:29.050 --> 00:16:34.190 align:middle line:84% So the first theorem states, or this term that kind of answers 00:16:34.190 --> 00:16:38.750 align:middle line:84% this question is the following is that limits respect order, 00:16:38.750 --> 00:16:40.470 align:middle line:90% basically. 00:16:40.470 --> 00:17:06.680 align:middle line:84% So if xn, yn, are convergent sequences and for all n, 00:17:06.680 --> 00:17:10.400 align:middle line:84% xn is less than or equal to yn, then 00:17:10.400 --> 00:17:11.650 align:middle line:90% what should be the conclusion? 00:17:11.650 --> 00:17:13.460 align:middle line:90% The limit as n goes to infinity. 00:17:13.460 --> 00:17:18.220 align:middle line:84% So what I said a minute ago as limits respect inequality. 00:17:18.220 --> 00:17:21.339 align:middle line:84% Then I should be able to take the limits of both sides 00:17:21.339 --> 00:17:23.349 align:middle line:90% and still have this inequality. 00:17:23.349 --> 00:17:27.440 align:middle line:90% 00:17:27.440 --> 00:17:37.530 align:middle line:84% Then limit as n goes to infinity is less than or equal to limit 00:17:37.530 --> 00:17:39.180 align:middle line:84% as n goes to infinity of y sub n. 00:17:39.180 --> 00:17:42.210 align:middle line:90% 00:17:42.210 --> 00:17:46.800 align:middle line:84% And a simple corollary that follows from this 00:17:46.800 --> 00:18:09.100 align:middle line:84% is that if x sub n is a convergent sequence, 00:18:09.100 --> 00:18:17.740 align:middle line:84% and for all n a natural number, you have two numbers a and b, 00:18:17.740 --> 00:18:20.950 align:middle line:84% such that a sub n is less than or equal to-- 00:18:20.950 --> 00:18:22.750 align:middle line:84% or a is less than or equal to x sub n is 00:18:22.750 --> 00:18:26.850 align:middle line:84% less than or equal to b, then this 00:18:26.850 --> 00:18:37.980 align:middle line:84% implies that the limit of x sub n is also between a and b. 00:18:37.980 --> 00:18:45.420 align:middle line:84% Let me say something very brief about what 00:18:45.420 --> 00:18:47.130 align:middle line:90% this says and doesn't say. 00:18:47.130 --> 00:18:49.670 align:middle line:90% 00:18:49.670 --> 00:18:50.960 align:middle line:90% So what does this not say? 00:18:50.960 --> 00:19:04.550 align:middle line:90% 00:19:04.550 --> 00:19:07.235 align:middle line:84% You may lose a strict inequality, meaning what? 00:19:07.235 --> 00:19:09.830 align:middle line:90% 00:19:09.830 --> 00:19:12.500 align:middle line:84% It can be the case that x sub n is less than y sub 00:19:12.500 --> 00:19:15.860 align:middle line:84% n for all of n, but the limit of x sub n 00:19:15.860 --> 00:19:18.770 align:middle line:90% equals the limit of y sub n. 00:19:18.770 --> 00:19:23.180 align:middle line:84% So simply having less than x sub n less than y sub n does not 00:19:23.180 --> 00:19:28.850 align:middle line:84% imply the limit of x sub n is less than the limit of y sub n. 00:19:28.850 --> 00:19:30.725 align:middle line:84% All right, so what do I mean by this? 00:19:30.725 --> 00:19:36.280 align:middle line:90% 00:19:36.280 --> 00:19:46.078 align:middle line:84% This does not imply that the limit is less than y sub n. 00:19:46.078 --> 00:19:48.120 align:middle line:84% Now, at this point in class, I would ask somebody 00:19:48.120 --> 00:19:51.720 align:middle line:90% to give me a counter example. 00:19:51.720 --> 00:19:56.540 align:middle line:84% So at this point, I'm going to take a bite of my cookie 00:19:56.540 --> 00:19:57.800 align:middle line:90% and let you think about that. 00:19:57.800 --> 00:20:02.350 align:middle line:90% 00:20:02.350 --> 00:20:04.100 align:middle line:84% I didn't have to take a bite of my cookie. 00:20:04.100 --> 00:20:07.940 align:middle line:84% You could have just paused the video and thought about it. 00:20:07.940 --> 00:20:12.950 align:middle line:90% 00:20:12.950 --> 00:20:15.130 align:middle line:84% But then I wouldn't get a bite of my cookie. 00:20:15.130 --> 00:20:23.170 align:middle line:84% OK, so what's an example of two sequences that satisfy this, 00:20:23.170 --> 00:20:24.240 align:middle line:90% but don't satisfy this? 00:20:24.240 --> 00:20:30.090 align:middle line:90% 00:20:30.090 --> 00:20:41.470 align:middle line:84% If x sub n equals 0 for all n, and y sub n equals 1 over n, 00:20:41.470 --> 00:20:46.400 align:middle line:84% then the x sub n is less than y sub n for all n. 00:20:46.400 --> 00:20:50.040 align:middle line:90% 00:20:50.040 --> 00:20:55.242 align:middle line:90% And what's the limit x sub n? 00:20:55.242 --> 00:20:56.075 align:middle line:90% Well, that's just 0. 00:20:56.075 --> 00:21:02.490 align:middle line:90% 00:21:02.490 --> 00:21:05.130 align:middle line:84% And before the limit of y sub n, that's just 0. 00:21:05.130 --> 00:21:06.940 align:middle line:84% And these two things equal each other. 00:21:06.940 --> 00:21:10.120 align:middle line:90% This is not less than that, OK. 00:21:10.120 --> 00:21:12.145 align:middle line:84% So just want to make that small point. 00:21:12.145 --> 00:21:16.760 align:middle line:90% 00:21:16.760 --> 00:21:20.600 align:middle line:84% Could pop up as a small question on one of the midterms-- 00:21:20.600 --> 00:21:23.695 align:middle line:84% I should say the midterm-- and possibly the final. 00:21:23.695 --> 00:21:26.760 align:middle line:90% 00:21:26.760 --> 00:21:32.286 align:middle line:90% All right, so let's prove-- 00:21:32.286 --> 00:21:35.580 align:middle line:90% [COUGHS] let's prove 1. 00:21:35.580 --> 00:21:37.335 align:middle line:90% 2 follows immediately from one. 00:21:37.335 --> 00:21:39.870 align:middle line:90% 00:21:39.870 --> 00:21:44.480 align:middle line:84% For two, we simply take, for example, 00:21:44.480 --> 00:21:46.210 align:middle line:84% the upper inequality x sub n is less 00:21:46.210 --> 00:21:47.650 align:middle line:84% than or equal to b, we take y sub 00:21:47.650 --> 00:21:50.890 align:middle line:90% n to be the constant sequence b. 00:21:50.890 --> 00:21:56.830 align:middle line:84% And for the other one, we would take the bigger sequence 00:21:56.830 --> 00:21:58.900 align:middle line:84% to be x sub n, and the smaller sequence 00:21:58.900 --> 00:22:00.670 align:middle line:90% to be the constant sequence a. 00:22:00.670 --> 00:22:05.620 align:middle line:84% So two follows immediately from 1, so we're just going to do 1. 00:22:05.620 --> 00:22:07.510 align:middle line:84% And we haven't done a proof by contradiction 00:22:07.510 --> 00:22:12.530 align:middle line:84% in a while, so why not do it by contradiction? 00:22:12.530 --> 00:22:13.340 align:middle line:90% OK, so the proof-- 00:22:13.340 --> 00:22:18.230 align:middle line:90% 00:22:18.230 --> 00:22:19.940 align:middle line:84% let's label these sequences so I don't 00:22:19.940 --> 00:22:26.510 align:middle line:84% have to write limit as n goes to infinity, and limit of x sub n, 00:22:26.510 --> 00:22:30.900 align:middle line:84% and limit as n goes to infinity of y sub n so much. 00:22:30.900 --> 00:22:32.720 align:middle line:84% So let's call their limits x and y. 00:22:32.720 --> 00:22:38.020 align:middle line:90% 00:22:38.020 --> 00:22:39.280 align:middle line:90% And what do we want to show? 00:22:39.280 --> 00:22:44.460 align:middle line:90% 00:22:44.460 --> 00:22:46.440 align:middle line:90% x is less than or equal to y. 00:22:46.440 --> 00:22:48.000 align:middle line:90% That's our goal. 00:22:48.000 --> 00:22:50.164 align:middle line:84% And we're going to prove this by contradiction. 00:22:50.164 --> 00:22:55.490 align:middle line:90% 00:22:55.490 --> 00:22:59.240 align:middle line:84% That is, let's assume y is less than x, 00:22:59.240 --> 00:23:03.040 align:middle line:84% and arrive at a false statement, which 00:23:03.040 --> 00:23:06.580 align:middle line:84% contradicts our setting our assumptions that we have. 00:23:06.580 --> 00:23:10.060 align:middle line:90% Assume y is less than x. 00:23:10.060 --> 00:23:15.520 align:middle line:84% Now, let me draw a picture here to go along 00:23:15.520 --> 00:23:18.310 align:middle line:90% with what's going to happen. 00:23:18.310 --> 00:23:22.550 align:middle line:90% 00:23:22.550 --> 00:23:32.050 align:middle line:84% So if y is less than x, all of the y sub n's 00:23:32.050 --> 00:23:35.650 align:middle line:84% have to be near y if I go far enough out. 00:23:35.650 --> 00:23:38.200 align:middle line:84% And all the x sub n's have to be near x as long 00:23:38.200 --> 00:23:40.600 align:middle line:90% as I go far enough out. 00:23:40.600 --> 00:23:43.960 align:middle line:84% So that would contradict eventually 00:23:43.960 --> 00:23:45.730 align:middle line:84% the fact that the x sub ns are supposed 00:23:45.730 --> 00:23:49.480 align:middle line:84% to be less than or equal to the y sub n's, if y is less than x. 00:23:49.480 --> 00:23:54.340 align:middle line:84% And let's say I go out, let's say, 00:23:54.340 --> 00:23:57.790 align:middle line:84% half the distance between y and x. 00:23:57.790 --> 00:24:04.200 align:middle line:90% So this is like the midpoint. 00:24:04.200 --> 00:24:11.660 align:middle line:84% And all the y sub n's are here, and all the x sub n's are here, 00:24:11.660 --> 00:24:18.760 align:middle line:84% and the y sub n's are here, then I cannot have x sub n less than 00:24:18.760 --> 00:24:24.470 align:middle line:84% or equal to y sub n, which is my assumption, 00:24:24.470 --> 00:24:28.430 align:middle line:84% which is the assumption in my theorem. 00:24:28.430 --> 00:24:31.340 align:middle line:84% Now, maybe I should have written this here. 00:24:31.340 --> 00:24:35.710 align:middle line:90% 00:24:35.710 --> 00:24:38.590 align:middle line:84% Sometimes it's good to reiterate what your assumptions are. 00:24:38.590 --> 00:24:41.620 align:middle line:84% Suppose for all n, x sub n is less than or-- y sub n. 00:24:41.620 --> 00:24:45.450 align:middle line:90% And-- like that. 00:24:45.450 --> 00:24:48.060 align:middle line:84% And so the thing we're trying to show is this, OK. 00:24:48.060 --> 00:24:50.580 align:middle line:84% So we assume the negation of this 00:24:50.580 --> 00:24:55.350 align:middle line:84% and arrive at a contradiction to the things we're assuming, 00:24:55.350 --> 00:25:00.915 align:middle line:90% or just to general true facts. 00:25:00.915 --> 00:25:07.440 align:middle line:90% 00:25:07.440 --> 00:25:10.223 align:middle line:84% I had to add the word true on the facts, because they're, 00:25:10.223 --> 00:25:11.640 align:middle line:84% like I said, at some point there's 00:25:11.640 --> 00:25:14.010 align:middle line:84% alternative facts flying around out there. 00:25:14.010 --> 00:25:17.180 align:middle line:90% 00:25:17.180 --> 00:25:21.260 align:middle line:84% OK, so this picture, we're going to turn this into a proof. 00:25:21.260 --> 00:25:30.280 align:middle line:84% So since yn converges to y, there exists a natural number 00:25:30.280 --> 00:25:39.320 align:middle line:84% M sub 0 such that for all n bigger than or equal to M 00:25:39.320 --> 00:25:50.527 align:middle line:84% sub 0, y sub n minus y is less than x minus y over 2. 00:25:50.527 --> 00:25:52.360 align:middle line:84% Now, that's a positive number, because we're 00:25:52.360 --> 00:25:55.510 align:middle line:90% assuming x is bigger than y. 00:25:55.510 --> 00:26:00.610 align:middle line:84% And by the definition of limit, given any positive number, 00:26:00.610 --> 00:26:03.130 align:middle line:84% I can find an integer so that for all n bigger 00:26:03.130 --> 00:26:04.960 align:middle line:84% than or equal to that integer, this thing 00:26:04.960 --> 00:26:06.280 align:middle line:90% is less than that small number. 00:26:06.280 --> 00:26:07.905 align:middle line:84% And I'm just choosing that small number 00:26:07.905 --> 00:26:09.790 align:middle line:84% to be this very special small number, 00:26:09.790 --> 00:26:14.968 align:middle line:84% because that's going to help me arrive at a contradiction. 00:26:14.968 --> 00:26:20.590 align:middle line:90% 00:26:20.590 --> 00:26:26.260 align:middle line:84% Since xn converges to x, there exist M1 natural number 00:26:26.260 --> 00:26:32.700 align:middle line:84% such that for all n bigger than or equal to M1, x 00:26:32.700 --> 00:26:37.120 align:middle line:84% sub n minus x is less than the same thing. 00:26:37.120 --> 00:26:42.360 align:middle line:84% OK, so this is putting into a precise form what 00:26:42.360 --> 00:26:44.010 align:middle line:84% I was saying that all the x sub n's 00:26:44.010 --> 00:26:45.870 align:middle line:90% to be close to x eventually. 00:26:45.870 --> 00:26:48.930 align:middle line:84% And all the y sub n's to be close to y eventually. 00:26:48.930 --> 00:26:51.090 align:middle line:90% And how eventually? 00:26:51.090 --> 00:26:55.860 align:middle line:84% Well, eventually enough so that I'm in these two disjoint 00:26:55.860 --> 00:26:59.215 align:middle line:90% intervals, OK. 00:26:59.215 --> 00:27:04.100 align:middle line:90% 00:27:04.100 --> 00:27:11.410 align:middle line:90% Let n be M sub 0 plus M sub 1. 00:27:11.410 --> 00:27:14.490 align:middle line:90% 00:27:14.490 --> 00:27:20.010 align:middle line:84% Then n is bigger than or equal to M sub 0. 00:27:20.010 --> 00:27:22.200 align:middle line:84% And n is bigger than or equal to M sub 1. 00:27:22.200 --> 00:27:27.510 align:middle line:84% So both of these inequalities are valid for this n 00:27:27.510 --> 00:27:29.710 align:middle line:90% and what does this mean? 00:27:29.710 --> 00:27:33.510 align:middle line:84% Well, then that implies that y sub n-- 00:27:33.510 --> 00:27:38.940 align:middle line:84% so let me tack on one more inequality. 00:27:38.940 --> 00:27:45.640 align:middle line:84% Just by removing the absolute values and adding y, 00:27:45.640 --> 00:27:53.660 align:middle line:84% this tells me y sub n is less than x plus y over 2. 00:27:53.660 --> 00:27:57.800 align:middle line:84% And this tells me x sub n is less than-- 00:27:57.800 --> 00:28:00.320 align:middle line:90% 00:28:00.320 --> 00:28:01.520 align:middle line:90% or the other way, sorry. 00:28:01.520 --> 00:28:09.570 align:middle line:90% 00:28:09.570 --> 00:28:14.870 align:middle line:84% Well, we'll just write this down here. 00:28:14.870 --> 00:28:15.370 align:middle line:90% So 00:28:15.370 --> 00:28:25.630 align:middle line:84% Then y sub n is less than y plus x minus y over 2, 00:28:25.630 --> 00:28:30.640 align:middle line:84% which equals x plus y over 2, which equals 00:28:30.640 --> 00:28:34.870 align:middle line:90% x minus x minus y over 2. 00:28:34.870 --> 00:28:39.890 align:middle line:84% And this is less than x sub n by the second inequality. 00:28:39.890 --> 00:28:43.840 align:middle line:84% So this follows from the first inequality. 00:28:43.840 --> 00:28:48.340 align:middle line:84% This follows from the second inequality, 00:28:48.340 --> 00:28:50.650 align:middle line:84% which implies for this specific n, 00:28:50.650 --> 00:28:53.890 align:middle line:90% y sub n is less than x sub n. 00:28:53.890 --> 00:28:56.410 align:middle line:84% And this is a contradiction to our assumption 00:28:56.410 --> 00:29:01.220 align:middle line:84% that y sub n is bigger than or equal to x sub n for all n. 00:29:01.220 --> 00:29:05.840 align:middle line:84% So we had just found, based on this assumption here, 00:29:05.840 --> 00:29:09.628 align:middle line:84% that we arrive at a contradiction 00:29:09.628 --> 00:29:10.670 align:middle line:90% to our other assumptions. 00:29:10.670 --> 00:29:14.175 align:middle line:84% And therefore this must be false. 00:29:14.175 --> 00:29:26.380 align:middle line:90% 00:29:26.380 --> 00:29:33.740 align:middle line:84% So that's how limits interact with inequalities. 00:29:33.740 --> 00:29:35.950 align:middle line:84% So if I have two sequences, one bigger 00:29:35.950 --> 00:29:40.570 align:middle line:84% than the other, then the limits respect that inequality. 00:29:40.570 --> 00:29:42.490 align:middle line:84% That has to deal with the order part of R 00:29:42.490 --> 00:29:43.550 align:middle line:90% being an ordered field. 00:29:43.550 --> 00:29:47.090 align:middle line:84% So what about the field part of R being an ordered field? 00:29:47.090 --> 00:30:09.220 align:middle line:84% So how does limits interact with algebraic operations? 00:30:09.220 --> 00:30:15.790 align:middle line:90% 00:30:15.790 --> 00:30:21.240 align:middle line:84% All right, quite well, it turns out. 00:30:21.240 --> 00:30:22.815 align:middle line:90% So let's in theorem. 00:30:22.815 --> 00:30:27.790 align:middle line:90% 00:30:27.790 --> 00:30:37.590 align:middle line:84% So suppose I have two convergence sequences, 00:30:37.590 --> 00:30:42.090 align:middle line:84% limit as n goes to infinity of x sub n equals x, 00:30:42.090 --> 00:30:47.430 align:middle line:84% and limit as n goes to infinity of y sub n 00:30:47.430 --> 00:30:56.570 align:middle line:84% equals y, then several things hold. 00:30:56.570 --> 00:31:00.860 align:middle line:84% The first is that, again, you should kind of read this 00:31:00.860 --> 00:31:04.910 align:middle line:84% as two statements written in one, limit as n 00:31:04.910 --> 00:31:08.510 align:middle line:84% goes to infinity of x sub n plus y sub n. 00:31:08.510 --> 00:31:11.150 align:middle line:84% So this is a new sequence that I formed 00:31:11.150 --> 00:31:14.270 align:middle line:84% by just taking the term-by-term sum of these two things. 00:31:14.270 --> 00:31:17.710 align:middle line:90% 00:31:17.710 --> 00:31:19.960 align:middle line:84% This sequence, this new sequence is convergent. 00:31:19.960 --> 00:31:24.700 align:middle line:84% And the limit equals the sum of the limits, OK? 00:31:24.700 --> 00:31:28.130 align:middle line:90% 00:31:28.130 --> 00:31:35.870 align:middle line:84% The second is, for all c in R, the limit 00:31:35.870 --> 00:31:39.170 align:middle line:84% as n goes to infinity of the new sequence obtained 00:31:39.170 --> 00:31:43.780 align:middle line:84% by taking every entry of the sequence x sub n 00:31:43.780 --> 00:31:45.820 align:middle line:84% and multiplying it by this fixed number 00:31:45.820 --> 00:31:53.120 align:middle line:84% c, the limit of that product is the product of c and x. 00:31:53.120 --> 00:31:56.990 align:middle line:84% So limits respect what one would call scalar multiplication. 00:31:56.990 --> 00:32:05.250 align:middle line:90% 00:32:05.250 --> 00:32:07.950 align:middle line:84% But we could be more general than that. 00:32:07.950 --> 00:32:11.160 align:middle line:84% c you can think of as just one example 00:32:11.160 --> 00:32:14.410 align:middle line:84% of a convergent sequence, just a constant sequence. 00:32:14.410 --> 00:32:17.100 align:middle line:84% But, in general, we have that the product 00:32:17.100 --> 00:32:20.640 align:middle line:84% of two convergent sequences is convergent. 00:32:20.640 --> 00:32:23.895 align:middle line:84% And the limit of the product is the product of the limit. 00:32:23.895 --> 00:32:27.870 align:middle line:90% 00:32:27.870 --> 00:32:30.610 align:middle line:84% And, finally, if we have something for-- 00:32:30.610 --> 00:32:32.700 align:middle line:90% write it over here-- 00:32:32.700 --> 00:32:36.540 align:middle line:84% if we have something for a product, 00:32:36.540 --> 00:32:39.300 align:middle line:84% then perhaps we have something by quotient. 00:32:39.300 --> 00:32:41.530 align:middle line:84% And that's as long as we can divide things. 00:32:41.530 --> 00:32:50.140 align:middle line:84% So if for all n, y sub n does not equal 0, 00:32:50.140 --> 00:32:59.790 align:middle line:84% and the limit y does not equal 0, then limit of the quotient 00:32:59.790 --> 00:33:07.160 align:middle line:84% x sub n over y sub n is the quotient of the limits, OK. 00:33:07.160 --> 00:33:37.710 align:middle line:90% 00:33:37.710 --> 00:33:49.830 align:middle line:84% OK, so we're going to prove this first one using 00:33:49.830 --> 00:33:54.630 align:middle line:84% this scheme of using both this simple theorem about limits 00:33:54.630 --> 00:33:56.190 align:middle line:90% and the squeeze theorem. 00:33:56.190 --> 00:34:01.710 align:middle line:90% 00:34:01.710 --> 00:34:02.910 align:middle line:90% And it's quite simple. 00:34:02.910 --> 00:34:07.200 align:middle line:84% So we just use the triangle inequality. 00:34:07.200 --> 00:34:12.070 align:middle line:90% 00:34:12.070 --> 00:34:16.820 align:middle line:84% By the triangle inequality, 0 is less than or equal to x sub n 00:34:16.820 --> 00:34:22.190 align:middle line:90% plus y sub n minus x plus . 00:34:22.190 --> 00:34:25.155 align:middle line:84% And x sub n minus x, y sub n minus y. 00:34:25.155 --> 00:34:26.780 align:middle line:84% And then I use the triangle inequality. 00:34:26.780 --> 00:34:32.460 align:middle line:84% I get x sub n minus x plus y sub n-- 00:34:32.460 --> 00:34:35.370 align:middle line:84% oh-- you know what, scratch that. 00:34:35.370 --> 00:34:39.840 align:middle line:84% Because, in fact, I was trying to be too clever 00:34:39.840 --> 00:34:41.735 align:middle line:84% and would have ended up using the theorem 00:34:41.735 --> 00:34:42.610 align:middle line:90% to prove the theorem. 00:34:42.610 --> 00:34:43.832 align:middle line:90% So let's not do that. 00:34:43.832 --> 00:34:45.540 align:middle line:84% It's never good to use the theorem you're 00:34:45.540 --> 00:34:47.540 align:middle line:84% trying to prove to prove the theorem that you're 00:34:47.540 --> 00:34:48.580 align:middle line:90% trying to prove. 00:34:48.580 --> 00:34:54.330 align:middle line:84% So let's go back to basics and use the definition. 00:34:54.330 --> 00:34:55.710 align:middle line:90% Let epsilon be positive. 00:34:55.710 --> 00:35:01.700 align:middle line:90% 00:35:01.700 --> 00:35:10.690 align:middle line:84% So since x sub n converges to x, there exists a natural number 00:35:10.690 --> 00:35:19.750 align:middle line:84% M such that for all n bigger than or equal to M0, x 00:35:19.750 --> 00:35:21.340 align:middle line:90% sub n minus x-- 00:35:21.340 --> 00:35:24.610 align:middle line:84% it's supposed to be a n, but it looks like a k-- 00:35:24.610 --> 00:35:26.530 align:middle line:90% less than epsilon over 2. 00:35:26.530 --> 00:35:27.040 align:middle line:90% Why the 2? 00:35:27.040 --> 00:35:28.248 align:middle line:90% Well, you'll see in a minute. 00:35:28.248 --> 00:35:34.230 align:middle line:90% 00:35:34.230 --> 00:35:37.110 align:middle line:84% And similarly for the sequence y, 00:35:37.110 --> 00:35:45.660 align:middle line:84% there exist M1 natural number such that 00:35:45.660 --> 00:35:51.420 align:middle line:84% for all n bigger than or equal to M1, y sub n minus y 00:35:51.420 --> 00:35:54.760 align:middle line:90% is less than epsilon over 2. 00:35:54.760 --> 00:35:57.400 align:middle line:84% So I have these two integers, which 00:35:57.400 --> 00:36:02.170 align:middle line:84% are given to me by the fact that x sub n converges to x, y sub n 00:36:02.170 --> 00:36:07.610 align:middle line:84% converges to y, and the definition of convergence. 00:36:07.610 --> 00:36:09.110 align:middle line:84% I can always find these two integers 00:36:09.110 --> 00:36:11.210 align:middle line:90% for any small tolerance. 00:36:11.210 --> 00:36:12.710 align:middle line:84% And I'm choosing the tolerance to be 00:36:12.710 --> 00:36:16.890 align:middle line:84% epsilon over 2 for some reason, which you'll see in a minute. 00:36:16.890 --> 00:36:19.310 align:middle line:84% And so what is the integer capital M 00:36:19.310 --> 00:36:22.910 align:middle line:84% that I choose for this epsilon for the sequence x sub n plus y 00:36:22.910 --> 00:36:23.480 align:middle line:90% sub n? 00:36:23.480 --> 00:36:26.252 align:middle line:90% 00:36:26.252 --> 00:36:33.010 align:middle line:84% I'm going to choose M to be M0 plus M1. 00:36:33.010 --> 00:36:37.130 align:middle line:84% And now I need to show that this choice of M works. 00:36:37.130 --> 00:36:39.070 align:middle line:84% And if n is bigger than or equal to M, 00:36:39.070 --> 00:36:44.370 align:middle line:84% this implies that n is bigger than or equal to M0. 00:36:44.370 --> 00:36:48.540 align:middle line:84% And M is bigger than or equal to M1. 00:36:48.540 --> 00:36:54.700 align:middle line:84% So both of these inequalities are valid for this n. 00:36:54.700 --> 00:37:02.242 align:middle line:84% And therefore I get x sub n plus y sub n minus x plus y. 00:37:02.242 --> 00:37:04.450 align:middle line:84% And now I do what I was going to do a minute ago when 00:37:04.450 --> 00:37:06.533 align:middle line:84% I was going to use the theorem to prove a theorem. 00:37:06.533 --> 00:37:09.950 align:middle line:84% x sub n minus x, y sub n minus y, I group those together. 00:37:09.950 --> 00:37:11.740 align:middle line:84% And then I used a triangle inequality. 00:37:11.740 --> 00:37:13.315 align:middle line:84% This is less than or equal to x sub 00:37:13.315 --> 00:37:18.360 align:middle line:90% n minus x plus y sub n minus y. 00:37:18.360 --> 00:37:21.000 align:middle line:84% And now this is less than epsilon over 2. 00:37:21.000 --> 00:37:29.230 align:middle line:84% This is less than epsilon over 2, so equals epsilon. 00:37:29.230 --> 00:37:31.510 align:middle line:84% And now you can see why I chose the 2. 00:37:31.510 --> 00:37:35.180 align:middle line:84% Because I wanted to show this was less than epsilon. 00:37:35.180 --> 00:37:37.510 align:middle line:84% And I had control over these two things. 00:37:37.510 --> 00:37:41.480 align:middle line:84% And the sum of controls gives me epsilon. 00:37:41.480 --> 00:37:43.570 align:middle line:84% So I choose the control to be epsilon over 2. 00:37:43.570 --> 00:37:47.650 align:middle line:84% If I had three sequences, then you 00:37:47.650 --> 00:37:50.260 align:middle line:84% could probably guess though which is epsilon over 3. 00:37:50.260 --> 00:37:54.220 align:middle line:84% Meaning if I had sequences x sun n y sub n, 00:37:54.220 --> 00:37:56.830 align:middle line:84% and z sub n, and I looked at sum x sub n 00:37:56.830 --> 00:38:00.250 align:middle line:84% plus y sub n plus z sub n, I could show that converges 00:38:00.250 --> 00:38:01.900 align:middle line:90% to the sum of the limits. 00:38:01.900 --> 00:38:05.620 align:middle line:84% And I would choose these integers M sub 0, 00:38:05.620 --> 00:38:08.573 align:middle line:84% M sub 1, M sub 2, so that I have epsilon over 3 here, 00:38:08.573 --> 00:38:09.865 align:middle line:90% so that they sum up to epsilon. 00:38:09.865 --> 00:38:24.280 align:middle line:90% 00:38:24.280 --> 00:38:30.892 align:middle line:84% So now we prove 2, that for this single scalar multiplication 00:38:30.892 --> 00:38:32.350 align:middle line:84% if you like where you just multiply 00:38:32.350 --> 00:38:35.260 align:middle line:84% each term by a single number, the limit 00:38:35.260 --> 00:38:38.020 align:middle line:90% respects that multiplication. 00:38:38.020 --> 00:38:40.270 align:middle line:90% Do an epsilon proof again. 00:38:40.270 --> 00:38:47.780 align:middle line:90% 00:38:47.780 --> 00:38:49.970 align:middle line:90% Since x sub n converges to x-- 00:38:49.970 --> 00:38:53.300 align:middle line:84% so now we're trying to show that second limit-- 00:38:53.300 --> 00:39:01.740 align:middle line:84% there exists M sub 0, a natural number, 00:39:01.740 --> 00:39:09.270 align:middle line:84% such that for all n than or equal to M sub 0, 00:39:09.270 --> 00:39:17.550 align:middle line:84% x sub n minus x is less than epsilon over the absolute value 00:39:17.550 --> 00:39:21.960 align:middle line:90% of c plus 1, OK. 00:39:21.960 --> 00:39:24.912 align:middle line:84% And now you have to trust me, why that thing? 00:39:24.912 --> 00:39:25.620 align:middle line:90% Well, you'll see. 00:39:25.620 --> 00:39:28.710 align:middle line:84% It'll come out just like this did. 00:39:28.710 --> 00:39:32.370 align:middle line:84% So for the sequence c times x sub n, 00:39:32.370 --> 00:39:37.080 align:middle line:84% we'll choose M to be just this M sub 0. 00:39:37.080 --> 00:39:42.630 align:middle line:84% Then if n is bigger than or equal to M, 00:39:42.630 --> 00:39:48.020 align:middle line:84% which is equal to M sub 0, this implies this inequality holds. 00:39:48.020 --> 00:39:54.050 align:middle line:84% And therefore c times x sub n minus c times x. 00:39:54.050 --> 00:39:57.230 align:middle line:84% This c pops out of the absolute value 00:39:57.230 --> 00:39:59.870 align:middle line:84% and becomes the absolute value of c times the absolute value 00:39:59.870 --> 00:40:01.025 align:middle line:90% of x sub n minus x. 00:40:01.025 --> 00:40:06.070 align:middle line:90% 00:40:06.070 --> 00:40:09.190 align:middle line:84% And which is less than-- so I'm writing less than here. 00:40:09.190 --> 00:40:10.780 align:middle line:84% But this thing is less than that. 00:40:10.780 --> 00:40:19.570 align:middle line:84% [INAUDIBLE] is less than c over c plus 1 epsilon. 00:40:19.570 --> 00:40:26.950 align:middle line:84% Now, this quotient here, this number over this number plus 1, 00:40:26.950 --> 00:40:30.240 align:middle line:90% is always less than 1. 00:40:30.240 --> 00:40:33.630 align:middle line:84% So this positive number which is less than 1 00:40:33.630 --> 00:40:36.600 align:middle line:84% or non-negative number which is less than 1 00:40:36.600 --> 00:40:39.630 align:middle line:84% is going to be less than 1 times epsilon, which 00:40:39.630 --> 00:40:40.350 align:middle line:90% gives me epsilon. 00:40:40.350 --> 00:40:44.050 align:middle line:90% 00:40:44.050 --> 00:40:48.740 align:middle line:84% And maybe you're wondering why didn't I choose-- 00:40:48.740 --> 00:40:54.900 align:middle line:84% so this is just a little smidgen of sophistication, not much, 00:40:54.900 --> 00:40:55.550 align:middle line:90% just a little. 00:40:55.550 --> 00:40:58.850 align:middle line:84% Why didn't I choose this so that it's 00:40:58.850 --> 00:41:01.310 align:middle line:84% epsilon over the absolute value of c 00:41:01.310 --> 00:41:05.870 align:middle line:84% so that when I stick in my inequality for this guy 00:41:05.870 --> 00:41:08.090 align:middle line:90% I just get epsilon? 00:41:08.090 --> 00:41:09.830 align:middle line:90% Well, what if c equals 0? 00:41:09.830 --> 00:41:13.310 align:middle line:84% So then I'm telling you to choose capital M sub 0 so 00:41:13.310 --> 00:41:16.670 align:middle line:84% that the absolute value of x sub n minus x is less than epsilon 00:41:16.670 --> 00:41:17.840 align:middle line:90% over 0. 00:41:17.840 --> 00:41:20.600 align:middle line:90% Division by 0 is a no no. 00:41:20.600 --> 00:41:24.650 align:middle line:84% But if we fudge it a little by adding 1 00:41:24.650 --> 00:41:27.300 align:middle line:84% we get something that still does the job. 00:41:27.300 --> 00:41:29.690 align:middle line:84% It still gives me some number which 00:41:29.690 --> 00:41:35.180 align:middle line:84% is non-negative and less than 1, which is enough, OK. 00:41:35.180 --> 00:41:41.760 align:middle line:90% 00:41:41.760 --> 00:41:46.737 align:middle line:84% So let's prove that the limit of the product 00:41:46.737 --> 00:41:47.945 align:middle line:90% is the product of the limits. 00:41:47.945 --> 00:41:54.700 align:middle line:90% 00:41:54.700 --> 00:41:57.388 align:middle line:84% Since the sequence y sub n converges to y, 00:41:57.388 --> 00:41:59.680 align:middle line:84% it's a convergent sequence, and therefore it's bounded. 00:41:59.680 --> 00:42:07.390 align:middle line:90% 00:42:07.390 --> 00:42:13.960 align:middle line:84% That means that there exists some non-negative real number 00:42:13.960 --> 00:42:25.387 align:middle line:84% b such that for all natural numbers n, 00:42:25.387 --> 00:42:28.740 align:middle line:84% y sub n is less than or equal to b in absolute value. 00:42:28.740 --> 00:42:33.850 align:middle line:90% 00:42:33.850 --> 00:42:41.210 align:middle line:84% Then I look at x sub n times y sub n minus x times y and I add 00:42:41.210 --> 00:42:48.410 align:middle line:90% and subtract x times y sub n. 00:42:48.410 --> 00:42:52.390 align:middle line:90% You can write this as plus. 00:42:52.390 --> 00:43:02.980 align:middle line:90% 00:43:02.980 --> 00:43:05.080 align:middle line:84% And now I use the triangle inequality 00:43:05.080 --> 00:43:10.220 align:middle line:84% that this is less than or equal to x sub n minus x times y sub 00:43:10.220 --> 00:43:17.630 align:middle line:84% n plus y sub n minus y times absolute value of x. 00:43:17.630 --> 00:43:20.150 align:middle line:84% And y sub n is bounded by b for all n. 00:43:20.150 --> 00:43:21.620 align:middle line:84% So this is less than or equal to-- 00:43:21.620 --> 00:43:25.790 align:middle line:90% 00:43:25.790 --> 00:43:33.365 align:middle line:84% plus y sub n minus y times the absolute value of x. 00:43:33.365 --> 00:43:36.122 align:middle line:90% 00:43:36.122 --> 00:43:42.710 align:middle line:84% Now, let me just state the obvious that we 00:43:42.710 --> 00:43:45.830 align:middle line:84% get 0 is less than or equal to x sub n times y 00:43:45.830 --> 00:43:48.080 align:middle line:90% sub n minus x times y. 00:43:48.080 --> 00:43:49.760 align:middle line:84% And this is less than or equal to, 00:43:49.760 --> 00:43:58.960 align:middle line:84% as we've shown here, plus x times y sub n minus y. 00:43:58.960 --> 00:44:04.670 align:middle line:90% 00:44:04.670 --> 00:44:06.920 align:middle line:84% Now, the right-hand side-- so the left side 00:44:06.920 --> 00:44:09.920 align:middle line:84% of this inequality converges to 0. 00:44:09.920 --> 00:44:11.930 align:middle line:84% The right side also converges to 0 00:44:11.930 --> 00:44:14.660 align:middle line:84% because we've just shown by 1 and 2. 00:44:14.660 --> 00:44:16.925 align:middle line:90% By 2, this converges to 0. 00:44:16.925 --> 00:44:21.290 align:middle line:90% 00:44:21.290 --> 00:44:25.340 align:middle line:84% And by that theorem, since xn converges to x, this product 00:44:25.340 --> 00:44:27.980 align:middle line:84% here converges to 0. b is a fixed number. 00:44:27.980 --> 00:44:31.580 align:middle line:84% And the same thing for this, that converges to 0. 00:44:31.580 --> 00:44:38.810 align:middle line:84% And therefore by 2, this sum converges to 0. 00:44:38.810 --> 00:44:42.920 align:middle line:84% OK, so these two arrows are by 2. 00:44:42.920 --> 00:44:44.300 align:middle line:90% And this is by 1. 00:44:44.300 --> 00:44:47.710 align:middle line:90% 00:44:47.710 --> 00:44:55.770 align:middle line:84% So let me just summarize by 1 and 2 00:44:55.770 --> 00:44:59.100 align:middle line:84% the right-hand side of this inequality, 00:44:59.100 --> 00:45:07.960 align:middle line:84% b times x sub n minus x plus an absolute value of x times y sub 00:45:07.960 --> 00:45:09.430 align:middle line:90% n minus y. 00:45:09.430 --> 00:45:11.680 align:middle line:90% This converges to 0. 00:45:11.680 --> 00:45:16.585 align:middle line:84% Which by the squeeze theorem implies that-- 00:45:16.585 --> 00:45:22.190 align:middle line:90% 00:45:22.190 --> 00:45:26.940 align:middle line:90% this is by squeeze theorem-- 00:45:26.940 --> 00:45:29.650 align:middle line:90% 00:45:29.650 --> 00:45:35.050 align:middle line:84% which implies that x sub n times y sub n converges to x times 00:45:35.050 --> 00:45:37.323 align:middle line:90% y by that first theorem. 00:45:37.323 --> 00:45:38.740 align:middle line:84% So I'm not going to keep referring 00:45:38.740 --> 00:45:39.657 align:middle line:90% to that first theorem. 00:45:39.657 --> 00:45:42.130 align:middle line:84% Because it's such a simple fact, I'm 00:45:42.130 --> 00:45:44.920 align:middle line:84% just going to keep using it without referencing it, 00:45:44.920 --> 00:45:48.580 align:middle line:84% namely that the sequence, a sequence converges to x if 00:45:48.580 --> 00:45:52.300 align:middle line:84% and only if the absolute value of this thing, this difference, 00:45:52.300 --> 00:45:54.340 align:middle line:90% converges to 0, OK. 00:45:54.340 --> 00:45:57.710 align:middle line:90% 00:45:57.710 --> 00:46:02.450 align:middle line:84% All right, so that proves the limit of the product 00:46:02.450 --> 00:46:04.190 align:middle line:90% is the product of the limits. 00:46:04.190 --> 00:46:10.920 align:middle line:84% Now, for the quotient we can use 3 00:46:10.920 --> 00:46:15.090 align:middle line:84% once we've proven it for just 1 over y sub n. 00:46:15.090 --> 00:46:20.510 align:middle line:90% So what do I mean? 00:46:20.510 --> 00:46:28.470 align:middle line:90% 00:46:28.470 --> 00:46:35.490 align:middle line:84% So now we're assuming the y sub n is not equal to 0 for all n. 00:46:35.490 --> 00:46:37.080 align:middle line:90% And y does not equal 0. 00:46:37.080 --> 00:46:40.410 align:middle line:90% 00:46:40.410 --> 00:46:43.560 align:middle line:84% So if we prove this statement that the limit as n 00:46:43.560 --> 00:46:52.950 align:middle line:84% goes to infinity of 1 over y sub n equals 1 over y, then by 3 00:46:52.950 --> 00:46:58.480 align:middle line:84% implies that the limit as n goes to infinity 00:46:58.480 --> 00:47:04.510 align:middle line:84% of x sub n over y sub n equals x over y. 00:47:04.510 --> 00:47:07.780 align:middle line:84% Because x sub n over y sub n is just a product of x sub n 00:47:07.780 --> 00:47:10.090 align:middle line:90% and 1 over y sub n. 00:47:10.090 --> 00:47:15.250 align:middle line:84% So we just need to prove this special case if you like. 00:47:15.250 --> 00:47:21.343 align:middle line:90% 00:47:21.343 --> 00:47:23.010 align:middle line:84% And we're to do it kind of the same way. 00:47:23.010 --> 00:47:28.930 align:middle line:84% Now because we're dividing by y sub n, 00:47:28.930 --> 00:47:31.900 align:middle line:84% so here we use that we had an upper bound for the product 00:47:31.900 --> 00:47:32.800 align:middle line:90% here. 00:47:32.800 --> 00:47:36.970 align:middle line:84% But when we take 1 over y sub n to get an upper bound of that, 00:47:36.970 --> 00:47:41.170 align:middle line:84% it means we need a lower bound on y sub n, 00:47:41.170 --> 00:47:43.630 align:middle line:84% on the absolute values of y sub n. 00:47:43.630 --> 00:47:46.750 align:middle line:84% And we get that by our assumption that the limit is 00:47:46.750 --> 00:47:52.240 align:middle line:84% non-zero for all n, and so is the y sub n's. 00:47:52.240 --> 00:48:01.650 align:middle line:84% So first thing we prove, or let me write this as a claim, 00:48:01.650 --> 00:48:06.000 align:middle line:84% there exists a positive number b, little b, 00:48:06.000 --> 00:48:10.890 align:middle line:84% such that for all natural numbers n, 00:48:10.890 --> 00:48:13.256 align:middle line:84% y sub n is bigger than or equal to b. 00:48:13.256 --> 00:48:15.830 align:middle line:90% 00:48:15.830 --> 00:48:17.990 align:middle line:84% And we know that a sequence is bounded. 00:48:17.990 --> 00:48:19.810 align:middle line:84% So we know that there's always a capital B 00:48:19.810 --> 00:48:23.080 align:middle line:84% so that absolute value of y sub n is less than or equal to b. 00:48:23.080 --> 00:48:27.700 align:middle line:84% But for sequences that are non-zero and converging 00:48:27.700 --> 00:48:37.330 align:middle line:84% to a non-zero limit, then you can bound them away from 0. 00:48:37.330 --> 00:48:39.670 align:middle line:84% And it's kind of the same proof that we 00:48:39.670 --> 00:48:43.240 align:middle line:84% gave for showing that a convergent sequence is bounded 00:48:43.240 --> 00:48:44.450 align:middle line:90% above. 00:48:44.450 --> 00:48:48.740 align:middle line:90% So let me draw a quick picture. 00:48:48.740 --> 00:48:52.238 align:middle line:84% And let's assume that the limit is positive just 00:48:52.238 --> 00:48:53.405 align:middle line:90% for the sake of the picture. 00:48:53.405 --> 00:48:56.010 align:middle line:90% 00:48:56.010 --> 00:48:58.190 align:middle line:84% So this is a little discussion on why this is true. 00:48:58.190 --> 00:49:01.885 align:middle line:84% And this picture is going to look-- 00:49:01.885 --> 00:49:03.260 align:middle line:84% at least the explanation is going 00:49:03.260 --> 00:49:08.840 align:middle line:84% to be kind of similar to why a sequence is bounded. 00:49:08.840 --> 00:49:14.030 align:middle line:84% And this one is why is it bounded below. 00:49:14.030 --> 00:49:17.890 align:middle line:84% So let's assume that the limit y is positive. 00:49:17.890 --> 00:49:24.450 align:middle line:84% And let's say I go out within distance, 00:49:24.450 --> 00:49:26.920 align:middle line:84% let's say, y over 2 so that I'm still positive. 00:49:26.920 --> 00:49:33.510 align:middle line:84% So this is y minus y over 2, absolute value. 00:49:33.510 --> 00:49:35.060 align:middle line:84% So in this picture y is positive. 00:49:35.060 --> 00:49:38.120 align:middle line:84% So that's just equal to y over 2. 00:49:38.120 --> 00:49:39.770 align:middle line:90% Then what can I say? 00:49:39.770 --> 00:49:49.970 align:middle line:84% That eventually all of the y sub n's are here in this interval 00:49:49.970 --> 00:49:50.960 align:middle line:90% away from 0. 00:49:50.960 --> 00:49:59.620 align:middle line:84% And, in fact, their absolute value is bounded by y over 2. 00:49:59.620 --> 00:50:03.220 align:middle line:84% So let me just write it that way. 00:50:03.220 --> 00:50:06.880 align:middle line:84% So all the y sub n's or n bigger than or equal to some M, 00:50:06.880 --> 00:50:09.580 align:middle line:84% all have to lie in this interval because they're converging to y 00:50:09.580 --> 00:50:12.160 align:middle line:90% and y is positive. 00:50:12.160 --> 00:50:15.220 align:middle line:84% And therefore in absolute value, they're all bounded above-- 00:50:15.220 --> 00:50:17.260 align:middle line:90% below I mean-- by y over 2. 00:50:17.260 --> 00:50:21.560 align:middle line:84% They're all at least distance y over 2 to 0. 00:50:21.560 --> 00:50:24.350 align:middle line:84% OK, and then all that's left to handle are-- 00:50:24.350 --> 00:50:30.560 align:middle line:84% maybe the finitely many that are left, y sub 1, 00:50:30.560 --> 00:50:38.190 align:middle line:84% y sub 2, y sub m minus 1, that are scattered on the real line 00:50:38.190 --> 00:50:39.440 align:middle line:90% but are non-zero. 00:50:39.440 --> 00:50:40.940 align:middle line:84% So we're just going to end up taking 00:50:40.940 --> 00:50:45.242 align:middle line:84% the minimum of this number and the absolute value 00:50:45.242 --> 00:50:45.950 align:middle line:90% of these numbers. 00:50:45.950 --> 00:51:01.780 align:middle line:90% 00:51:01.780 --> 00:51:14.490 align:middle line:84% So since the y sub n's converge to y, and y does not equal 0, 00:51:14.490 --> 00:51:24.270 align:middle line:84% there exist an integer M such that all n bigger 00:51:24.270 --> 00:51:26.430 align:middle line:84% than or equal to capital M. So this picture 00:51:26.430 --> 00:51:28.710 align:middle line:90% was why this claim is true. 00:51:28.710 --> 00:51:30.810 align:middle line:84% It was not the proof of why this claim is true. 00:51:30.810 --> 00:51:32.640 align:middle line:84% What I'm writing now is the actual proof 00:51:32.640 --> 00:51:36.440 align:middle line:90% of why this claim is true-- 00:51:36.440 --> 00:51:38.660 align:middle line:84% such that for all n bigger than or equal to a capital 00:51:38.660 --> 00:51:46.670 align:middle line:84% M, y sub n minus y in absolute value is less than y over 2. 00:51:46.670 --> 00:51:48.920 align:middle line:84% So for this picture I drew where y is positive. 00:51:48.920 --> 00:51:50.690 align:middle line:84% That would have just been y over 2. 00:51:50.690 --> 00:51:54.410 align:middle line:84% But you have to use an absolute value for the other case 00:51:54.410 --> 00:52:00.240 align:middle line:84% that y is negative, because this has to b a positive number. 00:52:00.240 --> 00:52:07.960 align:middle line:84% Then for all n bigger than or equal to M, 00:52:07.960 --> 00:52:11.080 align:middle line:84% this any inequality and the triangle inequality 00:52:11.080 --> 00:52:14.530 align:middle line:84% gives me-- so if I look at the absolute value of y, 00:52:14.530 --> 00:52:21.560 align:middle line:84% this is equal to the absolute value of y minus y sub n plus y 00:52:21.560 --> 00:52:22.310 align:middle line:90% sub n. 00:52:22.310 --> 00:52:29.670 align:middle line:84% And now I use a triangle inequality, 00:52:29.670 --> 00:52:32.480 align:middle line:84% which is less than-- this is less than y 00:52:32.480 --> 00:52:37.910 align:middle line:90% over 2 plus y sub n. 00:52:37.910 --> 00:52:41.060 align:middle line:84% And I started off with the absolute value of y . 00:52:41.060 --> 00:52:44.880 align:middle line:84% So when I subtract that over, that 00:52:44.880 --> 00:52:49.890 align:middle line:84% tells me that absolute value of y over 2 00:52:49.890 --> 00:52:56.320 align:middle line:84% is less than absolute value of y sub n for all n 00:52:56.320 --> 00:53:05.670 align:middle line:84% bigger than or equal to capital M. 00:53:05.670 --> 00:53:14.820 align:middle line:84% So then I let to b to be the minimum of several numbers. 00:53:14.820 --> 00:53:16.830 align:middle line:84% know I'm writing min, but I should write inf. 00:53:16.830 --> 00:53:32.420 align:middle line:84% But so if you like, let me write inf of y1, Ym minus 1, 00:53:32.420 --> 00:53:37.010 align:middle line:90% and y over 2. 00:53:37.010 --> 00:53:39.860 align:middle line:84% And by what you are doing on the assignment, 00:53:39.860 --> 00:53:43.220 align:middle line:84% I think it was the assignment 2, this inf 00:53:43.220 --> 00:53:45.560 align:middle line:90% always exists in a finite set. 00:53:45.560 --> 00:53:48.980 align:middle line:84% This is a finite set of positive numbers. 00:53:48.980 --> 00:53:51.470 align:middle line:84% And therefore it's infimum exists 00:53:51.470 --> 00:53:52.670 align:middle line:90% as one of these elements. 00:53:52.670 --> 00:53:56.870 align:middle line:84% One of these M numbers is the infimum. 00:53:56.870 --> 00:53:59.885 align:middle line:84% And they're all positive, so this is a positive number. 00:53:59.885 --> 00:54:06.400 align:middle line:90% 00:54:06.400 --> 00:54:10.260 align:middle line:84% And so then simply how this number is defined, 00:54:10.260 --> 00:54:22.020 align:middle line:84% it follows that for all n, y sub n is bigger than or equal to b. 00:54:22.020 --> 00:54:28.160 align:middle line:90% 00:54:28.160 --> 00:54:31.520 align:middle line:84% Because, again, if n is between-- 00:54:31.520 --> 00:54:35.060 align:middle line:84% little n is between 1 and capital M minus 1, 00:54:35.060 --> 00:54:38.570 align:middle line:84% then certainly that the absolute value of that thing 00:54:38.570 --> 00:54:43.680 align:middle line:84% is bigger than or equal to the smaller of all of these, which 00:54:43.680 --> 00:54:46.000 align:middle line:90% is bigger than or equal to b. 00:54:46.000 --> 00:54:48.610 align:middle line:84% And if n is bigger than or equal to capital M, 00:54:48.610 --> 00:54:51.220 align:middle line:84% then we proved over here that y sub n 00:54:51.220 --> 00:54:53.800 align:middle line:84% is bigger than the absolute value of y 00:54:53.800 --> 00:54:57.400 align:middle line:84% over 2, which is bigger than or equal to the minimum 00:54:57.400 --> 00:55:02.480 align:middle line:84% of these numbers and y over 2, which equals b. 00:55:02.480 --> 00:55:03.980 align:middle line:84% All right, so that proves the claim. 00:55:03.980 --> 00:55:07.933 align:middle line:90% 00:55:07.933 --> 00:55:08.850 align:middle line:90% That proves the claim. 00:55:08.850 --> 00:55:10.642 align:middle line:84% But we haven't proved what we wanted to yet 00:55:10.642 --> 00:55:13.020 align:middle line:84% that the limit as n goes to infinity of 1 over y sub n 00:55:13.020 --> 00:55:15.390 align:middle line:90% equals 1 over y. 00:55:15.390 --> 00:55:19.680 align:middle line:84% But this follows almost immediately 00:55:19.680 --> 00:55:23.280 align:middle line:90% from what we've done so far. 00:55:23.280 --> 00:55:26.640 align:middle line:84% So now we're going to use the claim to prove it. 00:55:26.640 --> 00:55:27.830 align:middle line:90% So we look at-- 00:55:27.830 --> 00:55:33.230 align:middle line:90% 00:55:33.230 --> 00:55:38.060 align:middle line:84% compute that the 0 is less than or equal to 1 00:55:38.060 --> 00:55:40.573 align:middle line:90% over y sub n minus 1 over y. 00:55:40.573 --> 00:55:41.990 align:middle line:84% We're going to show that this goes 00:55:41.990 --> 00:55:46.540 align:middle line:90% to 0 using those two theorems. 00:55:46.540 --> 00:55:51.840 align:middle line:84% And so by algebra that's equal to-- 00:55:51.840 --> 00:55:56.670 align:middle line:84% and using the absolute value, this is 1 over y sub n minus y. 00:55:56.670 --> 00:55:58.620 align:middle line:84% I mean, the absolute value of y sub 00:55:58.620 --> 00:56:05.055 align:middle line:84% n minus y over absolute value of y sub n absolute value of y. 00:56:05.055 --> 00:56:07.820 align:middle line:90% 00:56:07.820 --> 00:56:11.870 align:middle line:84% Now y sub n is bigger than or equal to b. 00:56:11.870 --> 00:56:23.710 align:middle line:84% So this is less than or equal to 1 over b times y times 00:56:23.710 --> 00:56:25.180 align:middle line:90% y sub n minus y. 00:56:25.180 --> 00:56:28.560 align:middle line:90% 00:56:28.560 --> 00:56:32.820 align:middle line:84% So just to summarize, we've shown that 0 is less than 1 00:56:32.820 --> 00:56:41.240 align:middle line:84% over y sub n minus 1 over y is less than 1 over b times 00:56:41.240 --> 00:56:42.380 align:middle line:90% 1 over-- 00:56:42.380 --> 00:56:49.760 align:middle line:84% b times the absolute value of y times y sub n minus y. 00:56:49.760 --> 00:56:56.740 align:middle line:84% Now, this goes to 0 because it's just a constant sequence. 00:56:56.740 --> 00:57:01.900 align:middle line:84% This converges to 0 because y sub n minus y in absolute value 00:57:01.900 --> 00:57:02.950 align:middle line:90% goes to 0. 00:57:02.950 --> 00:57:05.110 align:middle line:84% This is just a fixed number of times that. 00:57:05.110 --> 00:57:12.720 align:middle line:84% And by what we've proven for 2, this product converges to 0. 00:57:12.720 --> 00:57:23.900 align:middle line:84% So by the squeeze theorem, we get that 1 over y sub n, this 00:57:23.900 --> 00:57:26.375 align:middle line:90% converges to 0, which implies-- 00:57:26.375 --> 00:57:32.030 align:middle line:90% 00:57:32.030 --> 00:57:34.350 align:middle line:90% OK. 00:57:34.350 --> 00:57:37.020 align:middle line:84% So another big property about the real numbers 00:57:37.020 --> 00:57:39.840 align:middle line:84% that we proved after we stated the existence 00:57:39.840 --> 00:57:43.290 align:middle line:84% of the real numbers, which remember was this 00:57:43.290 --> 00:57:45.875 align:middle line:84% is defined as this ordered field with a least upper 00:57:45.875 --> 00:57:46.500 align:middle line:90% bound property. 00:57:46.500 --> 00:57:54.590 align:middle line:90% 00:57:54.590 --> 00:57:57.860 align:middle line:84% We proved that the square root of 2 exists as a real number. 00:57:57.860 --> 00:58:01.890 align:middle line:90% 00:58:01.890 --> 00:58:03.980 align:middle line:84% There was really nothing special about 2. 00:58:03.980 --> 00:58:08.430 align:middle line:84% In fact, you could prove that the square root of x 00:58:08.430 --> 00:58:10.380 align:middle line:84% exists as a real number for any x that's 00:58:10.380 --> 00:58:12.990 align:middle line:84% a positive or non-negative number. 00:58:12.990 --> 00:58:16.520 align:middle line:90% 00:58:16.520 --> 00:58:18.810 align:middle line:84% So the square root of a real number is well defined. 00:58:18.810 --> 00:58:22.738 align:middle line:90% 00:58:22.738 --> 00:58:24.530 align:middle line:84% Or the square root of a non-negative number 00:58:24.530 --> 00:58:29.740 align:middle line:84% is well define and always exists as a real number. 00:58:29.740 --> 00:58:38.575 align:middle line:84% So you can ask how does limits interact with square roots? 00:58:38.575 --> 00:58:41.680 align:middle line:90% 00:58:41.680 --> 00:58:43.810 align:middle line:84% And they interact just as you think they should. 00:58:43.810 --> 00:58:53.300 align:middle line:90% 00:58:53.300 --> 00:58:56.030 align:middle line:84% If I have a sequence so that for all n, 00:58:56.030 --> 00:59:02.460 align:middle line:84% x sub n is bigger than or equal to 0, 00:59:02.460 --> 00:59:06.630 align:middle line:84% and it's a convergent sequence, converging to some number x, 00:59:06.630 --> 00:59:14.890 align:middle line:84% then the limit of the square roots of these guys 00:59:14.890 --> 00:59:21.050 align:middle line:84% equals the square root of the limit, OK. 00:59:21.050 --> 00:59:25.340 align:middle line:84% Now, I want you to just take a second here and understand 00:59:25.340 --> 00:59:28.100 align:middle line:84% that this is a meaningful statement. 00:59:28.100 --> 00:59:34.840 align:middle line:84% Because since the x sub n's are all non-negative by a theorem 00:59:34.840 --> 00:59:35.410 align:middle line:90% that-- 00:59:35.410 --> 00:59:38.310 align:middle line:84% let's see, did I erase it already? 00:59:38.310 --> 00:59:46.070 align:middle line:84% The one that had to deal with limits and the order, 00:59:46.070 --> 00:59:48.220 align:middle line:84% so since the x sub n's are all non-negative, 00:59:48.220 --> 00:59:49.930 align:middle line:84% that implies that x is non-negative 00:59:49.930 --> 00:59:51.730 align:middle line:84% so that the square root is meaningful. 00:59:51.730 --> 00:59:56.210 align:middle line:84% OK, so first check whenever somebody says 00:59:56.210 --> 01:00:00.980 align:middle line:84% here's this theorem, or this theorem is-- 01:00:00.980 --> 01:00:03.920 align:middle line:90% I think this theorem is true-- 01:00:03.920 --> 01:00:09.060 align:middle line:84% is to check to make sure that a theorem is meaningful. 01:00:09.060 --> 01:00:12.160 align:middle line:90% 01:00:12.160 --> 01:00:15.680 align:middle line:90% So let's prove this. 01:00:15.680 --> 01:00:19.640 align:middle line:84% So there's two cases to consider, x is equal to 0 01:00:19.640 --> 01:00:20.810 align:middle line:90% or x is non-zero. 01:00:20.810 --> 01:00:21.935 align:middle line:90% So let's do the first case. 01:00:21.935 --> 01:00:26.280 align:middle line:90% 01:00:26.280 --> 01:00:27.570 align:middle line:90% So the limit is 0. 01:00:27.570 --> 01:00:33.390 align:middle line:84% So we'll do this proof using the definition of limits, 01:00:33.390 --> 01:00:37.073 align:middle line:84% meaning the epsilon M definition. 01:00:37.073 --> 01:00:39.240 align:middle line:84% So we want to show that the limit of the square root 01:00:39.240 --> 01:00:41.700 align:middle line:90% of x sub n equals 0. 01:00:41.700 --> 01:00:43.410 align:middle line:90% So let epsilon be positive. 01:00:43.410 --> 01:00:46.120 align:middle line:90% 01:00:46.120 --> 01:00:55.550 align:middle line:84% And since x sub n converges to 0, 01:00:55.550 --> 01:00:58.690 align:middle line:84% there exists natural number n sub 01:00:58.690 --> 01:01:06.120 align:middle line:84% 0 such that if n is bigger than or equal to M0, 01:01:06.120 --> 01:01:15.430 align:middle line:84% then x sub n minus the limit, which is just 0, 01:01:15.430 --> 01:01:17.470 align:middle line:84% and taking the absolute value, which is just 01:01:17.470 --> 01:01:21.720 align:middle line:84% x sub n, which is equal to x sub n 01:01:21.720 --> 01:01:28.640 align:middle line:84% because x sub n is non-negative, is less than epsilon squared. 01:01:28.640 --> 01:01:30.530 align:middle line:84% Remember, I can always find, no matter 01:01:30.530 --> 01:01:34.520 align:middle line:84% what is underneath my hand, since x sub n converges to 0, 01:01:34.520 --> 01:01:37.250 align:middle line:84% I can find a natural number so that that thing is less 01:01:37.250 --> 01:01:38.567 align:middle line:90% than what's underneath my hand. 01:01:38.567 --> 01:01:41.150 align:middle line:84% And the thing that I'm going to have underneath my hand that's 01:01:41.150 --> 01:01:43.192 align:middle line:84% going to make things work out for the square root 01:01:43.192 --> 01:01:46.200 align:middle line:90% is epsilon squared, OK. 01:01:46.200 --> 01:01:51.230 align:middle line:90% 01:01:51.230 --> 01:01:53.870 align:middle line:90% Choose M to be M sub 0. 01:01:53.870 --> 01:01:56.550 align:middle line:84% So I'm going to show this M works for the sequence 01:01:56.550 --> 01:01:59.630 align:middle line:90% square root of x sub n. 01:01:59.630 --> 01:02:03.950 align:middle line:84% And n bigger than or equal to M. The square root 01:02:03.950 --> 01:02:12.610 align:middle line:84% of x sub n minus 0, which is just x sub n. 01:02:12.610 --> 01:02:15.490 align:middle line:84% Now, it's also-- so we didn't strictly speaking prove this-- 01:02:15.490 --> 01:02:25.660 align:middle line:84% but it's not too hard to show that square roots respect 01:02:25.660 --> 01:02:26.750 align:middle line:90% inequalities. 01:02:26.750 --> 01:02:29.540 align:middle line:84% So x sub n is less than epsilon squared. 01:02:29.540 --> 01:02:31.690 align:middle line:84% So the square root is less than epsilon 01:02:31.690 --> 01:02:33.910 align:middle line:90% squared, which equals epsilon. 01:02:33.910 --> 01:02:43.570 align:middle line:90% 01:02:43.570 --> 01:02:50.055 align:middle line:84% So the second case is x not equal to 0. 01:02:50.055 --> 01:02:52.630 align:middle line:90% 01:02:52.630 --> 01:02:58.280 align:middle line:84% And to do this case, we'll use those two theorems again. 01:02:58.280 --> 01:03:02.360 align:middle line:90% 01:03:02.360 --> 01:03:07.970 align:middle line:84% And let's look at square root of x sub n minus square root of x. 01:03:07.970 --> 01:03:12.485 align:middle line:90% Now, if I write this as-- 01:03:12.485 --> 01:03:20.310 align:middle line:90% 01:03:20.310 --> 01:03:23.670 align:middle line:84% and multiply top and bottom by x sub n 01:03:23.670 --> 01:03:27.780 align:middle line:84% plus square root of x, square root of x sub n 01:03:27.780 --> 01:03:30.510 align:middle line:84% plus square root of x, which is a positive number. 01:03:30.510 --> 01:03:35.010 align:middle line:84% It's fine to divide by it as well, because x is non-zero. 01:03:35.010 --> 01:03:37.290 align:middle line:84% So here not just non-zero, it's positive. 01:03:37.290 --> 01:03:40.780 align:middle line:84% Because x has to be non-negative. 01:03:40.780 --> 01:03:43.420 align:middle line:84% Now this is the product of something 01:03:43.420 --> 01:03:46.805 align:middle line:84% minus something else with something plus something else. 01:03:46.805 --> 01:03:49.180 align:middle line:84% So then that's going to be the difference of the squares. 01:03:49.180 --> 01:03:52.480 align:middle line:90% So that's equal to-- 01:03:52.480 --> 01:03:55.460 align:middle line:84% over-- and, again, these are positive numbers 01:03:55.460 --> 01:03:59.770 align:middle line:84% so that they come out of the absolute value. 01:03:59.770 --> 01:04:02.400 align:middle line:90% And this is non-negative. 01:04:02.400 --> 01:04:05.820 align:middle line:84% So, in fact, it's only making things bigger on the bottom 01:04:05.820 --> 01:04:08.290 align:middle line:84% and therefore things smaller overall. 01:04:08.290 --> 01:04:11.430 align:middle line:84% So this is less than or equal to-- 01:04:11.430 --> 01:04:13.875 align:middle line:84% just if I replace this by square root of x. 01:04:13.875 --> 01:04:17.990 align:middle line:90% 01:04:17.990 --> 01:04:21.620 align:middle line:90% So what did I prove? 01:04:21.620 --> 01:04:32.860 align:middle line:90% 01:04:32.860 --> 01:04:34.750 align:middle line:90% 1 over square root of x-- 01:04:34.750 --> 01:04:37.780 align:middle line:90% 01:04:37.780 --> 01:04:38.380 align:middle line:90% all right. 01:04:38.380 --> 01:04:42.010 align:middle line:84% And so by assumption, x sub n converges to x. 01:04:42.010 --> 01:04:43.690 align:middle line:90% So this goes to 0. 01:04:43.690 --> 01:04:45.970 align:middle line:84% This is a fixed number multiplied 01:04:45.970 --> 01:04:47.230 align:middle line:90% by this thing going to 0. 01:04:47.230 --> 01:04:50.310 align:middle line:84% So by number 2 and the theorem we 01:04:50.310 --> 01:04:54.190 align:middle line:84% proved before, this whole product converges to 0. 01:04:54.190 --> 01:04:56.650 align:middle line:84% So, and of course this converges to 0. 01:04:56.650 --> 01:04:59.110 align:middle line:84% So this thing in the middle must converge to 0 01:04:59.110 --> 01:05:00.070 align:middle line:90% by the squeeze theorem. 01:05:00.070 --> 01:05:29.110 align:middle line:90% 01:05:29.110 --> 01:05:39.440 align:middle line:90% So that's the square root. 01:05:39.440 --> 01:06:04.995 align:middle line:90% And let me just remark that-- 01:06:04.995 --> 01:06:08.320 align:middle line:90% 01:06:08.320 --> 01:06:12.160 align:middle line:84% so this number 3 up here, that the limit of the product 01:06:12.160 --> 01:06:14.590 align:middle line:84% is a product of the limits, this thus 01:06:14.590 --> 01:06:18.740 align:middle line:84% implies that the limit of xn squared converges to x squared. 01:06:18.740 --> 01:06:21.490 align:middle line:84% So the square of x sub n converges 01:06:21.490 --> 01:06:23.560 align:middle line:90% to the square of the limit. 01:06:23.560 --> 01:06:28.030 align:middle line:84% And by induction, you can show that the cube fourth power, 01:06:28.030 --> 01:06:31.330 align:middle line:84% fifth power of x sub n converges to the fourth power, 01:06:31.330 --> 01:06:32.530 align:middle line:90% fifth power of x. 01:06:32.530 --> 01:06:38.260 align:middle line:84% And not only that, you can also prove-- 01:06:38.260 --> 01:06:40.282 align:middle line:90% so I'm not going to do this. 01:06:40.282 --> 01:06:41.990 align:middle line:84% And I'm not going to force you to either. 01:06:41.990 --> 01:06:45.023 align:middle line:84% But just know these as facts, that I 01:06:45.023 --> 01:06:46.690 align:middle line:84% don't have to just take the square root, 01:06:46.690 --> 01:06:48.460 align:middle line:90% I could take the k-th root. 01:06:48.460 --> 01:06:51.150 align:middle line:90% 01:06:51.150 --> 01:06:53.820 align:middle line:84% And this statement still is true that if x sub n is 01:06:53.820 --> 01:06:58.320 align:middle line:84% bigger than or equal to 0 for all n, and I have a limit, 01:06:58.320 --> 01:07:00.930 align:middle line:84% then the k-th root of x sub n converges 01:07:00.930 --> 01:07:02.580 align:middle line:90% to the k-th root of x. 01:07:02.580 --> 01:07:08.340 align:middle line:90% 01:07:08.340 --> 01:07:12.850 align:middle line:84% So the final theorem we'll prove for today, 01:07:12.850 --> 01:07:22.440 align:middle line:84% which will conclude our facts about limits, is we have-- 01:07:22.440 --> 01:07:24.240 align:middle line:84% I mean, we've been using this all along, 01:07:24.240 --> 01:07:27.060 align:middle line:84% although I haven't made special attention about it. 01:07:27.060 --> 01:07:29.160 align:middle line:84% The real numbers there, this, again, 01:07:29.160 --> 01:07:31.380 align:middle line:84% like I said, an ordered field with a least upper 01:07:31.380 --> 01:07:32.760 align:middle line:90% bound property. 01:07:32.760 --> 01:07:38.790 align:middle line:84% So we've seen how limits interact with this structure 01:07:38.790 --> 01:07:39.750 align:middle line:90% of the real numbers. 01:07:39.750 --> 01:07:41.940 align:middle line:84% But they also have a distance associated 01:07:41.940 --> 01:07:43.780 align:middle line:90% to them, the absolute value. 01:07:43.780 --> 01:07:47.460 align:middle line:84% The distance from two numbers a and b 01:07:47.460 --> 01:07:49.440 align:middle line:84% is the absolute value of a minus b. 01:07:49.440 --> 01:07:52.650 align:middle line:90% 01:07:52.650 --> 01:07:56.500 align:middle line:84% Or the distance from a number to 0 01:07:56.500 --> 01:07:58.940 align:middle line:84% is just the absolute value of that number. 01:07:58.940 --> 01:08:03.250 align:middle line:84% So one could ask, how does the limit 01:08:03.250 --> 01:08:05.620 align:middle line:84% interact with this additional structure 01:08:05.620 --> 01:08:08.410 align:middle line:90% of the absolute value? 01:08:08.410 --> 01:08:13.420 align:middle line:84% And so just like everything's been good so far with limits, 01:08:13.420 --> 01:08:15.580 align:middle line:84% it's the, same with the absolute value namely 01:08:15.580 --> 01:08:17.635 align:middle line:84% that limits respect absolute value. 01:08:17.635 --> 01:08:21.970 align:middle line:90% 01:08:21.970 --> 01:08:42.810 align:middle line:84% So if x sub n is convergent sequence, 01:08:42.810 --> 01:08:52.410 align:middle line:84% the limit x, then the sequence of absolute values 01:08:52.410 --> 01:08:53.759 align:middle line:90% is also a convergent sequence. 01:08:53.759 --> 01:09:06.210 align:middle line:90% 01:09:06.210 --> 01:09:12.270 align:middle line:84% And the limit as n goes to infinity of the absolute values 01:09:12.270 --> 01:09:16.410 align:middle line:84% equals the absolute value of the limit. 01:09:16.410 --> 01:09:22.740 align:middle line:84% Now let me, again, let's try to think a little more deeply 01:09:22.740 --> 01:09:25.390 align:middle line:90% about this real quick. 01:09:25.390 --> 01:09:27.810 align:middle line:84% If I have a convergence sequence and the absolute values 01:09:27.810 --> 01:09:31.040 align:middle line:84% converge, does the converse hold? 01:09:31.040 --> 01:09:33.229 align:middle line:84% If the absolute values converge, does this 01:09:33.229 --> 01:09:35.540 align:middle line:84% imply that the original sequence converges? 01:09:35.540 --> 01:09:38.407 align:middle line:84% And the answer is, of course, no. 01:09:38.407 --> 01:09:39.240 align:middle line:90% Well, not of course. 01:09:39.240 --> 01:09:41.323 align:middle line:84% I didn't even give you a minute to think about it. 01:09:41.323 --> 01:09:45.636 align:middle line:84% But why is the converse not true? 01:09:45.636 --> 01:09:46.844 align:middle line:90% So let me make this a remark. 01:09:46.844 --> 01:09:58.600 align:middle line:90% 01:09:58.600 --> 01:10:01.930 align:middle line:84% Converse is not true because you could look at x 01:10:01.930 --> 01:10:03.922 align:middle line:90% sub n equals minus 1 to the n. 01:10:03.922 --> 01:10:06.480 align:middle line:90% 01:10:06.480 --> 01:10:14.280 align:middle line:84% Then the absolute values of these guys 01:10:14.280 --> 01:10:16.890 align:middle line:84% converges, but the original sequence does not converge. 01:10:16.890 --> 01:10:25.030 align:middle line:90% 01:10:25.030 --> 01:10:32.190 align:middle line:84% So this is a one-way street for convergence 01:10:32.190 --> 01:10:33.990 align:middle line:84% and the convergence of the absolute values. 01:10:33.990 --> 01:10:44.350 align:middle line:90% 01:10:44.350 --> 01:10:45.920 align:middle line:84% So before I prove this theorem, let 01:10:45.920 --> 01:10:53.983 align:middle line:84% me just prove a quick inequality which 01:10:53.983 --> 01:10:56.150 align:middle line:84% I think I said I was going to put on the assignment, 01:10:56.150 --> 01:10:57.858 align:middle line:84% and then forgot to put on the assignment. 01:10:57.858 --> 01:10:59.600 align:middle line:90% But you should know it. 01:10:59.600 --> 01:11:07.400 align:middle line:84% So this theorem is the reverse triangle inequality 01:11:07.400 --> 01:11:12.710 align:middle line:84% which states for all a, b real number, the absolute value 01:11:12.710 --> 01:11:15.980 align:middle line:84% of the difference in absolute values is less than 01:11:15.980 --> 01:11:19.810 align:middle line:84% or equal to the absolute value of the difference. 01:11:19.810 --> 01:11:28.360 align:middle line:84% OK, so the proof of this reverse triangle inequality 01:11:28.360 --> 01:11:31.280 align:middle line:84% just uses the original triangle inequality. 01:11:31.280 --> 01:11:39.060 align:middle line:84% So absolute value of a, this is equal to the absolute value 01:11:39.060 --> 01:11:42.320 align:middle line:90% of a minus b plus b. 01:11:42.320 --> 01:11:45.230 align:middle line:84% And this is less than or equal to the absolute value 01:11:45.230 --> 01:11:50.420 align:middle line:84% of a minus b plus absolute value of b. 01:11:50.420 --> 01:11:55.970 align:middle line:84% And thus the absolute value of a minus the absolute value of b 01:11:55.970 --> 01:12:00.350 align:middle line:84% is less than or equal to the absolute value of a minus b. 01:12:00.350 --> 01:12:04.040 align:middle line:84% Now, these are just two numbers, I mean two letters, 01:12:04.040 --> 01:12:04.925 align:middle line:90% reverse the letters. 01:12:04.925 --> 01:12:11.580 align:middle line:90% 01:12:11.580 --> 01:12:15.570 align:middle line:84% In this argument replace a with b and b with a. 01:12:15.570 --> 01:12:20.060 align:middle line:84% So then I get b minus the absolute value. 01:12:20.060 --> 01:12:22.290 align:middle line:84% The absolute value of b minus the absolute value of a 01:12:22.290 --> 01:12:28.860 align:middle line:84% is less than or equal to the absolute value of b minus a, 01:12:28.860 --> 01:12:30.000 align:middle line:90% which is the same as this. 01:12:30.000 --> 01:12:34.080 align:middle line:90% 01:12:34.080 --> 01:12:40.260 align:middle line:84% And therefore-- let me multiply through by minus 1, 01:12:40.260 --> 01:12:41.430 align:middle line:90% and that tells me-- 01:12:41.430 --> 01:12:47.650 align:middle line:90% 01:12:47.650 --> 01:12:51.550 align:middle line:84% OK, so I have this is less than or equal to the absolute value 01:12:51.550 --> 01:12:52.600 align:middle line:90% of a minus b. 01:12:52.600 --> 01:12:54.220 align:middle line:84% I also have it's bigger than or equal 01:12:54.220 --> 01:12:56.440 align:middle line:84% to minus the absolute value of a minus b. 01:12:56.440 --> 01:13:00.810 align:middle line:90% 01:13:00.810 --> 01:13:03.870 align:middle line:84% And therefore the absolute value of a minus 01:13:03.870 --> 01:13:10.470 align:middle line:84% the absolute value of b is less than or equal to absolute value 01:13:10.470 --> 01:13:11.370 align:middle line:90% of a minus b. 01:13:11.370 --> 01:13:17.680 align:middle line:90% 01:13:17.680 --> 01:13:19.840 align:middle line:84% So I it's taboo to write on the very back board, 01:13:19.840 --> 01:13:22.450 align:middle line:90% but I'm going to do it anyway. 01:13:22.450 --> 01:13:24.970 align:middle line:84% So that was the proof of the reverse triangle inequality. 01:13:24.970 --> 01:13:29.910 align:middle line:84% Here's the proof of the theorem before it. 01:13:29.910 --> 01:13:33.230 align:middle line:84% It just follows from the reverse triangle inequality 01:13:33.230 --> 01:13:38.050 align:middle line:84% and that combination of those two theorems over there. 01:13:38.050 --> 01:13:42.150 align:middle line:90% 01:13:42.150 --> 01:13:45.240 align:middle line:84% We have that the absolute value of x sub n 01:13:45.240 --> 01:13:48.090 align:middle line:90% minus the absolute value of x. 01:13:48.090 --> 01:13:50.760 align:middle line:84% This is less than or equal to, by the reverse triangle 01:13:50.760 --> 01:13:54.000 align:middle line:84% inequality, the absolute value of x sub n minus x. 01:13:54.000 --> 01:13:57.340 align:middle line:90% 01:13:57.340 --> 01:14:05.650 align:middle line:84% So this is by the reverse triangle inequality. 01:14:05.650 --> 01:14:10.390 align:middle line:84% And by assumption, this converges to 0 01:14:10.390 --> 01:14:11.770 align:middle line:90% as n goes to infinity. 01:14:11.770 --> 01:14:18.730 align:middle line:90% 01:14:18.730 --> 01:14:27.820 align:middle line:84% So by the squeeze theorem, this goes to 0. 01:14:27.820 --> 01:14:33.500 align:middle line:84% And therefore the absolute value of x sub n converges 01:14:33.500 --> 01:14:37.180 align:middle line:90% to the absolute value of x. 01:14:37.180 --> 01:14:44.470 align:middle line:84% And that's it for that proof for this lecture and for this week. 01:14:44.470 --> 01:14:52.110 align:middle line:90%