WEBVTT 00:00:00.000 --> 00:00:01.479 align:middle line:90% [SQUEAKING] 00:00:01.479 --> 00:00:02.465 align:middle line:90% 00:00:02.465 --> 00:00:03.451 align:middle line:90% [RUSTLING] 00:00:03.451 --> 00:00:04.437 align:middle line:90% 00:00:04.437 --> 00:00:05.423 align:middle line:90% [CLICKING] 00:00:05.423 --> 00:00:11.178 align:middle line:90% 00:00:11.178 --> 00:00:11.970 align:middle line:90% TOBIAS COLDING: OK. 00:00:11.970 --> 00:00:16.140 align:middle line:84% So last time, we talked about at towards the end, 00:00:16.140 --> 00:00:17.496 align:middle line:90% we talked about sequences. 00:00:17.496 --> 00:00:21.290 align:middle line:90% 00:00:21.290 --> 00:00:25.610 align:middle line:84% And so in particular, we talked about square root of 2 00:00:25.610 --> 00:00:27.740 align:middle line:90% by 2 square root of 2. 00:00:27.740 --> 00:00:30.890 align:middle line:84% We could think about this if we wanted 00:00:30.890 --> 00:00:34.710 align:middle line:84% to think about the square root of 2 on the real line. 00:00:34.710 --> 00:00:39.170 align:middle line:84% Then we have here 1 and we have two. 00:00:39.170 --> 00:00:44.780 align:middle line:84% And we could define a sequence of numbers where the first one 00:00:44.780 --> 00:00:55.260 align:middle line:84% is 1, the second one is 1.4, the third is a 1.4.1, 00:00:55.260 --> 00:01:00.650 align:middle line:84% and the next one is then the next digit in square root of 2, 00:01:00.650 --> 00:01:06.880 align:middle line:90% so 1.4.1.4, et cetera. 00:01:06.880 --> 00:01:14.890 align:middle line:84% And so this is like a sequence of numbers that are increasing. 00:01:14.890 --> 00:01:18.160 align:middle line:84% And it is square root of 2 where you're 00:01:18.160 --> 00:01:20.270 align:middle line:90% taking more and more digits. 00:01:20.270 --> 00:01:26.710 align:middle line:84% So square root of 2 is, in some sense, 00:01:26.710 --> 00:01:32.910 align:middle line:90% the limit of the sequence. 00:01:32.910 --> 00:01:38.800 align:middle line:90% 00:01:38.800 --> 00:01:43.123 align:middle line:84% And so precisely, a sequence is-- 00:01:43.123 --> 00:01:44.540 align:middle line:84% so we talked about that last time. 00:01:44.540 --> 00:01:47.060 align:middle line:84% But let me just remind you what that is. 00:01:47.060 --> 00:02:01.830 align:middle line:84% So a sequence is a function from the natural numbers-- 00:02:01.830 --> 00:02:07.250 align:middle line:84% so f goes from the natural numbers into the real. 00:02:07.250 --> 00:02:12.260 align:middle line:84% And the value at some natural number n-- 00:02:12.260 --> 00:02:15.830 align:middle line:84% the value of this function at a natural number n. 00:02:15.830 --> 00:02:19.670 align:middle line:84% This we denote-- usually, it's a standard notation 00:02:19.670 --> 00:02:24.170 align:middle line:84% that one denoted like this where the n is a subscript. 00:02:24.170 --> 00:02:24.670 align:middle line:90% OK. 00:02:24.670 --> 00:02:28.160 align:middle line:90% 00:02:28.160 --> 00:02:33.540 align:middle line:84% And so let's take a look at a couple of different sequences. 00:02:33.540 --> 00:02:37.160 align:middle line:84% So we talked-- so this here really is a sequence. 00:02:37.160 --> 00:02:46.310 align:middle line:84% Another example is where you have a n equal to minus 1 00:02:46.310 --> 00:02:48.740 align:middle line:90% to the power n. 00:02:48.740 --> 00:02:52.740 align:middle line:84% So n again is this natural number. 00:02:52.740 --> 00:02:57.580 align:middle line:84% So this means that this sequence a1 is equal to minus 1, 00:02:57.580 --> 00:03:02.740 align:middle line:84% a2 is equal to minus 1 times minus 1, so that's one. 00:03:02.740 --> 00:03:10.460 align:middle line:84% a3 is minus 1 times minus 1 times minus 1. 00:03:10.460 --> 00:03:13.220 align:middle line:90% So this is minus 1. 00:03:13.220 --> 00:03:20.620 align:middle line:84% And a4 is minus 1 multiplied itself four 4 So that's 1. 00:03:20.620 --> 00:03:21.200 align:middle line:90% Right. 00:03:21.200 --> 00:03:34.240 align:middle line:84% So this sequence here, it's alternating between the values 00:03:34.240 --> 00:03:37.630 align:middle line:90% minus 1 and 1. 00:03:37.630 --> 00:03:40.490 align:middle line:90% So it's starting off at minus 1. 00:03:40.490 --> 00:03:41.960 align:middle line:90% And then it goes to 1. 00:03:41.960 --> 00:03:46.570 align:middle line:84% Then it goes to minus 1, 1, minus 1, et cetera. 00:03:46.570 --> 00:03:47.941 align:middle line:90% OK. 00:03:47.941 --> 00:03:53.060 align:middle line:84% And so this sequence here, right, it doesn't really-- 00:03:53.060 --> 00:03:56.420 align:middle line:84% it's not-- it doesn't really convert. 00:03:56.420 --> 00:03:58.770 align:middle line:84% It's not-- it doesn't really have a limit. 00:03:58.770 --> 00:04:02.040 align:middle line:84% So this here-- we'll make this precise in just a minute. 00:04:02.040 --> 00:04:04.285 align:middle line:84% So this sequence does not have a limit. 00:04:04.285 --> 00:04:13.670 align:middle line:90% 00:04:13.670 --> 00:04:19.850 align:middle line:84% Another an example of a sequence is a decaying sequence 00:04:19.850 --> 00:04:20.519 align:middle line:90% like this. 00:04:20.519 --> 00:04:25.760 align:middle line:90% So a n is equal to 1 over n. 00:04:25.760 --> 00:04:34.460 align:middle line:84% So a1 is equal to 1, a2 is equal to 1/2, a3 is equal to 1/2, 00:04:34.460 --> 00:04:38.070 align:middle line:90% and then a4 is equal to 1/4. 00:04:38.070 --> 00:04:41.000 align:middle line:84% So these numbers here are all positive. 00:04:41.000 --> 00:04:41.520 align:middle line:90% Right? 00:04:41.520 --> 00:04:45.480 align:middle line:84% The values of a is all-- all of them are all positive. 00:04:45.480 --> 00:04:49.820 align:middle line:84% And they become smaller and smaller. 00:04:49.820 --> 00:04:53.350 align:middle line:84% So you could write that suggestively. 00:04:53.350 --> 00:04:57.190 align:middle line:84% This here is going down like that. 00:04:57.190 --> 00:05:03.940 align:middle line:84% And so it seems obvious that this limit here-- 00:05:03.940 --> 00:05:06.050 align:middle line:90% that this sequence has a limit. 00:05:06.050 --> 00:05:07.155 align:middle line:90% This here has a limit. 00:05:07.155 --> 00:05:10.960 align:middle line:90% 00:05:10.960 --> 00:05:13.590 align:middle line:90% And the limit is 0. 00:05:13.590 --> 00:05:20.530 align:middle line:90% 00:05:20.530 --> 00:05:25.280 align:middle line:84% So now let me make the formal definition. 00:05:25.280 --> 00:05:30.880 align:middle line:84% So what does it mean that a sequence is convergent? 00:05:30.880 --> 00:05:33.120 align:middle line:90% So a sequence. 00:05:33.120 --> 00:05:37.930 align:middle line:90% 00:05:37.930 --> 00:05:48.530 align:middle line:90% a n is convergent to a. 00:05:48.530 --> 00:05:50.515 align:middle line:90% So a is another real number. 00:05:50.515 --> 00:05:53.330 align:middle line:90% 00:05:53.330 --> 00:05:59.630 align:middle line:84% A sequence a n is convergent to a if the following is the case-- 00:05:59.630 --> 00:06:01.045 align:middle line:90% if the following holds. 00:06:01.045 --> 00:06:10.220 align:middle line:90% 00:06:10.220 --> 00:06:24.170 align:middle line:84% For all epsilon-- for all epsilon greater than 0, 00:06:24.170 --> 00:06:31.550 align:middle line:84% there exists a natural number such 00:06:31.550 --> 00:06:42.170 align:middle line:84% that if little n is bigger than this natural number, 00:06:42.170 --> 00:06:48.500 align:middle line:84% then the difference between a n and the supposed limit 00:06:48.500 --> 00:06:52.810 align:middle line:90% is bounded by epsilon. 00:06:52.810 --> 00:06:58.210 align:middle line:84% So if you think about it-- so if you think about it, 00:06:58.210 --> 00:07:01.920 align:middle line:84% here you have on the real line here you have a. 00:07:01.920 --> 00:07:06.940 align:middle line:84% And so if you give me an epsilon, 00:07:06.940 --> 00:07:09.010 align:middle line:90% so then I'm looking at a here. 00:07:09.010 --> 00:07:11.990 align:middle line:84% I'm looking at an interval around a. 00:07:11.990 --> 00:07:16.420 align:middle line:84% So this is an interval here from a minus 00:07:16.420 --> 00:07:20.320 align:middle line:90% epsilon to a plus epsilon. 00:07:20.320 --> 00:07:23.230 align:middle line:84% And the claim is that the sequence a n 00:07:23.230 --> 00:07:28.180 align:middle line:84% is converging to a if there exists 00:07:28.180 --> 00:07:33.680 align:middle line:84% some n so that if little n is larger than this capital N, 00:07:33.680 --> 00:07:38.350 align:middle line:84% then all of these a n lands somewhere here. 00:07:38.350 --> 00:07:46.630 align:middle line:84% All of them a n is in this little interval for little n 00:07:46.630 --> 00:07:49.140 align:middle line:84% bigger than this capital N. Right? 00:07:49.140 --> 00:07:52.340 align:middle line:84% That's what it means to be convergent. 00:07:52.340 --> 00:07:54.600 align:middle line:90% But it's like for all epsilon. 00:07:54.600 --> 00:07:59.790 align:middle line:84% So you allow to-- someone else is allowed to pick the epsilon. 00:07:59.790 --> 00:08:03.620 align:middle line:84% And then you have to prove that there exists a capital N so 00:08:03.620 --> 00:08:07.370 align:middle line:84% that from there on and out, all of the a n 00:08:07.370 --> 00:08:11.480 align:middle line:90% lie in this little interval. 00:08:11.480 --> 00:08:16.380 align:middle line:84% So this is what it means for a sequence to be convergent. 00:08:16.380 --> 00:08:23.370 align:middle line:84% So we say that a n is convergent to a if this is the case. 00:08:23.370 --> 00:08:26.030 align:middle line:90% And in that case-- 00:08:26.030 --> 00:08:34.789 align:middle line:84% in that case, one also writes that the limit 00:08:34.789 --> 00:08:39.459 align:middle line:84% as n goes to infinity of a n is equal to a. 00:08:39.459 --> 00:08:46.070 align:middle line:90% 00:08:46.070 --> 00:09:00.876 align:middle line:84% So if you have a sequence a sequence that is not convergent, 00:09:00.876 --> 00:09:02.990 align:middle line:90% it is said to be divergent. 00:09:02.990 --> 00:09:12.735 align:middle line:90% 00:09:12.735 --> 00:09:19.960 align:middle line:84% And so now let me look at an example. 00:09:19.960 --> 00:09:30.890 align:middle line:84% So suppose you're looking at the number 0.99999, et cetera. 00:09:30.890 --> 00:09:35.260 align:middle line:84% This number here, this is really-- 00:09:35.260 --> 00:09:40.120 align:middle line:84% this number here, we will think about this number here as 1. 00:09:40.120 --> 00:09:44.550 align:middle line:84% This is indistinguishable from 1. 00:09:44.550 --> 00:09:46.980 align:middle line:84% And in fact, what you really think 00:09:46.980 --> 00:09:53.790 align:middle line:84% about is that you think about that you have a sequence where 00:09:53.790 --> 00:10:08.400 align:middle line:84% a1 is 0.9, a2 is 0.99, a3 is 0.999, 00:10:08.400 --> 00:10:15.040 align:middle line:84% a4 is equal to 0.9999, et cetera. 00:10:15.040 --> 00:10:25.955 align:middle line:84% So a n is where you have zero point and then 9 n times. 00:10:25.955 --> 00:10:31.260 align:middle line:90% 00:10:31.260 --> 00:10:36.840 align:middle line:84% If you're looking at that sequence here, 00:10:36.840 --> 00:10:43.330 align:middle line:84% then you have that this sequence a n is convergent to 1. 00:10:43.330 --> 00:10:50.240 align:middle line:84% And a n-- this is one also that is convergent to 1 like this. 00:10:50.240 --> 00:10:57.810 align:middle line:90% So a n is convergent to 1. 00:10:57.810 --> 00:10:58.310 align:middle line:90% Right? 00:10:58.310 --> 00:11:01.160 align:middle line:84% So there's various ways of writing that it's convergent. 00:11:01.160 --> 00:11:03.770 align:middle line:90% You can write it like this. 00:11:03.770 --> 00:11:05.630 align:middle line:90% This means that it's convergent. 00:11:05.630 --> 00:11:07.190 align:middle line:90% And the limit is a. 00:11:07.190 --> 00:11:08.720 align:middle line:90% You can write it like this. 00:11:08.720 --> 00:11:13.010 align:middle line:84% This means that this sequence is convergent and the limit is 1. 00:11:13.010 --> 00:11:15.165 align:middle line:84% Or you can write it out in words. 00:11:15.165 --> 00:11:20.800 align:middle line:90% 00:11:20.800 --> 00:11:24.220 align:middle line:84% So this sequence-- I claim that this sequence here 00:11:24.220 --> 00:11:25.700 align:middle line:90% defined like this. 00:11:25.700 --> 00:11:28.990 align:middle line:84% And this is really how you think about this. 00:11:28.990 --> 00:11:30.470 align:middle line:84% When you think about this number, 00:11:30.470 --> 00:11:37.640 align:middle line:84% you think about it that this is 0.99 with more and more numbers, 00:11:37.640 --> 00:11:40.480 align:middle line:84% nines, added and the limits, which 00:11:40.480 --> 00:11:44.520 align:middle line:84% is really what this number here in itself is is 1. 00:11:44.520 --> 00:11:48.540 align:middle line:84% Now let me explain why the sequence defined 00:11:48.540 --> 00:11:53.255 align:middle line:84% this way, why this is convergent as a limit 1. 00:11:53.255 --> 00:12:02.730 align:middle line:90% 00:12:02.730 --> 00:12:07.840 align:middle line:84% So now, of course, now you give me an epsilon. 00:12:07.840 --> 00:12:13.530 align:middle line:84% So given epsilon given epsilon greater than 0, 00:12:13.530 --> 00:12:18.000 align:middle line:84% then I need to find a capital N So that from there 00:12:18.000 --> 00:12:21.970 align:middle line:84% on and out the An a n within epsilon. 00:12:21.970 --> 00:12:23.255 align:middle line:90% So we need to find-- 00:12:23.255 --> 00:12:26.700 align:middle line:90% 00:12:26.700 --> 00:12:31.620 align:middle line:84% for this epsilon, we need to find capital N 00:12:31.620 --> 00:12:35.020 align:middle line:84% so that if little n is bigger than this capital N, 00:12:35.020 --> 00:12:40.270 align:middle line:84% then a n minus 1 is less than epsilon. 00:12:40.270 --> 00:12:42.100 align:middle line:90% That's what we need to do. 00:12:42.100 --> 00:12:43.960 align:middle line:90% Right? 00:12:43.960 --> 00:12:51.100 align:middle line:84% But for your epsilon, so to do that, 00:12:51.100 --> 00:12:54.400 align:middle line:84% we need to choose this capital N. 00:12:54.400 --> 00:12:57.640 align:middle line:90% So I'm just choosing-- 00:12:57.640 --> 00:13:00.310 align:middle line:84% I just choose capital N. So to do 00:13:00.310 --> 00:13:16.660 align:middle line:84% that, choose capital N such that 1 over n is less than epsilon. 00:13:16.660 --> 00:13:17.530 align:middle line:90% Right? 00:13:17.530 --> 00:13:21.260 align:middle line:84% We can do this by the Archimedean property. 00:13:21.260 --> 00:13:37.870 align:middle line:84% So this is-- so we can find such an m and capital 00:13:37.870 --> 00:13:43.160 align:middle line:84% N by the Archimedean property that we discussed last time. 00:13:43.160 --> 00:13:48.990 align:middle line:90% 00:13:48.990 --> 00:13:56.880 align:middle line:84% But now you see if you're looking at-- 00:13:56.880 --> 00:14:07.770 align:middle line:84% if take-- if you now look at little n that 00:14:07.770 --> 00:14:16.500 align:middle line:84% is bigger than capital N, then you have that 1 minus a n. 00:14:16.500 --> 00:14:19.320 align:middle line:84% Well, 1 minus a n, actually, for all n, 00:14:19.320 --> 00:14:29.940 align:middle line:84% this here is less than 1/10 to the power n because if n is 00:14:29.940 --> 00:14:36.820 align:middle line:84% equal to 1, this here, this is within 1/10 of one. 00:14:36.820 --> 00:14:40.940 align:middle line:84% And likewise, this here is in 1/100 of 1, et cetera. 00:14:40.940 --> 00:14:43.820 align:middle line:90% So we have this inequality. 00:14:43.820 --> 00:14:46.010 align:middle line:84% It's actually equality in this case. 00:14:46.010 --> 00:14:50.260 align:middle line:90% This here is an equality. 00:14:50.260 --> 00:14:54.170 align:middle line:84% But now you see that this thing here, 00:14:54.170 --> 00:15:00.310 align:middle line:84% this is obviously 1/10 to the power n is less than or equal 00:15:00.310 --> 00:15:01.930 align:middle line:90% to-- 00:15:01.930 --> 00:15:06.160 align:middle line:90% is less than 1 over n. 00:15:06.160 --> 00:15:09.790 align:middle line:84% And so if little n is bigger than capital N, 00:15:09.790 --> 00:15:13.490 align:middle line:84% this is therefore less than 1 over capital N. 00:15:13.490 --> 00:15:15.940 align:middle line:90% But this is less than epsilon. 00:15:15.940 --> 00:15:18.520 align:middle line:84% And that's what we need to prove. 00:15:18.520 --> 00:15:19.660 align:middle line:90% Right? 00:15:19.660 --> 00:15:23.200 align:middle line:84% So for the given, for your given epsilon, 00:15:23.200 --> 00:15:27.160 align:middle line:84% I specify a capital N. And then we 00:15:27.160 --> 00:15:30.850 align:middle line:84% need to prove that if little n is bigger than our capital N, 00:15:30.850 --> 00:15:34.135 align:middle line:84% then we come within epsilon of the limit. 00:15:34.135 --> 00:15:34.635 align:middle line:90% OK? 00:15:34.635 --> 00:15:41.550 align:middle line:90% 00:15:41.550 --> 00:15:45.390 align:middle line:84% Now let me prove a little theorem 00:15:45.390 --> 00:15:49.166 align:middle line:90% about convergence sequences. 00:15:49.166 --> 00:15:56.340 align:middle line:90% 00:15:56.340 --> 00:16:01.110 align:middle line:84% And this is like, if you will, this little Lemma 00:16:01.110 --> 00:16:06.120 align:middle line:84% is like the first test for whether a sequence is 00:16:06.120 --> 00:16:07.560 align:middle line:90% convergent. 00:16:07.560 --> 00:16:08.253 align:middle line:90% So theorem. 00:16:08.253 --> 00:16:13.620 align:middle line:90% 00:16:13.620 --> 00:16:37.990 align:middle line:84% If a n is a convergent sequence, then a n is bounded. 00:16:37.990 --> 00:16:38.710 align:middle line:90% Sorry. 00:16:38.710 --> 00:16:40.370 align:middle line:84% Let me explain what I mean by that. 00:16:40.370 --> 00:16:46.700 align:middle line:84% I mean that then the set of a n-- so let me just explain that. 00:16:46.700 --> 00:17:00.970 align:middle line:84% So then the set a1, a2, a3, et cetera, this set here 00:17:00.970 --> 00:17:01.785 align:middle line:90% is bounded. 00:17:01.785 --> 00:17:07.240 align:middle line:90% 00:17:07.240 --> 00:17:13.670 align:middle line:84% It's not enough to be bounded to be convergent 00:17:13.670 --> 00:17:16.060 align:middle line:90% but it is a necessary criterion. 00:17:16.060 --> 00:17:21.280 align:middle line:90% 00:17:21.280 --> 00:17:23.980 align:middle line:84% Let me, just as a side remark, say 00:17:23.980 --> 00:17:27.970 align:middle line:84% that if you're looking at the sequence a n 00:17:27.970 --> 00:17:30.260 align:middle line:84% that we looked at just a minute ago, 00:17:30.260 --> 00:17:34.770 align:middle line:84% this here that is alternating between minus 1 and 1, 00:17:34.770 --> 00:17:37.740 align:middle line:84% this is a bounded sequence because the value-- there's 00:17:37.740 --> 00:17:38.740 align:middle line:90% only two values. 00:17:38.740 --> 00:17:42.250 align:middle line:84% So it's obviously bounded but it's not convergent. 00:17:42.250 --> 00:17:47.700 align:middle line:84% But it is a necessary for a sequence to be convergent. 00:17:47.700 --> 00:17:55.890 align:middle line:84% It's necessary that the sequence of numbers here is bounded. 00:17:55.890 --> 00:18:00.138 align:middle line:90% 00:18:00.138 --> 00:18:02.015 align:middle line:90% So now let's try to prove that. 00:18:02.015 --> 00:18:06.870 align:middle line:90% 00:18:06.870 --> 00:18:17.730 align:middle line:84% So we have-- so since a n is convergent, 00:18:17.730 --> 00:18:23.730 align:middle line:84% then there is some a so that a n converges to this a. 00:18:23.730 --> 00:18:26.370 align:middle line:90% Right. 00:18:26.370 --> 00:18:31.950 align:middle line:84% And so now I can just focus on one single epsilon. 00:18:31.950 --> 00:18:53.780 align:middle line:84% So in particular, if we choose epsilon to be equal to 1, 00:18:53.780 --> 00:18:59.720 align:middle line:84% then there exist capital N so that if little n is bigger 00:18:59.720 --> 00:19:08.460 align:middle line:84% than this capital N, then a little n minus a is less than 1. 00:19:08.460 --> 00:19:12.200 align:middle line:84% This is there is convergence just applied 00:19:12.200 --> 00:19:14.070 align:middle line:90% to this single epsilon. 00:19:14.070 --> 00:19:15.795 align:middle line:90% Right? 00:19:15.795 --> 00:19:16.295 align:middle line:90% OK. 00:19:16.295 --> 00:19:21.120 align:middle line:90% 00:19:21.120 --> 00:19:23.130 align:middle line:84% So this is the first observation. 00:19:23.130 --> 00:19:27.590 align:middle line:84% The second observation is that if we now look at all the 00:19:27.590 --> 00:19:31.060 align:middle line:84% a n that are not considered here-- 00:19:31.060 --> 00:19:38.010 align:middle line:84% so if you're looking at the set, this 00:19:38.010 --> 00:19:46.130 align:middle line:84% is the first a capital N minus 1 a n. 00:19:46.130 --> 00:19:46.630 align:middle line:90% Right. 00:19:46.630 --> 00:19:49.530 align:middle line:84% Those are the ones that doesn't fall in under, 00:19:49.530 --> 00:19:53.470 align:middle line:84% that this little n is bigger than the capital N. 00:19:53.470 --> 00:19:55.685 align:middle line:90% So this set here is finite. 00:19:55.685 --> 00:19:58.920 align:middle line:90% 00:19:58.920 --> 00:20:02.640 align:middle line:84% So obviously, so for any finite set, 00:20:02.640 --> 00:20:19.110 align:middle line:84% you have that so they exist c, a real number such that a n is 00:20:19.110 --> 00:20:23.020 align:middle line:84% less than or equal to c for any of these guys, 00:20:23.020 --> 00:20:33.100 align:middle line:84% for any a in this, any a n in this finite set. 00:20:33.100 --> 00:20:43.170 align:middle line:90% 00:20:43.170 --> 00:20:45.770 align:middle line:84% In other words, this means that a n 00:20:45.770 --> 00:20:53.485 align:middle line:84% is less than or equal to c if n here is between 1 and n minus 1. 00:20:53.485 --> 00:20:57.924 align:middle line:90% 00:20:57.924 --> 00:21:02.330 align:middle line:90% So now I look at-- 00:21:02.330 --> 00:21:05.240 align:middle line:90% so now I look at-- 00:21:05.240 --> 00:21:12.860 align:middle line:84% now I want to look at the numbers a n and this number 00:21:12.860 --> 00:21:14.690 align:middle line:90% here. 00:21:14.690 --> 00:21:18.500 align:middle line:84% And I want to look at-- so I'm thinking 00:21:18.500 --> 00:21:27.300 align:middle line:84% about the set a n absolute value and where in here 00:21:27.300 --> 00:21:33.990 align:middle line:90% is one of two, 1, 2, cetera. 00:21:33.990 --> 00:21:37.650 align:middle line:84% Well, I claim that this thing here, I 00:21:37.650 --> 00:21:40.300 align:middle line:84% can now prove that this set here is bounded. 00:21:40.300 --> 00:21:46.230 align:middle line:84% And this is just because that the first n minus 1, those 00:21:46.230 --> 00:21:48.030 align:middle line:90% are bounded by c. 00:21:48.030 --> 00:21:50.410 align:middle line:90% So I claim that this set here-- 00:21:50.410 --> 00:21:51.410 align:middle line:90% so claim. 00:21:51.410 --> 00:21:54.750 align:middle line:90% 00:21:54.750 --> 00:22:11.490 align:middle line:84% So that's the sup of this set where n from 1 right over all 00:22:11.490 --> 00:22:13.080 align:middle line:90% n's. 00:22:13.080 --> 00:22:22.980 align:middle line:84% The sup of this set is less than or equal to the max 00:22:22.980 --> 00:22:29.100 align:middle line:84% of c and a plus normal, absolute value of a plus 1. 00:22:29.100 --> 00:22:31.790 align:middle line:90% So I claim that. 00:22:31.790 --> 00:22:37.580 align:middle line:84% Now this is clearly, if you're taking the first n minus 1, 00:22:37.580 --> 00:22:42.750 align:middle line:84% then we already know that they are bounded by c. 00:22:42.750 --> 00:22:45.410 align:middle line:84% So you just need to worry about the one 00:22:45.410 --> 00:22:48.380 align:middle line:90% starting at capital N. Right? 00:22:48.380 --> 00:22:53.160 align:middle line:84% So we only need to worry about when. 00:22:53.160 --> 00:22:58.050 align:middle line:84% So this is obviously the case for the first capital N minus 1. 00:22:58.050 --> 00:23:00.620 align:middle line:90% So we only need to worry about-- 00:23:00.620 --> 00:23:15.170 align:middle line:84% only need to show this for little n 00:23:15.170 --> 00:23:17.870 align:middle line:90% larger than capital N. Larger. 00:23:17.870 --> 00:23:18.785 align:middle line:90% Yeah. 00:23:18.785 --> 00:23:20.870 align:middle line:84% AUDIENCE: Can you explain in the max over there 00:23:20.870 --> 00:23:23.665 align:middle line:84% why there's the absolute value of a plus-- like, the plus 1? 00:23:23.665 --> 00:23:24.540 align:middle line:90% TOBIAS COLDING: Yeah. 00:23:24.540 --> 00:23:25.590 align:middle line:90% I will explain this. 00:23:25.590 --> 00:23:27.450 align:middle line:90% And I'll get to this. 00:23:27.450 --> 00:23:28.690 align:middle line:90% This is part of the proof. 00:23:28.690 --> 00:23:29.190 align:middle line:90% Yeah. 00:23:29.190 --> 00:23:31.900 align:middle line:84% So it hopefully will be clear in a minute. 00:23:31.900 --> 00:23:34.500 align:middle line:84% If it's not, you should complain. 00:23:34.500 --> 00:23:40.330 align:middle line:84% So I only need to prove that when I'm just looking at the set 00:23:40.330 --> 00:23:40.830 align:middle line:90% here. 00:23:40.830 --> 00:23:43.800 align:middle line:84% And I'm starting at capital N, then 00:23:43.800 --> 00:23:47.460 align:middle line:90% the sup is bounded like that. 00:23:47.460 --> 00:23:49.590 align:middle line:90% And so then I-- 00:23:49.590 --> 00:23:53.910 align:middle line:90% so I'm looking at a n. 00:23:53.910 --> 00:23:56.770 align:middle line:90% But a n I can write as a n. 00:23:56.770 --> 00:24:00.120 align:middle line:84% I can write in a stupid way I can write 00:24:00.120 --> 00:24:06.170 align:middle line:90% a n as a n minus a plus a. 00:24:06.170 --> 00:24:06.670 align:middle line:90% Right. 00:24:06.670 --> 00:24:07.880 align:middle line:90% So I can write it like that. 00:24:07.880 --> 00:24:10.548 align:middle line:90% 00:24:10.548 --> 00:24:12.770 align:middle line:90% And then I can just-- 00:24:12.770 --> 00:24:13.270 align:middle line:90% right. 00:24:13.270 --> 00:24:15.460 align:middle line:90% So I have the sum of two things. 00:24:15.460 --> 00:24:20.630 align:middle line:84% The absolute value is less than the absolute value-- 00:24:20.630 --> 00:24:23.990 align:middle line:90% the sum of the absolute values. 00:24:23.990 --> 00:24:25.520 align:middle line:90% So I have this. 00:24:25.520 --> 00:24:26.840 align:middle line:90% Right. 00:24:26.840 --> 00:24:30.500 align:middle line:84% But now you see that since little n is bigger 00:24:30.500 --> 00:24:33.200 align:middle line:84% than this capital N, this capital N 00:24:33.200 --> 00:24:36.530 align:middle line:84% was chosen so that this here was bounded by 1. 00:24:36.530 --> 00:24:39.690 align:middle line:84% So this thing here is less than 1. 00:24:39.690 --> 00:24:43.760 align:middle line:84% And this here is less than the absolute value of a. 00:24:43.760 --> 00:24:45.350 align:middle line:90% Right. 00:24:45.350 --> 00:24:50.190 align:middle line:84% And so we have for the whole tail starting at capital N, 00:24:50.190 --> 00:24:53.810 align:middle line:84% we have that this thing here is bounded by 1 00:24:53.810 --> 00:24:56.130 align:middle line:90% plus the absolute value of a. 00:24:56.130 --> 00:25:03.240 align:middle line:84% And for the remaining capital N minus 1, those are bounded by C. 00:25:03.240 --> 00:25:09.110 align:middle line:84% And so together, all of this is bounded by the maximum 00:25:09.110 --> 00:25:12.170 align:middle line:90% of these two things. 00:25:12.170 --> 00:25:14.040 align:middle line:90% OK. 00:25:14.040 --> 00:25:16.100 align:middle line:84% Now the next thing I want to talk about 00:25:16.100 --> 00:25:22.470 align:middle line:84% is some sort of algebraic property of limits 00:25:22.470 --> 00:25:24.140 align:middle line:90% that is often very useful. 00:25:24.140 --> 00:25:26.940 align:middle line:90% 00:25:26.940 --> 00:25:29.555 align:middle line:84% And I will prove-- there's four of these. 00:25:29.555 --> 00:25:32.070 align:middle line:90% 00:25:32.070 --> 00:25:34.810 align:middle line:90% And I will prove two of them. 00:25:34.810 --> 00:25:41.165 align:middle line:84% And the other one I will outline the proof of. 00:25:41.165 --> 00:25:43.595 align:middle line:90% 00:25:43.595 --> 00:25:44.970 align:middle line:84% AUDIENCE: [INAUDIBLE], professor. 00:25:44.970 --> 00:25:45.803 align:middle line:90% TOBIAS COLDING: Yep. 00:25:45.803 --> 00:25:50.580 align:middle line:84% AUDIENCE: So in that the claim was that the sup is-- 00:25:50.580 --> 00:25:51.670 align:middle line:90% the sup is less than. 00:25:51.670 --> 00:25:53.010 align:middle line:90% So we don't need to-- 00:25:53.010 --> 00:25:53.370 align:middle line:90% TOBIAS COLDING: Yeah. 00:25:53.370 --> 00:25:53.580 align:middle line:90% No. 00:25:53.580 --> 00:25:55.340 align:middle line:84% You don't need to actually find the sup. 00:25:55.340 --> 00:25:55.840 align:middle line:90% Right? 00:25:55.840 --> 00:25:56.950 align:middle line:90% Yeah. 00:25:56.950 --> 00:26:01.050 align:middle line:84% And the reason I want to write sup and not maximum 00:26:01.050 --> 00:26:04.080 align:middle line:84% is that, of course, if it's infinite numbers, 00:26:04.080 --> 00:26:06.850 align:middle line:84% it's not clear there is a maximum. 00:26:06.850 --> 00:26:11.340 align:middle line:84% So we need to use the terminology sup. 00:26:11.340 --> 00:26:15.780 align:middle line:84% So that max of C and S a plus 1 is not the sup. 00:26:15.780 --> 00:26:17.320 align:middle line:90% No no that's right. 00:26:17.320 --> 00:26:19.260 align:middle line:90% It's a bound for the sup. 00:26:19.260 --> 00:26:22.923 align:middle line:90% But the sup presumably is less. 00:26:22.923 --> 00:26:24.840 align:middle line:84% AUDIENCE: And one that really is just epsilon. 00:26:24.840 --> 00:26:27.390 align:middle line:84% TOBIAS COLDING: And the one really, to some extent, 00:26:27.390 --> 00:26:28.800 align:middle line:90% could be almost anything. 00:26:28.800 --> 00:26:29.300 align:middle line:90% Yeah. 00:26:29.300 --> 00:26:31.380 align:middle line:90% It came from the epsilon. 00:26:31.380 --> 00:26:34.820 align:middle line:84% But you see if you made that epsilon smaller 00:26:34.820 --> 00:26:40.500 align:middle line:84% then probably the c would jump up because that is-- right. 00:26:40.500 --> 00:26:44.310 align:middle line:84% Because the c was the maximum of the first n minus 1. 00:26:44.310 --> 00:26:46.220 align:middle line:84% So if you are further out, then there 00:26:46.220 --> 00:26:49.340 align:middle line:84% are more numbers that you're taking the maximum over. 00:26:49.340 --> 00:26:50.630 align:middle line:90% So anyway. 00:26:50.630 --> 00:26:53.150 align:middle line:90% Yeah. 00:26:53.150 --> 00:26:54.380 align:middle line:90% OK. 00:26:54.380 --> 00:26:58.106 align:middle line:84% So we want to prove some algebraic properties. 00:26:58.106 --> 00:27:08.390 align:middle line:90% 00:27:08.390 --> 00:27:09.115 align:middle line:90% Of limits. 00:27:09.115 --> 00:27:14.430 align:middle line:90% 00:27:14.430 --> 00:27:17.440 align:middle line:90% So the idea is that we take-- 00:27:17.440 --> 00:27:20.470 align:middle line:90% so we have four of these. 00:27:20.470 --> 00:27:24.260 align:middle line:84% And the first one is that you take a sequence. 00:27:24.260 --> 00:27:29.170 align:middle line:84% So maybe I'll just say throughout a n and bn 00:27:29.170 --> 00:27:30.306 align:middle line:90% are sequences. 00:27:30.306 --> 00:27:35.230 align:middle line:90% 00:27:35.230 --> 00:27:45.012 align:middle line:84% And a n is convergent to a and bn is convergent to b. 00:27:45.012 --> 00:27:48.130 align:middle line:90% 00:27:48.130 --> 00:27:50.950 align:middle line:84% The first one, the first property 00:27:50.950 --> 00:27:53.330 align:middle line:84% is that if you take a real number-- sorry. 00:27:53.330 --> 00:27:57.430 align:middle line:84% If you take a real number-- c here is a real number-- 00:27:57.430 --> 00:28:00.760 align:middle line:84% and you form a new sequence where 00:28:00.760 --> 00:28:05.770 align:middle line:84% CN is just this real number times a n 00:28:05.770 --> 00:28:10.060 align:middle line:84% so each cn is somehow almost like a n except 00:28:10.060 --> 00:28:13.700 align:middle line:84% that they're multiplied by this fixed constant c. 00:28:13.700 --> 00:28:21.200 align:middle line:84% So this here-- so the first one is that cn is also convergent. 00:28:21.200 --> 00:28:26.360 align:middle line:90% And the limit is just c times a. 00:28:26.360 --> 00:28:28.490 align:middle line:90% OK. 00:28:28.490 --> 00:28:35.210 align:middle line:84% The second one is that if you take-- 00:28:35.210 --> 00:28:38.450 align:middle line:84% if you define a sequence-- so this 00:28:38.450 --> 00:28:40.470 align:middle line:84% is a different sequence than this one. 00:28:40.470 --> 00:28:46.670 align:middle line:84% But if you define a sequence cn by its a n, the n-th element 00:28:46.670 --> 00:28:51.050 align:middle line:84% is a n plus bn, then this sequence 00:28:51.050 --> 00:28:54.710 align:middle line:84% is also convergent. cn is convergent 00:28:54.710 --> 00:28:57.380 align:middle line:90% and the limit is a plus b. 00:28:57.380 --> 00:28:58.960 align:middle line:90% So that was the second property. 00:28:58.960 --> 00:29:12.320 align:middle line:90% 00:29:12.320 --> 00:29:14.041 align:middle line:90% The third property is-- 00:29:14.041 --> 00:29:19.790 align:middle line:90% 00:29:19.790 --> 00:29:22.220 align:middle line:84% the third property is that if you take a product. 00:29:22.220 --> 00:29:22.840 align:middle line:90% Sorry. 00:29:22.840 --> 00:29:25.030 align:middle line:90% So if you take a n-- 00:29:25.030 --> 00:29:31.100 align:middle line:84% if you define cn as a n times bn-- 00:29:31.100 --> 00:29:33.100 align:middle line:90% so this is a new sequence. 00:29:33.100 --> 00:29:40.870 align:middle line:84% Then cn is convergent and the limit is a times b. 00:29:40.870 --> 00:29:53.290 align:middle line:84% And the last property is that if the a n are-- 00:29:53.290 --> 00:29:56.980 align:middle line:84% all of these are different from 0. 00:29:56.980 --> 00:29:59.205 align:middle line:90% So this here is for all n. 00:29:59.205 --> 00:30:02.500 align:middle line:90% 00:30:02.500 --> 00:30:09.460 align:middle line:84% And also you have that the limit here is not equal to 0. 00:30:09.460 --> 00:30:17.300 align:middle line:84% Then if you define a sequence to be 1 over a n, 00:30:17.300 --> 00:30:22.310 align:middle line:84% then cn here converge and the limit is 1 over n. 00:30:22.310 --> 00:30:25.520 align:middle line:90% 00:30:25.520 --> 00:30:29.820 align:middle line:84% And so I will prove rigorously the first two. 00:30:29.820 --> 00:30:33.300 align:middle line:84% And I will basically tell you how to prove the other two. 00:30:33.300 --> 00:30:38.510 align:middle line:84% But the first two I will write out 00:30:38.510 --> 00:30:43.070 align:middle line:84% formally, as we talked about last time, how you write 00:30:43.070 --> 00:30:49.070 align:middle line:84% a formal proof, whereas what I'm going to discuss for three 00:30:49.070 --> 00:30:52.410 align:middle line:84% and four is very, very close to a formal proof 00:30:52.410 --> 00:30:53.870 align:middle line:84% but it's more like what you would 00:30:53.870 --> 00:30:55.525 align:middle line:90% have on a piece of scrap paper. 00:30:55.525 --> 00:31:10.660 align:middle line:90% 00:31:10.660 --> 00:31:12.480 align:middle line:84% So now let's prove the first one. 00:31:12.480 --> 00:31:15.130 align:middle line:90% 00:31:15.130 --> 00:31:16.620 align:middle line:90% So proof of one. 00:31:16.620 --> 00:31:30.940 align:middle line:90% 00:31:30.940 --> 00:31:35.145 align:middle line:84% So given epsilon greater than 0-- 00:31:35.145 --> 00:31:39.800 align:middle line:90% 00:31:39.800 --> 00:31:41.330 align:middle line:90% so yeah. 00:31:41.330 --> 00:31:45.850 align:middle line:84% So first of all, of course, if this capital C-- 00:31:45.850 --> 00:31:50.600 align:middle line:84% if this capital C is 0, then everything is 0. 00:31:50.600 --> 00:31:53.300 align:middle line:90% So it's trivially the case. 00:31:53.300 --> 00:31:55.570 align:middle line:90% So we may as well assume. 00:31:55.570 --> 00:31:58.790 align:middle line:84% So may as well assume that capital C-- 00:31:58.790 --> 00:32:03.700 align:middle line:90% 00:32:03.700 --> 00:32:05.265 align:middle line:90% that capital C is not 0. 00:32:05.265 --> 00:32:09.170 align:middle line:90% 00:32:09.170 --> 00:32:12.390 align:middle line:84% So now, given epsilon greater than 0, 00:32:12.390 --> 00:32:16.680 align:middle line:84% since this sequence a n is converging to a, 00:32:16.680 --> 00:32:26.790 align:middle line:84% this means that there exists n such that-- 00:32:26.790 --> 00:32:29.430 align:middle line:84% and now I'm not going to use that epsilon. 00:32:29.430 --> 00:32:31.640 align:middle line:90% You gave me that epsilon. 00:32:31.640 --> 00:32:35.600 align:middle line:84% I'm not going to use that epsilon in the definition 00:32:35.600 --> 00:32:38.190 align:middle line:90% that a n is converging to a. 00:32:38.190 --> 00:32:40.310 align:middle line:84% I'm not going to use that one, but I'm 00:32:40.310 --> 00:32:42.840 align:middle line:84% going to use one that is closely related. 00:32:42.840 --> 00:32:47.120 align:middle line:84% I'm going to use epsilon divided by the norm of z 00:32:47.120 --> 00:32:50.000 align:middle line:90% because that's another positive. 00:32:50.000 --> 00:32:53.250 align:middle line:90% So you gave me an epsilon. 00:32:53.250 --> 00:32:56.990 align:middle line:84% I'm going to look at this number here. 00:32:56.990 --> 00:32:59.630 align:middle line:90% That's a positive number. 00:32:59.630 --> 00:33:03.230 align:middle line:84% And I can use that in the definition 00:33:03.230 --> 00:33:05.610 align:middle line:90% that a n converges to a. 00:33:05.610 --> 00:33:09.680 align:middle line:84% So I'm going to say that because an converts to a, 00:33:09.680 --> 00:33:13.720 align:middle line:84% there exists a capital N such that if little n is bigger 00:33:13.720 --> 00:33:23.470 align:middle line:84% than this capital N, then a n minus a is less than epsilon 00:33:23.470 --> 00:33:26.950 align:middle line:90% over capital S-- 00:33:26.950 --> 00:33:31.940 align:middle line:84% or C, sorry, capital C, a normal capital C. 00:33:31.940 --> 00:33:35.540 align:middle line:84% But now I just multiply on both sides. 00:33:35.540 --> 00:33:46.540 align:middle line:84% So multiply by the absolute value of capital C 00:33:46.540 --> 00:33:47.610 align:middle line:90% on both sides. 00:33:47.610 --> 00:33:52.240 align:middle line:90% 00:33:52.240 --> 00:33:58.050 align:middle line:84% And so if we do that, then this here pops in. 00:33:58.050 --> 00:33:58.550 align:middle line:90% Right? 00:33:58.550 --> 00:34:04.880 align:middle line:84% It actually goes in as c times a n minus c times a 00:34:04.880 --> 00:34:07.880 align:middle line:90% is less than epsilon. 00:34:07.880 --> 00:34:11.090 align:middle line:84% And this is as long as this n is-- 00:34:11.090 --> 00:34:13.070 align:middle line:84% little n is bigger than that capital 00:34:13.070 --> 00:34:15.770 align:middle line:84% N. This is exactly what we wanted to prove. 00:34:15.770 --> 00:34:18.350 align:middle line:90% 00:34:18.350 --> 00:34:19.940 align:middle line:90% OK. 00:34:19.940 --> 00:34:23.253 align:middle line:90% Now the second one is-- 00:34:23.253 --> 00:34:33.080 align:middle line:90% 00:34:33.080 --> 00:34:39.050 align:middle line:84% so the reason why I'm doing the first two-- 00:34:39.050 --> 00:34:42.030 align:middle line:84% you see that those are pretty straightforward. 00:34:42.030 --> 00:34:47.929 align:middle line:84% I mean, it's not like any of these is complicated to prove. 00:34:47.929 --> 00:34:52.074 align:middle line:84% The next two is a little bit more-- 00:34:52.074 --> 00:34:54.199 align:middle line:84% I mean, I wouldn't say complicated but a little bit 00:34:54.199 --> 00:34:55.860 align:middle line:90% more involved, these two. 00:34:55.860 --> 00:34:58.490 align:middle line:84% But I want to give you the idea because it's 00:34:58.490 --> 00:35:04.390 align:middle line:84% kind of a very simple but useful idea in these two. 00:35:04.390 --> 00:35:06.920 align:middle line:84% So I don't want to skip the proof. 00:35:06.920 --> 00:35:11.050 align:middle line:90% 00:35:11.050 --> 00:35:17.040 align:middle line:84% So we want to approve of the second property. 00:35:17.040 --> 00:35:27.920 align:middle line:90% 00:35:27.920 --> 00:35:30.760 align:middle line:84% So we have that a n is converging 00:35:30.760 --> 00:35:34.660 align:middle line:90% to a and bn is converging to b. 00:35:34.660 --> 00:35:37.240 align:middle line:84% And so again, you give me an epsilon 00:35:37.240 --> 00:35:39.040 align:middle line:90% that I have to work with. 00:35:39.040 --> 00:35:46.840 align:middle line:84% So given epsilon-- given epsilon greater than zero, 00:35:46.840 --> 00:35:52.060 align:middle line:84% then I'm going to not again use this epsilon in the definition 00:35:52.060 --> 00:35:53.960 align:middle line:90% that a n converge to a. 00:35:53.960 --> 00:35:56.470 align:middle line:84% And I'm not going to use it in the definition 00:35:56.470 --> 00:36:01.920 align:middle line:84% that bn is converging to b, but I'm going to use epsilon over 2 00:36:01.920 --> 00:36:03.400 align:middle line:90% in each of them. 00:36:03.400 --> 00:36:17.310 align:middle line:84% So since a n converts to a, there exists a capital N-- 00:36:17.310 --> 00:36:21.560 align:middle line:84% and let me call it capital N subscript a-- 00:36:21.560 --> 00:36:29.310 align:middle line:84% such that if little n is bigger than this capital N 00:36:29.310 --> 00:36:38.835 align:middle line:84% subscript a, then a n minus a is less than epsilon over 2. 00:36:38.835 --> 00:36:39.335 align:middle line:90% Right? 00:36:39.335 --> 00:37:00.340 align:middle line:90% 00:37:00.340 --> 00:37:16.000 align:middle line:84% Likewise, since being converted to b, then 00:37:16.000 --> 00:37:21.170 align:middle line:84% there exists a capital N, but denoted by this, 00:37:21.170 --> 00:37:24.370 align:middle line:84% it may be different than the one coming from the sequence 00:37:24.370 --> 00:37:35.140 align:middle line:84% a n such that if little n is bigger than this capital N 00:37:35.140 --> 00:37:45.280 align:middle line:84% subscript b, then bn minus b is smaller than epsilon over 2. 00:37:45.280 --> 00:37:50.600 align:middle line:90% And so now I have to choose. 00:37:50.600 --> 00:37:52.060 align:middle line:90% So you gave me the epsilon. 00:37:52.060 --> 00:37:55.630 align:middle line:84% And I have to choose a capital N. I haven't chosen that yet 00:37:55.630 --> 00:37:58.750 align:middle line:90% but I will do that now. 00:37:58.750 --> 00:38:08.670 align:middle line:84% So set capital N equal to the maximum of these two N's. 00:38:08.670 --> 00:38:12.510 align:middle line:84% So you're setting it equal to the maximum of these two. 00:38:12.510 --> 00:38:15.000 align:middle line:90% Now if little n-- 00:38:15.000 --> 00:38:21.210 align:middle line:84% so then for little n bigger than this capital N, 00:38:21.210 --> 00:38:25.360 align:middle line:84% if you're looking at a-- so now we want to look at-- 00:38:25.360 --> 00:38:30.880 align:middle line:84% so our sequence cn was the sum of these two. 00:38:30.880 --> 00:38:33.160 align:middle line:90% so this here was cn. 00:38:33.160 --> 00:38:36.310 align:middle line:90% So we want to prove that-- 00:38:36.310 --> 00:38:40.150 align:middle line:84% so we want to look at cn minus a times b. 00:38:40.150 --> 00:38:43.890 align:middle line:84% And we want to prove that if little n is bigger than this, 00:38:43.890 --> 00:38:47.470 align:middle line:84% then this here is strictly less than epsilon. 00:38:47.470 --> 00:38:54.510 align:middle line:84% But this thing here is cn was, by definition, a n plus bn. 00:38:54.510 --> 00:38:55.110 align:middle line:90% Sorry. 00:38:55.110 --> 00:38:56.470 align:middle line:90% And this was not the product. 00:38:56.470 --> 00:38:57.115 align:middle line:90% It was the sum. 00:38:57.115 --> 00:39:00.613 align:middle line:90% 00:39:00.613 --> 00:39:05.740 align:middle line:90% Minus a plus b. 00:39:05.740 --> 00:39:09.190 align:middle line:84% But this thing here you can write 00:39:09.190 --> 00:39:16.620 align:middle line:90% as a n minus a plus bn minus b. 00:39:16.620 --> 00:39:19.540 align:middle line:90% 00:39:19.540 --> 00:39:22.465 align:middle line:84% And now you can just take the absolute value of this. 00:39:22.465 --> 00:39:24.465 align:middle line:84% It's bounded by the absolute value of the first. 00:39:24.465 --> 00:39:37.420 align:middle line:90% 00:39:37.420 --> 00:39:40.280 align:middle line:84% So plus the absolute value of this thing. 00:39:40.280 --> 00:39:45.400 align:middle line:90% 00:39:45.400 --> 00:39:54.310 align:middle line:84% And now you see that if little a-- 00:39:54.310 --> 00:39:59.150 align:middle line:90% let me keep-- so if little a-- 00:39:59.150 --> 00:39:59.650 align:middle line:90% sorry. 00:39:59.650 --> 00:40:05.070 align:middle line:84% If little n-- since little n is bigger than capital N, which 00:40:05.070 --> 00:40:09.310 align:middle line:84% is bigger than capital N of a, in particular, 00:40:09.310 --> 00:40:13.690 align:middle line:84% then this thing here is less than epsilon over 2. 00:40:13.690 --> 00:40:23.940 align:middle line:84% So you have that cn minus a plus b is less than epsilon 00:40:23.940 --> 00:40:26.280 align:middle line:90% over 2 that comes from this. 00:40:26.280 --> 00:40:34.440 align:middle line:84% But you also have that capital N is bigger or equal to n sub b. 00:40:34.440 --> 00:40:37.590 align:middle line:84% So you also have that this thing here is less than epsilon 00:40:37.590 --> 00:40:38.430 align:middle line:90% over 2. 00:40:38.430 --> 00:40:40.710 align:middle line:90% Right? 00:40:40.710 --> 00:40:44.550 align:middle line:84% So you have that this thing here with little a 00:40:44.550 --> 00:40:48.550 align:middle line:84% is big N-- is bigger than this capital N. This thing here, 00:40:48.550 --> 00:40:51.580 align:middle line:84% this difference is strictly less than epsilon. 00:40:51.580 --> 00:40:53.740 align:middle line:84% That's what we needed to prove in order 00:40:53.740 --> 00:40:56.380 align:middle line:84% to prove that the sequence that was the sum of the two 00:40:56.380 --> 00:40:59.660 align:middle line:84% sequences converging to the sum of the limits. 00:40:59.660 --> 00:41:00.580 align:middle line:90% Yeah. 00:41:00.580 --> 00:41:05.050 align:middle line:84% AUDIENCE: We initially defined the n minus [? e ?] or n minus a 00:41:05.050 --> 00:41:06.880 align:middle line:90% be less than epsilon over 2. 00:41:06.880 --> 00:41:08.800 align:middle line:84% What was the justification for why 00:41:08.800 --> 00:41:13.510 align:middle line:84% we would say if we know that the n converge to be within? 00:41:13.510 --> 00:41:14.530 align:middle line:90% TOBIAS COLDING: Right. 00:41:14.530 --> 00:41:15.170 align:middle line:90% That's right. 00:41:15.170 --> 00:41:21.230 align:middle line:84% So because a n or bn, because they converge. 00:41:21.230 --> 00:41:25.090 align:middle line:90% So let's say a n converges to a. 00:41:25.090 --> 00:41:25.820 align:middle line:90% Right. 00:41:25.820 --> 00:41:29.360 align:middle line:84% Then for all epsilon, you gave me the epsilon. 00:41:29.360 --> 00:41:31.960 align:middle line:84% And I'm using that it's converging 00:41:31.960 --> 00:41:35.420 align:middle line:84% not just for that choice of epsilon 00:41:35.420 --> 00:41:38.030 align:middle line:84% but also even for epsilon over 2. 00:41:38.030 --> 00:41:41.980 align:middle line:84% That epsilon, you gave me the epsilon. 00:41:41.980 --> 00:41:44.090 align:middle line:84% Then I'm using that it's converging. 00:41:44.090 --> 00:41:46.210 align:middle line:84% So I will have to choose probably 00:41:46.210 --> 00:41:49.780 align:middle line:84% an n that is much larger if I'm using epsilon over 2 00:41:49.780 --> 00:41:52.400 align:middle line:84% than if I use directly the epsilon. 00:41:52.400 --> 00:41:54.120 align:middle line:84% AUDIENCE: So I'd assume that it increases 00:41:54.120 --> 00:41:56.460 align:middle line:90% like epsilon gets smaller. 00:41:56.460 --> 00:41:58.450 align:middle line:84% TOBIAS COLDING: As epsilon gets smaller, 00:41:58.450 --> 00:42:02.078 align:middle line:84% presumably you need to choose capital N larger. 00:42:02.078 --> 00:42:02.620 align:middle line:90% AUDIENCE: OK. 00:42:02.620 --> 00:42:04.470 align:middle line:84% So you couldn't run into a case where 00:42:04.470 --> 00:42:08.220 align:middle line:84% we're assuming a convergence is like smooth as opposed 00:42:08.220 --> 00:42:10.273 align:middle line:90% to oscillating about a point? 00:42:10.273 --> 00:42:11.190 align:middle line:90% TOBIAS COLDING: Right. 00:42:11.190 --> 00:42:11.732 align:middle line:90% That's right. 00:42:11.732 --> 00:42:19.170 align:middle line:84% I mean, so-- but I'm using that for-- 00:42:19.170 --> 00:42:21.510 align:middle line:84% I'm basically using that for all epsilon, 00:42:21.510 --> 00:42:24.010 align:middle line:84% that it wasn't just for a fixed epsilon. 00:42:24.010 --> 00:42:28.170 align:middle line:84% And so that's why, when you came to this epsilon, then 00:42:28.170 --> 00:42:30.850 align:middle line:84% I'm actually using it for epsilon over 2. 00:42:30.850 --> 00:42:31.350 align:middle line:90% Yeah. 00:42:31.350 --> 00:42:33.900 align:middle line:90% 00:42:33.900 --> 00:42:34.570 align:middle line:90% OK. 00:42:34.570 --> 00:42:37.740 align:middle line:84% So that was the first two properties. 00:42:37.740 --> 00:42:42.060 align:middle line:84% It's not like the other one is super difficult. 00:42:42.060 --> 00:42:44.590 align:middle line:84% But those two are the most straightforward. 00:42:44.590 --> 00:42:47.430 align:middle line:90% Let's look at the other two. 00:42:47.430 --> 00:42:52.730 align:middle line:84% And so in particular, because it's like, for the next one, 00:42:52.730 --> 00:42:56.620 align:middle line:84% there's a little trick that is useful 00:42:56.620 --> 00:42:59.113 align:middle line:90% in various different settings. 00:42:59.113 --> 00:43:04.270 align:middle line:90% 00:43:04.270 --> 00:43:11.110 align:middle line:84% And so prove-- but the next two I will just outline. 00:43:11.110 --> 00:43:13.690 align:middle line:90% OK. 00:43:13.690 --> 00:43:21.060 align:middle line:90% So proof of 3 so outline. 00:43:21.060 --> 00:43:26.170 align:middle line:90% 00:43:26.170 --> 00:43:31.450 align:middle line:84% So we need to look at a n times bn. 00:43:31.450 --> 00:43:32.050 align:middle line:90% Right? 00:43:32.050 --> 00:43:35.030 align:middle line:90% So cn is a n times bn. 00:43:35.030 --> 00:43:40.870 align:middle line:84% And we want to compare cn with a times b. 00:43:40.870 --> 00:43:44.620 align:middle line:84% We want to show that this thing here, if little n is 00:43:44.620 --> 00:43:45.740 align:middle line:90% sufficiently large-- 00:43:45.740 --> 00:43:49.870 align:middle line:84% so given an epsilon, if little n is sufficiently large, 00:43:49.870 --> 00:43:54.000 align:middle line:84% then this difference is bounded by epsilon. 00:43:54.000 --> 00:44:01.650 align:middle line:84% But this thing here is anbn minus a b. 00:44:01.650 --> 00:44:03.690 align:middle line:90% OK. 00:44:03.690 --> 00:44:07.530 align:middle line:84% And so the little trick is now that, in itself, it's 00:44:07.530 --> 00:44:11.440 align:middle line:84% kind of difficult to compare these two things. 00:44:11.440 --> 00:44:17.040 align:middle line:84% But the little trick is that if you're looking at anbn, 00:44:17.040 --> 00:44:20.280 align:middle line:90% if you're sort of-- 00:44:20.280 --> 00:44:22.450 align:middle line:84% instead of comparing them directly, 00:44:22.450 --> 00:44:25.260 align:middle line:84% then comparing this with something that 00:44:25.260 --> 00:44:30.270 align:middle line:84% is halfway between this and that, then it gets a lot easier. 00:44:30.270 --> 00:44:34.470 align:middle line:84% So what you do is you're looking at anbn, 00:44:34.470 --> 00:44:39.930 align:middle line:84% and you subtract a and b, and then 00:44:39.930 --> 00:44:41.450 align:middle line:90% you add it immediately again. 00:44:41.450 --> 00:44:45.000 align:middle line:90% 00:44:45.000 --> 00:44:47.900 align:middle line:90% So this is certainly the same. 00:44:47.900 --> 00:44:53.840 align:middle line:84% And now I can write this thing here as a n. 00:44:53.840 --> 00:44:57.030 align:middle line:84% I can just concentrate on these two first. 00:44:57.030 --> 00:44:59.900 align:middle line:90% And then I can factor out a n. 00:44:59.900 --> 00:45:01.630 align:middle line:90% So I can write that as this. 00:45:01.630 --> 00:45:04.940 align:middle line:90% 00:45:04.940 --> 00:45:08.460 align:middle line:84% And this thing here I can factor out a b. 00:45:08.460 --> 00:45:12.410 align:middle line:90% 00:45:12.410 --> 00:45:16.340 align:middle line:90% So I can write it like that. 00:45:16.340 --> 00:45:19.340 align:middle line:90% And now you see this. 00:45:19.340 --> 00:45:21.030 align:middle line:90% It looks much more promising. 00:45:21.030 --> 00:45:23.250 align:middle line:84% This one looks like the problematic one. 00:45:23.250 --> 00:45:25.790 align:middle line:84% This one looks like no problem at all 00:45:25.790 --> 00:45:29.310 align:middle line:84% because this is just some fixed constant times 00:45:29.310 --> 00:45:30.800 align:middle line:90% this difference here. 00:45:30.800 --> 00:45:32.900 align:middle line:84% This one looks a little bit more problematic 00:45:32.900 --> 00:45:34.320 align:middle line:84% because it's not a fixed constant. 00:45:34.320 --> 00:45:36.780 align:middle line:84% But we can deal with this in just a second. 00:45:36.780 --> 00:45:39.230 align:middle line:84% But the first thing I can do is I just take 00:45:39.230 --> 00:45:48.590 align:middle line:84% the absolute value of this plus the absolute value of that. 00:45:48.590 --> 00:45:52.830 align:middle line:90% 00:45:52.830 --> 00:45:55.000 align:middle line:90% And now I just take-- 00:45:55.000 --> 00:46:00.190 align:middle line:84% so then this here is actually the same for a product. 00:46:00.190 --> 00:46:04.717 align:middle line:84% You can just take the product of the absolute value. 00:46:04.717 --> 00:46:05.300 align:middle line:90% It's the same. 00:46:05.300 --> 00:46:17.540 align:middle line:90% 00:46:17.540 --> 00:46:24.060 align:middle line:84% Now the next thing is that we're going 00:46:24.060 --> 00:46:32.730 align:middle line:84% to use the theorem that we proved earlier 00:46:32.730 --> 00:46:41.850 align:middle line:90% that since this sequence here-- 00:46:41.850 --> 00:46:47.090 align:middle line:84% since these numbers here, all of these numbers for all, 00:46:47.090 --> 00:46:48.310 align:middle line:90% when you take the set-- 00:46:48.310 --> 00:46:51.830 align:middle line:90% 00:46:51.830 --> 00:46:58.233 align:middle line:84% when you're looking at this set, since this set here is bounded-- 00:46:58.233 --> 00:47:01.190 align:middle line:90% 00:47:01.190 --> 00:47:04.890 align:middle line:84% and it's bounded because the sequence was convergent. 00:47:04.890 --> 00:47:09.350 align:middle line:84% And so we already proved that then the values, the set 00:47:09.350 --> 00:47:11.590 align:middle line:90% of values is a bounded set. 00:47:11.590 --> 00:47:12.090 align:middle line:90% Right. 00:47:12.090 --> 00:47:13.620 align:middle line:90% So it's bounded. 00:47:13.620 --> 00:47:15.500 align:middle line:90% And so we can assume-- 00:47:15.500 --> 00:47:33.080 align:middle line:84% so that exists C. So that's a n is the absolute value is bounded 00:47:33.080 --> 00:47:34.095 align:middle line:90% by the c. 00:47:34.095 --> 00:47:38.870 align:middle line:90% 00:47:38.870 --> 00:47:40.025 align:middle line:90% So now I have-- 00:47:40.025 --> 00:47:42.790 align:middle line:90% 00:47:42.790 --> 00:47:51.120 align:middle line:84% and so now I have that cn minus a times b. 00:47:51.120 --> 00:47:53.670 align:middle line:90% This is what I had up there. 00:47:53.670 --> 00:47:55.300 align:middle line:90% And I'm just using that. 00:47:55.300 --> 00:47:58.110 align:middle line:84% This is now bounded by just because these things here 00:47:58.110 --> 00:48:10.650 align:middle line:84% are bounded by c, I get cbn minus b plus b over a n minus a. 00:48:10.650 --> 00:48:12.750 align:middle line:90% Right. 00:48:12.750 --> 00:48:15.910 align:middle line:84% And now you see now it's going to follow like-- 00:48:15.910 --> 00:48:18.660 align:middle line:84% so this here was just some observations. 00:48:18.660 --> 00:48:19.210 align:middle line:90% Right? 00:48:19.210 --> 00:48:21.280 align:middle line:84% And now when you come to the proof, 00:48:21.280 --> 00:48:34.980 align:middle line:84% so to prove three, then given epsilon greater than 0, 00:48:34.980 --> 00:48:38.130 align:middle line:84% now we want to make sure that this thing here 00:48:38.130 --> 00:48:41.960 align:middle line:84% is less than epsilon over 2 and that this thing here 00:48:41.960 --> 00:48:44.090 align:middle line:90% is less than epsilon over 2. 00:48:44.090 --> 00:48:49.790 align:middle line:84% And so I don't want to get in trouble if this set was 0 00:48:49.790 --> 00:48:51.510 align:middle line:90% or if this set was 0. 00:48:51.510 --> 00:49:00.590 align:middle line:84% So I'm just-- so given epsilon since a n converges to a, then I 00:49:00.590 --> 00:49:04.820 align:middle line:90% can find-- 00:49:04.820 --> 00:49:06.840 align:middle line:84% now I'm basically writing out the whole proof, 00:49:06.840 --> 00:49:10.610 align:middle line:90% but I can find capital An-- 00:49:10.610 --> 00:49:15.290 align:middle line:84% sorry-- capital Na so that if little n is bigger than capital 00:49:15.290 --> 00:49:24.270 align:middle line:84% Na, then you have that a n minus a is less than-- 00:49:24.270 --> 00:49:26.520 align:middle line:90% I don't quite use that epsilon. 00:49:26.520 --> 00:49:29.160 align:middle line:84% And I don't quite use epsilon over 2, 00:49:29.160 --> 00:49:32.460 align:middle line:90% but I take absolute value. 00:49:32.460 --> 00:49:37.820 align:middle line:84% I multiply it like this because I don't want to have something 00:49:37.820 --> 00:49:40.770 align:middle line:84% where if b was was zero, then it really 00:49:40.770 --> 00:49:42.490 align:middle line:84% didn't make any sense what I wrote. 00:49:42.490 --> 00:49:44.050 align:middle line:90% So now there's no problem. 00:49:44.050 --> 00:49:46.200 align:middle line:84% It definitely makes sense even if b is 0. 00:49:46.200 --> 00:50:06.450 align:middle line:90% 00:50:06.450 --> 00:50:14.525 align:middle line:84% Likewise since bn converges to b, then there exists-- 00:50:14.525 --> 00:50:17.460 align:middle line:90% 00:50:17.460 --> 00:50:20.560 align:middle line:84% so again, you gave me an epsilon. 00:50:20.560 --> 00:50:31.080 align:middle line:84% So there exists an nb such that if little n is bigger than nb, 00:50:31.080 --> 00:50:35.730 align:middle line:84% then actually you have that bn minus 00:50:35.730 --> 00:50:40.260 align:middle line:84% b is less than epsilon again over 2. 00:50:40.260 --> 00:50:42.380 align:middle line:84% And this is not quite what I use because I 00:50:42.380 --> 00:50:44.450 align:middle line:84% don't want to be in trouble that c-- 00:50:44.450 --> 00:50:47.240 align:middle line:84% c is already non-negative because it's bigger 00:50:47.240 --> 00:50:48.600 align:middle line:90% than this absolute value. 00:50:48.600 --> 00:50:52.370 align:middle line:84% But I'm just adding one for the same reason. 00:50:52.370 --> 00:50:59.930 align:middle line:84% And so now you see that if little n-- so therefore, 00:50:59.930 --> 00:51:04.130 align:middle line:84% if little n is bigger than the maximum of these two, 00:51:04.130 --> 00:51:08.150 align:middle line:84% then both of these inequalities hold. 00:51:08.150 --> 00:51:09.030 align:middle line:90% Right. 00:51:09.030 --> 00:51:10.500 align:middle line:90% Both of them hold. 00:51:10.500 --> 00:51:16.370 align:middle line:84% And so this means that you now have that cn minus a times 00:51:16.370 --> 00:51:19.400 align:middle line:90% b, this thing here. 00:51:19.400 --> 00:51:20.880 align:middle line:90% So we had it here. 00:51:20.880 --> 00:51:24.380 align:middle line:90% This is less than c. 00:51:24.380 --> 00:51:28.160 align:middle line:84% And then this difference here was now 00:51:28.160 --> 00:51:35.460 align:middle line:84% assumed to be epsilon over 2, 1 plus c. 00:51:35.460 --> 00:51:40.980 align:middle line:84% And likewise, this thing here is bounded 00:51:40.980 --> 00:51:46.290 align:middle line:90% by this number here times-- 00:51:46.290 --> 00:51:53.190 align:middle line:84% and then this thing here, so epsilon 00:51:53.190 --> 00:51:58.680 align:middle line:84% over 2 absolute value of b plus 1. 00:51:58.680 --> 00:52:00.905 align:middle line:90% OK. 00:52:00.905 --> 00:52:03.180 align:middle line:84% But now you see that this thing here, 00:52:03.180 --> 00:52:08.250 align:middle line:84% that c divided by 1 plus c, that's bounded by 1. 00:52:08.250 --> 00:52:08.920 align:middle line:90% Right. 00:52:08.920 --> 00:52:12.635 align:middle line:84% So this thing here is bounded by epsilon over 2. 00:52:12.635 --> 00:52:15.180 align:middle line:90% 00:52:15.180 --> 00:52:18.360 align:middle line:84% And this thing here, same reason is that this thing here 00:52:18.360 --> 00:52:21.570 align:middle line:90% is bounded by epsilon over 2. 00:52:21.570 --> 00:52:25.650 align:middle line:84% And so the whole thing was bounded by epsilon 00:52:25.650 --> 00:52:31.300 align:middle line:84% when little n was bigger than the maximum of these two. 00:52:31.300 --> 00:52:36.360 align:middle line:84% So in the end, I wrote out the whole proof. 00:52:36.360 --> 00:52:38.420 align:middle line:84% I already posted the lecture notes. 00:52:38.420 --> 00:52:41.040 align:middle line:84% In the lecture notes, I didn't write out the whole proof 00:52:41.040 --> 00:52:42.240 align:middle line:90% but I gave the idea. 00:52:42.240 --> 00:52:46.100 align:middle line:84% So I stopped once we were here and somewhere 00:52:46.100 --> 00:52:49.080 align:middle line:84% here after this one somewhere anyway. 00:52:49.080 --> 00:52:49.795 align:middle line:90% But you see. 00:52:49.795 --> 00:52:54.530 align:middle line:90% 00:52:54.530 --> 00:52:57.080 align:middle line:90% OK. 00:52:57.080 --> 00:52:57.690 align:middle line:90% Right. 00:52:57.690 --> 00:53:02.630 align:middle line:84% So the next thing I want to talk about is that if you-- 00:53:02.630 --> 00:53:05.100 align:middle line:84% No, no, I still need property four. 00:53:05.100 --> 00:53:06.120 align:middle line:90% And that's also useful. 00:53:06.120 --> 00:53:06.620 align:middle line:90% Sorry. 00:53:06.620 --> 00:53:11.045 align:middle line:84% So let's go through that and then we'll come to subsequences. 00:53:11.045 --> 00:53:24.710 align:middle line:90% 00:53:24.710 --> 00:53:30.050 align:middle line:84% So I want to prove that if you take a sequence of real numbers 00:53:30.050 --> 00:53:31.740 align:middle line:90% none of them is zero. 00:53:31.740 --> 00:53:37.090 align:middle line:84% They converge and the limit is not 0, then 1 over the sequence 00:53:37.090 --> 00:53:38.973 align:middle line:90% is another convergence sequence. 00:53:38.973 --> 00:53:54.850 align:middle line:90% 00:53:54.850 --> 00:53:58.130 align:middle line:84% But again, just the main takeaway. 00:53:58.130 --> 00:54:01.040 align:middle line:84% The main takeaway, I think, from property, 00:54:01.040 --> 00:54:06.310 align:middle line:84% this third property was that this little idea about inserting 00:54:06.310 --> 00:54:08.950 align:middle line:84% something that was halfway between the two things you 00:54:08.950 --> 00:54:10.450 align:middle line:84% were comparing because they weren't 00:54:10.450 --> 00:54:12.440 align:middle line:90% very easy to compare directly. 00:54:12.440 --> 00:54:14.200 align:middle line:84% And that's like a little trick that 00:54:14.200 --> 00:54:15.980 align:middle line:90% is useful in various places. 00:54:15.980 --> 00:54:18.410 align:middle line:84% It comes up almost the same thing. 00:54:18.410 --> 00:54:21.430 align:middle line:84% One place it comes up is what's called Leibniz 00:54:21.430 --> 00:54:23.590 align:middle line:84% rule that if you take two functions, 00:54:23.590 --> 00:54:26.510 align:middle line:84% and you take the product of these two functions, 00:54:26.510 --> 00:54:31.040 align:middle line:84% and you try to prove that one is the derivative of the product, 00:54:31.040 --> 00:54:32.850 align:middle line:90% it's exactly the same trick. 00:54:32.850 --> 00:54:38.420 align:middle line:84% And it comes up various other-- more fancy ways. 00:54:38.420 --> 00:54:44.990 align:middle line:84% So now let's look at-- let's prove property four. 00:54:44.990 --> 00:54:47.270 align:middle line:90% So property four. 00:54:47.270 --> 00:54:49.975 align:middle line:90% Proof of property four. 00:54:49.975 --> 00:54:56.900 align:middle line:90% 00:54:56.900 --> 00:54:59.000 align:middle line:90% So we have this. 00:54:59.000 --> 00:55:01.610 align:middle line:90% So cn is 1 over n. 00:55:01.610 --> 00:55:05.250 align:middle line:84% And we know that a n are not equal to 0. 00:55:05.250 --> 00:55:07.140 align:middle line:84% And we know that a is not equal to 0. 00:55:07.140 --> 00:55:08.400 align:middle line:90% This here makes sense. 00:55:08.400 --> 00:55:11.370 align:middle line:90% And also 1 over a makes sense. 00:55:11.370 --> 00:55:15.210 align:middle line:84% So we want to compare this here with 1 over a. 00:55:15.210 --> 00:55:18.020 align:middle line:90% 00:55:18.020 --> 00:55:26.360 align:middle line:84% And so this thing here is 1 over a n minus 1 over n 00:55:26.360 --> 00:55:30.110 align:middle line:84% And now I can put it on a common denominator. 00:55:30.110 --> 00:55:40.750 align:middle line:84% So this is a minus a n divided by a n times a so like that. 00:55:40.750 --> 00:55:42.080 align:middle line:90% You can write it like this. 00:55:42.080 --> 00:55:45.640 align:middle line:84% Now already the denominator looks very promising 00:55:45.640 --> 00:55:48.655 align:middle line:84% because we know that this goes to 0. 00:55:48.655 --> 00:55:49.155 align:middle line:90% Right. 00:55:49.155 --> 00:55:53.500 align:middle line:90% 00:55:53.500 --> 00:55:55.970 align:middle line:90% But now I need to bound this. 00:55:55.970 --> 00:55:59.070 align:middle line:90% So I need to bound-- 00:55:59.070 --> 00:55:59.570 align:middle line:90% Right. 00:55:59.570 --> 00:56:00.500 align:middle line:90% So I need to bound. 00:56:00.500 --> 00:56:02.470 align:middle line:90% We need to bound. 00:56:02.470 --> 00:56:07.030 align:middle line:84% So we want to bound this here by epsilon when 00:56:07.030 --> 00:56:09.680 align:middle line:90% little n is sufficiently large. 00:56:09.680 --> 00:56:13.070 align:middle line:84% The top, the [? nominator ?] here looks great. 00:56:13.070 --> 00:56:17.680 align:middle line:84% The denominator maybe we're a little bit worried about. 00:56:17.680 --> 00:56:19.460 align:middle line:90% But the thing is that-- 00:56:19.460 --> 00:56:33.620 align:middle line:84% so we need to bound a n times a, this here from below. 00:56:33.620 --> 00:56:36.830 align:middle line:84% But we want to bound it from below because then 00:56:36.830 --> 00:56:40.400 align:middle line:84% when you're looking at 1 over it, then that's an upper bound. 00:56:40.400 --> 00:56:44.720 align:middle line:84% And really, because a here is just 00:56:44.720 --> 00:56:50.160 align:middle line:84% a fixed number that is not 0, then it's equivalent to proving. 00:56:50.160 --> 00:57:05.080 align:middle line:84% So this is equivalent to bound the norm of a n from below. 00:57:05.080 --> 00:57:08.930 align:middle line:90% 00:57:08.930 --> 00:57:13.650 align:middle line:84% Remember, we don't need to bound it from below for all a n. 00:57:13.650 --> 00:57:20.550 align:middle line:84% But we need to bounded from below for n sufficiently large. 00:57:20.550 --> 00:57:25.440 align:middle line:84% And so now, you make the trivial observation. 00:57:25.440 --> 00:57:35.140 align:middle line:84% So that a n, you can write a n as a n minus a plus a. 00:57:35.140 --> 00:57:39.970 align:middle line:84% And so therefore, therefore, the norm of a n 00:57:39.970 --> 00:57:46.710 align:middle line:84% is bounded by the norm of a n minus a plus the norm of a. 00:57:46.710 --> 00:57:52.210 align:middle line:90% 00:57:52.210 --> 00:57:55.855 align:middle line:90% Sorry, this was not the-- 00:57:55.855 --> 00:57:58.100 align:middle line:84% this was not how I wanted to write it. 00:57:58.100 --> 00:58:00.940 align:middle line:84% I'm going to write it slightly differently. 00:58:00.940 --> 00:58:03.790 align:middle line:90% I'm going to write a. 00:58:03.790 --> 00:58:05.630 align:middle line:84% I'm going to do the symmetric thing. 00:58:05.630 --> 00:58:09.620 align:middle line:84% But I'm doing it-- so I'm writing a in as a minus-- 00:58:09.620 --> 00:58:14.620 align:middle line:84% so in writing a as a minus a n plus a n. 00:58:14.620 --> 00:58:17.920 align:middle line:90% 00:58:17.920 --> 00:58:22.480 align:middle line:84% And then, if I do that, then I can just 00:58:22.480 --> 00:58:25.640 align:middle line:84% take the absolute value of a is bounded 00:58:25.640 --> 00:58:28.700 align:middle line:84% by the absolute value of this difference 00:58:28.700 --> 00:58:33.890 align:middle line:90% plus the absolute value of a n. 00:58:33.890 --> 00:58:37.800 align:middle line:84% And now I can move this here over on the other side. 00:58:37.800 --> 00:58:45.380 align:middle line:84% So this here implies that norm of a minus 00:58:45.380 --> 00:58:53.060 align:middle line:84% the norm of a minus a n is less or equal to this here. 00:58:53.060 --> 00:58:56.240 align:middle line:90% 00:58:56.240 --> 00:59:00.450 align:middle line:84% And now, you see that this thing here is-- 00:59:00.450 --> 00:59:07.110 align:middle line:84% so now we're going to use that this here was not equal to 0. 00:59:07.110 --> 00:59:10.430 align:middle line:84% So this here is bounded away from 0. 00:59:10.430 --> 00:59:18.600 align:middle line:84% And so now we can take, say, a here over 2. 00:59:18.600 --> 00:59:21.560 align:middle line:90% So we can use this here. 00:59:21.560 --> 00:59:25.310 align:middle line:90% We can use norm of a over 2. 00:59:25.310 --> 00:59:28.750 align:middle line:84% This is a positive number because a was not 0. 00:59:28.750 --> 00:59:34.070 align:middle line:84% And we can use this in the definition of that a n-- 00:59:34.070 --> 00:59:50.600 align:middle line:84% so use this in the definition of that a n converges to a, right? 00:59:50.600 --> 00:59:52.960 align:middle line:84% So this means that if you are-- this 00:59:52.960 --> 01:00:00.580 align:middle line:84% means that there exists a capital 01:00:00.580 --> 01:00:02.480 align:middle line:90% N. Let's call it capital N1. 01:00:02.480 --> 01:00:05.540 align:middle line:84% It's not the final N that we want to use. 01:00:05.540 --> 01:00:09.850 align:middle line:84% But there exists a capital N1 so that if little n is bigger 01:00:09.850 --> 01:00:16.720 align:middle line:84% than this capital N1, then, actually, we have that a n-- 01:00:16.720 --> 01:00:23.065 align:middle line:84% a minus a n is less than this thing here. 01:00:23.065 --> 01:00:29.345 align:middle line:90% 01:00:29.345 --> 01:00:31.280 align:middle line:90% But now, you see that-- 01:00:31.280 --> 01:00:34.220 align:middle line:84% this means that when you're looking at-- when you go up 01:00:34.220 --> 01:00:36.890 align:middle line:90% here, and you're looking at-- 01:00:36.890 --> 01:00:39.870 align:middle line:84% when you go up to that line before-- 01:00:39.870 --> 01:00:41.620 align:middle line:90% so this is the line from before. 01:00:41.620 --> 01:00:45.890 align:middle line:90% 01:00:45.890 --> 01:00:46.610 align:middle line:90% Sorry. 01:00:46.610 --> 01:00:50.480 align:middle line:84% This is minus this here-- this and this. 01:00:50.480 --> 01:00:55.020 align:middle line:84% When you're looking at this line here, well, this thing here-- 01:00:55.020 --> 01:00:59.210 align:middle line:84% this here is at most half of this. 01:00:59.210 --> 01:01:08.771 align:middle line:84% So this thing here is bigger than this thing here over 2. 01:01:08.771 --> 01:01:14.871 align:middle line:84% And so now you see that we have bounded these guys from below. 01:01:14.871 --> 01:01:17.670 align:middle line:90% And so now we go back up to-- 01:01:17.670 --> 01:01:21.610 align:middle line:84% so now we go back up to-- we're looking at C minus-- 01:01:21.610 --> 01:01:22.370 align:middle line:90% oh, sorry. 01:01:22.370 --> 01:01:26.715 align:middle line:84% We're looking at Cn minus 1 over a. 01:01:26.715 --> 01:01:29.418 align:middle line:90% 01:01:29.418 --> 01:01:34.960 align:middle line:90% This is this up here. 01:01:34.960 --> 01:01:39.280 align:middle line:84% I'm just transferring that equation down to this board 01:01:39.280 --> 01:01:40.600 align:middle line:90% here. 01:01:40.600 --> 01:01:48.020 align:middle line:84% So this here is a n, a, minus a n, divided by-- 01:01:48.020 --> 01:01:50.620 align:middle line:84% and I can just take the absolute value separately 01:01:50.620 --> 01:01:55.905 align:middle line:84% of the denominator-- and the denominator, a n times a. 01:01:55.905 --> 01:01:58.600 align:middle line:90% 01:01:58.600 --> 01:01:59.560 align:middle line:90% OK. 01:01:59.560 --> 01:02:08.110 align:middle line:84% So now you see that this thing here, as long as-- so when 01:02:08.110 --> 01:02:13.540 align:middle line:84% n here, little n, is bigger than N1, well, then we have 01:02:13.540 --> 01:02:16.870 align:middle line:84% that this thing here is at the very least 01:02:16.870 --> 01:02:21.480 align:middle line:90% the absolute value of a over 2. 01:02:21.480 --> 01:02:23.670 align:middle line:84% So this means that this thing here, we 01:02:23.670 --> 01:02:28.030 align:middle line:84% can now bound from above, like this. 01:02:28.030 --> 01:02:32.040 align:middle line:84% This is as long as you have this inequality here. 01:02:32.040 --> 01:02:36.730 align:middle line:84% And this thing here-- this is just a norm of a. 01:02:36.730 --> 01:02:39.850 align:middle line:84% This here will give you another norm of a over 2. 01:02:39.850 --> 01:02:41.770 align:middle line:90% So the 2 jumps up. 01:02:41.770 --> 01:02:43.265 align:middle line:90% So it's like this thing here. 01:02:43.265 --> 01:02:49.804 align:middle line:90% 01:02:49.804 --> 01:02:53.370 align:middle line:90% And now you see that-- 01:02:53.370 --> 01:02:57.702 align:middle line:84% now you see that this a is just like a constant. 01:02:57.702 --> 01:03:02.530 align:middle line:84% So this means that I may have to go even further out than N1. 01:03:02.530 --> 01:03:04.830 align:middle line:84% But if I go even further out, then I 01:03:04.830 --> 01:03:09.435 align:middle line:84% can make this thing here smaller than a given epsilon, OK? 01:03:09.435 --> 01:03:12.762 align:middle line:90% 01:03:12.762 --> 01:03:13.780 align:middle line:90% All right. 01:03:13.780 --> 01:03:15.580 align:middle line:84% So this was like the outline, if you will. 01:03:15.580 --> 01:03:21.180 align:middle line:84% But that's not-- at this point, that's not very much to fill in. 01:03:21.180 --> 01:03:21.680 align:middle line:90% OK. 01:03:21.680 --> 01:03:24.944 align:middle line:84% So now I want to talk about subsequences. 01:03:24.944 --> 01:03:38.410 align:middle line:90% 01:03:38.410 --> 01:03:40.750 align:middle line:90% And then I want to-- 01:03:40.750 --> 01:03:44.420 align:middle line:84% then I want to prove a little theorem about subsequences. 01:03:44.420 --> 01:04:00.370 align:middle line:90% 01:04:00.370 --> 01:04:03.940 align:middle line:90% So I have a sequence. 01:04:03.940 --> 01:04:10.580 align:middle line:84% And so I have a sequence and then a subsequence. 01:04:10.580 --> 01:04:12.560 align:middle line:84% So I'll give you first some examples. 01:04:12.560 --> 01:04:15.030 align:middle line:90% And then I will explain. 01:04:15.030 --> 01:04:17.520 align:middle line:84% Then I will give you the proper definition 01:04:17.520 --> 01:04:21.880 align:middle line:90% of what a subsequence is. 01:04:21.880 --> 01:04:28.870 align:middle line:84% So I'll start with this sequence from before. 01:04:28.870 --> 01:04:33.690 align:middle line:84% So this was the sequence where a1 was minus 1, a2 was 1, 01:04:33.690 --> 01:04:35.740 align:middle line:90% a3 was minus 1. 01:04:35.740 --> 01:04:40.770 align:middle line:84% It was alternating between minus 1 and 1. 01:04:40.770 --> 01:04:43.770 align:middle line:90% And so, in this case here-- 01:04:43.770 --> 01:04:49.950 align:middle line:84% in this case here, if you take the sequence-- 01:04:49.950 --> 01:04:57.180 align:middle line:90% let's call it-- let's call it b. 01:04:57.180 --> 01:05:01.030 align:middle line:84% Let's call it-- let's use a different index. 01:05:01.030 --> 01:05:01.985 align:middle line:90% Let's call it bk. 01:05:01.985 --> 01:05:04.590 align:middle line:90% 01:05:04.590 --> 01:05:07.030 align:middle line:84% So k here plays the same role as n before. 01:05:07.030 --> 01:05:10.440 align:middle line:84% But I want to use a different symbol to not somehow confuse 01:05:10.440 --> 01:05:13.060 align:middle line:90% with the n over here. 01:05:13.060 --> 01:05:15.230 align:middle line:90% So if I take-- 01:05:15.230 --> 01:05:19.720 align:middle line:84% so I could take a sequence that was like 1 all the time, 01:05:19.720 --> 01:05:20.990 align:middle line:90% this here, for all k. 01:05:20.990 --> 01:05:23.920 align:middle line:90% 01:05:23.920 --> 01:05:26.390 align:middle line:84% This here would be a subsequence. 01:05:26.390 --> 01:05:28.210 align:middle line:84% This would correspond to that you 01:05:28.210 --> 01:05:35.030 align:middle line:84% took all the even ones, that you just take all the even ones. 01:05:35.030 --> 01:05:37.780 align:middle line:90% So this is like a subsequence. 01:05:37.780 --> 01:05:40.550 align:middle line:84% You could also have another example. 01:05:40.550 --> 01:05:43.150 align:middle line:90% So this is example 1. 01:05:43.150 --> 01:05:47.580 align:middle line:84% Another example is where you took all the odd ones. 01:05:47.580 --> 01:05:50.260 align:middle line:90% 01:05:50.260 --> 01:05:55.450 align:middle line:84% So this here is also a subsequence. 01:05:55.450 --> 01:06:00.190 align:middle line:84% And you could do something more elaborate, where you take-- 01:06:00.190 --> 01:06:05.050 align:middle line:84% say, first, you take the first two odd, 01:06:05.050 --> 01:06:09.160 align:middle line:84% and then you take the next two even. 01:06:09.160 --> 01:06:11.440 align:middle line:84% So you could have a sequence that was 01:06:11.440 --> 01:06:14.980 align:middle line:90% like b1 is equal to minus 1. 01:06:14.980 --> 01:06:19.440 align:middle line:90% b2 is equal to minus 1. 01:06:19.440 --> 01:06:25.540 align:middle line:84% b3-- now it becomes the next two even ones. 01:06:25.540 --> 01:06:29.650 align:middle line:84% And then you go back to the next two odd ones, et cetera. 01:06:29.650 --> 01:06:32.656 align:middle line:84% That would be another subsequence. 01:06:32.656 --> 01:06:36.840 align:middle line:90% 01:06:36.840 --> 01:06:38.863 align:middle line:84% Let me look at a few more examples. 01:06:38.863 --> 01:06:49.350 align:middle line:90% 01:06:49.350 --> 01:06:53.760 align:middle line:84% So, for instance, if your sequence, 01:06:53.760 --> 01:06:55.920 align:middle line:90% your original sequence-- 01:06:55.920 --> 01:06:58.980 align:middle line:90% so here's another example. 01:06:58.980 --> 01:07:07.470 align:middle line:84% If your original sequence was just that a n was equal to n, 01:07:07.470 --> 01:07:10.820 align:middle line:90% then you could take a sequence. 01:07:10.820 --> 01:07:17.110 align:middle line:84% So then the sequence where you just take the odd numbers, 01:07:17.110 --> 01:07:19.220 align:middle line:90% this here is a subsequence. 01:07:19.220 --> 01:07:23.200 align:middle line:90% 01:07:23.200 --> 01:07:28.840 align:middle line:84% And likewise, if you take just the even numbers, 01:07:28.840 --> 01:07:33.500 align:middle line:84% this is also a sequence, a subsequence. 01:07:33.500 --> 01:07:38.160 align:middle line:90% 01:07:38.160 --> 01:07:40.270 align:middle line:90% But if you're looking at-- 01:07:40.270 --> 01:07:45.470 align:middle line:84% if you look at, say, something like this, where you're taking-- 01:07:45.470 --> 01:07:50.834 align:middle line:84% so if you define a sequence where it was 1, 1-- 01:07:50.834 --> 01:07:53.770 align:middle line:90% 01:07:53.770 --> 01:07:56.523 align:middle line:90% 1, 1, 2, even like this-- 01:07:56.523 --> 01:08:00.850 align:middle line:90% 01:08:00.850 --> 01:08:02.990 align:middle line:90% this here is not a subsequence. 01:08:02.990 --> 01:08:12.030 align:middle line:84% This is not a subsequence, because you have-- 01:08:12.030 --> 01:08:15.930 align:middle line:84% what you have done here is that you have taken the same number 01:08:15.930 --> 01:08:17.140 align:middle line:90% several times. 01:08:17.140 --> 01:08:18.880 align:middle line:84% And it doesn't appear several times. 01:08:18.880 --> 01:08:22.080 align:middle line:84% The value doesn't appear several times in the original one. 01:08:22.080 --> 01:08:24.450 align:middle line:90% So that's not allowed. 01:08:24.450 --> 01:08:28.930 align:middle line:84% It's also not allowed to do something like this. 01:08:28.930 --> 01:08:32.470 align:middle line:84% This here is also not a-- so not a subsequence, 01:08:32.470 --> 01:08:35.490 align:middle line:84% another thing that's not a subsequence-- 01:08:35.490 --> 01:08:37.750 align:middle line:84% when you take things out of order. 01:08:37.750 --> 01:08:42.640 align:middle line:84% So if this here was the original sequence, 01:08:42.640 --> 01:08:48.540 align:middle line:84% if you're now looking at a sequence that was 3, 2, 01:08:48.540 --> 01:08:52.710 align:middle line:84% and then 4, 5, this here is-- et cetera-- 01:08:52.710 --> 01:08:55.710 align:middle line:84% this here is not a subsequence either, 01:08:55.710 --> 01:08:59.819 align:middle line:84% because things are taken out of order. 01:08:59.819 --> 01:09:03.346 align:middle line:84% So now, what is the formal definition of a subsequence? 01:09:03.346 --> 01:09:05.880 align:middle line:90% 01:09:05.880 --> 01:09:07.906 align:middle line:90% So here is a formal definition. 01:09:07.906 --> 01:09:30.100 align:middle line:90% 01:09:30.100 --> 01:09:33.669 align:middle line:90% So you have a sequence, a n. 01:09:33.669 --> 01:09:49.210 align:middle line:84% A subsequence of a n is where you have a map, g, from N 01:09:49.210 --> 01:09:53.890 align:middle line:84% into N. This is an increasing map. 01:09:53.890 --> 01:09:55.520 align:middle line:90% This is strictly increasing. 01:09:55.520 --> 01:10:02.920 align:middle line:90% 01:10:02.920 --> 01:10:06.780 align:middle line:84% And then, if you're thinking about this thing here, 01:10:06.780 --> 01:10:10.950 align:middle line:84% you think about this as a function, f, 01:10:10.950 --> 01:10:15.450 align:middle line:84% then you now could think about the composition of these two 01:10:15.450 --> 01:10:19.380 align:middle line:90% functions, so this thing here. 01:10:19.380 --> 01:10:22.950 align:middle line:84% And maybe I will use a different notation 01:10:22.950 --> 01:10:27.070 align:middle line:84% like this, a different index like this. 01:10:27.070 --> 01:10:34.170 align:middle line:84% This thing here, bk, this here is a subsequence. 01:10:34.170 --> 01:10:38.470 align:middle line:84% So you see that this here, when you pick-- 01:10:38.470 --> 01:10:42.870 align:middle line:84% if you have a sequence like this, 01:10:42.870 --> 01:10:46.750 align:middle line:84% the first one could be anything in the original one. 01:10:46.750 --> 01:10:50.400 align:middle line:84% But once you have fixed the first one, you can't look back, 01:10:50.400 --> 01:10:52.170 align:middle line:90% or you can't go to this. 01:10:52.170 --> 01:10:55.740 align:middle line:84% You have to-- the next element have to be somewhat further 01:10:55.740 --> 01:10:57.427 align:middle line:90% out in the sequence. 01:10:57.427 --> 01:11:01.682 align:middle line:90% And that continues. 01:11:01.682 --> 01:11:05.660 align:middle line:84% And this here is often just to indicate that it's 01:11:05.660 --> 01:11:07.980 align:middle line:90% a subsequence of this sequence. 01:11:07.980 --> 01:11:15.440 align:middle line:84% Then the standard notation is a n and then another subscript. 01:11:15.440 --> 01:11:17.930 align:middle line:84% So anyway, so of course, hopefully, 01:11:17.930 --> 01:11:20.480 align:middle line:84% there's not too many subsequences of sequences, 01:11:20.480 --> 01:11:25.730 align:middle line:84% because each time, it's supposed to be another subscript. 01:11:25.730 --> 01:11:28.250 align:middle line:84% Now, there's a little theorem that we can just 01:11:28.250 --> 01:11:30.140 align:middle line:90% prove here at the end. 01:11:30.140 --> 01:11:34.130 align:middle line:84% And that is that if you-- and it's-- 01:11:34.130 --> 01:11:38.940 align:middle line:84% yeah, it's sometimes-- it's a very simple fact. 01:11:38.940 --> 01:11:43.280 align:middle line:84% But it is sometimes a useful fact. 01:11:43.280 --> 01:11:48.300 align:middle line:84% And that is that if you take a sequence-- 01:11:48.300 --> 01:11:58.280 align:middle line:84% so a sequence-- so theorem, that-- 01:11:58.280 --> 01:12:11.340 align:middle line:84% so a sequence, a n, is convergent to a if and only 01:12:11.340 --> 01:12:37.380 align:middle line:84% if all subsequences of a n converge to a. 01:12:37.380 --> 01:12:38.140 align:middle line:90% OK. 01:12:38.140 --> 01:12:43.500 align:middle line:84% Now, to prove this, there's two things we need to prove, 01:12:43.500 --> 01:12:46.720 align:middle line:84% because it's an "if and only if" statement. 01:12:46.720 --> 01:12:49.930 align:middle line:84% So we need to prove that if it is convergent, 01:12:49.930 --> 01:12:52.870 align:middle line:84% then all subsequences are also convergent. 01:12:52.870 --> 01:12:55.890 align:middle line:84% And then we need to prove that if all subsequences are 01:12:55.890 --> 01:12:59.080 align:middle line:84% convergent, then the original sequence is convergent. 01:12:59.080 --> 01:13:01.800 align:middle line:84% But one of the two implications is trivial. 01:13:01.800 --> 01:13:13.960 align:middle line:84% So one of the two implications is trivial. 01:13:13.960 --> 01:13:16.880 align:middle line:90% 01:13:16.880 --> 01:13:25.940 align:middle line:84% And that is that, namely, since-- 01:13:25.940 --> 01:13:30.920 align:middle line:84% so the map-- remember that the map to define a subsequence 01:13:30.920 --> 01:13:34.620 align:middle line:90% has to be strictly increasing. 01:13:34.620 --> 01:13:42.470 align:middle line:84% So remember that a sequence was a function like this. 01:13:42.470 --> 01:13:44.280 align:middle line:90% That was a sequence. 01:13:44.280 --> 01:13:47.600 align:middle line:84% And a subsequence-- there, you have another map 01:13:47.600 --> 01:13:50.600 align:middle line:90% that is strictly increasing. 01:13:50.600 --> 01:13:53.100 align:middle line:84% And then you're looking at the composition. 01:13:53.100 --> 01:13:57.440 align:middle line:84% But if you take this other map to be the identity, 01:13:57.440 --> 01:14:03.250 align:middle line:84% that the other map is like g of k is equal to k, of course, 01:14:03.250 --> 01:14:05.130 align:middle line:90% this is strictly increasing. 01:14:05.130 --> 01:14:09.810 align:middle line:84% And if you did-- if this here map was the identity, then 01:14:09.810 --> 01:14:14.004 align:middle line:84% the subsequence would just be the original sequence. 01:14:14.004 --> 01:14:16.980 align:middle line:84% So one of the two implications here 01:14:16.980 --> 01:14:27.400 align:middle line:84% is trivial, since a n itself is admittedly trivial. 01:14:27.400 --> 01:14:28.900 align:middle line:90% But it is a subsequence. 01:14:28.900 --> 01:14:35.860 align:middle line:90% 01:14:35.860 --> 01:14:41.280 align:middle line:84% So if all subsequences converge to a, well, then 01:14:41.280 --> 01:14:45.610 align:middle line:84% the original sequence, since it is a subsequence, 01:14:45.610 --> 01:14:47.250 align:middle line:90% certainly, it will converge. 01:14:47.250 --> 01:14:49.050 align:middle line:90% So that was trivial. 01:14:49.050 --> 01:14:54.780 align:middle line:84% OK, so we just need to prove that if a n converged to a, then 01:14:54.780 --> 01:14:57.230 align:middle line:90% any subsequence. 01:14:57.230 --> 01:15:02.900 align:middle line:84% But this is not much more difficult, because you see, 01:15:02.900 --> 01:15:07.310 align:middle line:84% if you're-- so given an epsilon-- 01:15:07.310 --> 01:15:08.945 align:middle line:90% so now, given epsilon-- 01:15:08.945 --> 01:15:12.260 align:middle line:90% 01:15:12.260 --> 01:15:21.335 align:middle line:84% so now we have-- so we now have that a n converges to a. 01:15:21.335 --> 01:15:24.960 align:middle line:84% And we're looking at a subsequence, ank. 01:15:24.960 --> 01:15:35.180 align:middle line:84% And we want to show that ank also converges to a. 01:15:35.180 --> 01:15:41.540 align:middle line:84% So given epsilon, given epsilon greater than 0, 01:15:41.540 --> 01:15:51.590 align:middle line:84% there exists a capital N so that if little n is bigger 01:15:51.590 --> 01:15:57.960 align:middle line:84% than capital N, then a n minus a is less than epsilon. 01:15:57.960 --> 01:16:06.030 align:middle line:84% But now you see that if k is bigger than capital N, 01:16:06.030 --> 01:16:10.740 align:middle line:90% then what corresponds to-- 01:16:10.740 --> 01:16:17.730 align:middle line:84% this here is-- this here, how to think about it-- this here, nk, 01:16:17.730 --> 01:16:20.520 align:middle line:84% you should really think about that this thing here 01:16:20.520 --> 01:16:24.742 align:middle line:84% is the value of this strictly increasing function. 01:16:24.742 --> 01:16:26.700 align:middle line:90% But this thing here is-- 01:16:26.700 --> 01:16:30.240 align:middle line:84% because it's strictly increasing, it's bigger than k. 01:16:30.240 --> 01:16:35.080 align:middle line:84% So you see that if this thing here is bigger than this, 01:16:35.080 --> 01:16:38.910 align:middle line:84% then means that this index here is further out than, 01:16:38.910 --> 01:16:43.860 align:middle line:84% in particular, the capital N. And so this 01:16:43.860 --> 01:16:47.880 align:middle line:84% means that if you have this here, then ank, 01:16:47.880 --> 01:16:51.310 align:middle line:90% this is some element like this. 01:16:51.310 --> 01:16:54.810 align:middle line:84% But it's further out than capital N. 01:16:54.810 --> 01:16:57.770 align:middle line:84% So this thing here surely will also 01:16:57.770 --> 01:17:02.180 align:middle line:84% be less than epsilon because it's falling under here. 01:17:02.180 --> 01:17:09.450 align:middle line:84% So that implication wasn't too much harder. 01:17:09.450 --> 01:17:12.904 align:middle line:84% And that's sometimes useful, that-- 01:17:12.904 --> 01:17:19.370 align:middle line:84% so if-- yeah, it's sometimes useful that you're looking 01:17:19.370 --> 01:17:28.238 align:middle line:84% at a sequence, and you see that if it is converging, then-- 01:17:28.238 --> 01:17:29.780 align:middle line:84% you're looking at a sequence, and you 01:17:29.780 --> 01:17:32.670 align:middle line:84% see that there's a subsequence that is not converging, 01:17:32.670 --> 01:17:36.140 align:middle line:84% or you have that there are two different subsequences 01:17:36.140 --> 01:17:39.440 align:middle line:84% of a given sequence that converge to different limits. 01:17:39.440 --> 01:17:42.780 align:middle line:84% Then the original sequence wasn't converging either. 01:17:42.780 --> 01:17:43.850 align:middle line:90% Yeah? 01:17:43.850 --> 01:17:46.310 align:middle line:84% AUDIENCE: Can you not have a finite subsequence 01:17:46.310 --> 01:17:49.910 align:middle line:84% of-- a convergent subsequence that doesn't get to n, 01:17:49.910 --> 01:17:54.700 align:middle line:84% so it hasn't arrived at the point of convergence? 01:17:54.700 --> 01:17:56.460 align:middle line:84% TOBIAS COLDING: So convergence is always 01:17:56.460 --> 01:18:00.310 align:middle line:84% something about the tail, about the end, right? 01:18:00.310 --> 01:18:05.282 align:middle line:84% It never cares about what happened in the beginning. 01:18:05.282 --> 01:18:08.070 align:middle line:84% So there might be finitely many n's 01:18:08.070 --> 01:18:11.890 align:middle line:84% initially where things are crazy, 01:18:11.890 --> 01:18:13.948 align:middle line:90% but then from there on out. 01:18:13.948 --> 01:18:14.490 align:middle line:90% AUDIENCE: OK. 01:18:14.490 --> 01:18:16.800 align:middle line:84% The subsequences that we care about in this case 01:18:16.800 --> 01:18:18.900 align:middle line:84% are those subsequences that also-- 01:18:18.900 --> 01:18:20.025 align:middle line:90% TOBIAS COLDING: Yeah, yeah. 01:18:20.025 --> 01:18:22.200 align:middle line:84% So all sequences, including subsequences, 01:18:22.200 --> 01:18:24.730 align:middle line:90% are infinite things, right? 01:18:24.730 --> 01:18:25.350 align:middle line:90% Yeah. 01:18:25.350 --> 01:18:28.590 align:middle line:84% So like-- that's right, because the thing is 01:18:28.590 --> 01:18:32.820 align:middle line:90% that you had to have this map. 01:18:32.820 --> 01:18:34.000 align:middle line:90% Maybe I erased it. 01:18:34.000 --> 01:18:40.270 align:middle line:84% But g here had to be a map from all the natural numbers N to N. 01:18:40.270 --> 01:18:44.950 align:middle line:84% So it means that for each element here, you get something. 01:18:44.950 --> 01:18:46.930 align:middle line:84% So there's really infinitely many of them. 01:18:46.930 --> 01:18:48.480 align:middle line:90% Yeah. 01:18:48.480 --> 01:18:52.000 align:middle line:90%