WEBVTT 00:00:00.000 --> 00:00:01.494 align:middle line:90% [SQUEAKING] 00:00:01.494 --> 00:00:02.490 align:middle line:90% [RUSTLING] 00:00:02.490 --> 00:00:04.980 align:middle line:90% [CLICKING] 00:00:04.980 --> 00:00:12.877 align:middle line:90% 00:00:12.877 --> 00:00:14.460 align:middle line:84% CASEY RODRIGUEZ: So we ended last time 00:00:14.460 --> 00:00:18.127 align:middle line:84% with these elementary properties that you 00:00:18.127 --> 00:00:19.605 align:middle line:90% know about the absolute value. 00:00:19.605 --> 00:00:22.420 align:middle line:90% 00:00:22.420 --> 00:00:29.400 align:middle line:84% So let me prove one more theorem about the absolute value, which 00:00:29.400 --> 00:00:31.680 align:middle line:84% maybe you didn't cover in calculus. 00:00:31.680 --> 00:00:42.070 align:middle line:84% But it's of fundamental importance 00:00:42.070 --> 00:00:47.770 align:middle line:84% in real analysis, which is the triangle inequality, which 00:00:47.770 --> 00:00:55.630 align:middle line:84% states that for all xy in R, the absolute value of x plus y 00:00:55.630 --> 00:01:01.000 align:middle line:84% is less than or equal to the absolute value of x 00:01:01.000 --> 00:01:02.590 align:middle line:90% plus the absolute value of y. 00:01:02.590 --> 00:01:05.610 align:middle line:90% 00:01:05.610 --> 00:01:07.650 align:middle line:84% So why is it called the triangle inequality? 00:01:07.650 --> 00:01:14.410 align:middle line:84% Well, let me-- instead of x and y being real numbers, 00:01:14.410 --> 00:01:18.650 align:middle line:90% let's think of these as vectors. 00:01:18.650 --> 00:01:22.670 align:middle line:90% So there's x. 00:01:22.670 --> 00:01:30.310 align:middle line:90% Let's say y. 00:01:30.310 --> 00:01:38.275 align:middle line:84% And then vector x plus y is this side. 00:01:38.275 --> 00:01:43.000 align:middle line:90% 00:01:43.000 --> 00:01:47.380 align:middle line:84% Then what this is stating is that the length 00:01:47.380 --> 00:01:50.500 align:middle line:84% of one of these sides of this triangle 00:01:50.500 --> 00:01:52.780 align:middle line:84% is less than or equal to the sum of the two 00:01:52.780 --> 00:01:55.863 align:middle line:84% sides, the sum of the lengths of the other two sides 00:01:55.863 --> 00:01:56.530 align:middle line:90% of the triangle. 00:01:56.530 --> 00:02:00.980 align:middle line:90% 00:02:00.980 --> 00:02:06.120 align:middle line:84% So that's why it's called the triangle inequality. 00:02:06.120 --> 00:02:14.070 align:middle line:84% OK, so the proof is not very difficult, 00:02:14.070 --> 00:02:19.690 align:middle line:84% although this inequality is probably 00:02:19.690 --> 00:02:21.880 align:middle line:84% one of the three most useful things you'll 00:02:21.880 --> 00:02:25.400 align:middle line:84% learn in this class if you go on to study more analysis-- 00:02:25.400 --> 00:02:26.740 align:middle line:90% the other one being-- 00:02:26.740 --> 00:02:29.380 align:middle line:90% OK, so I'm coming-- 00:02:29.380 --> 00:02:32.020 align:middle line:90% this is knowledge from PDE. 00:02:32.020 --> 00:02:35.560 align:middle line:84% So I guess I'm viewing it through that lens of what 00:02:35.560 --> 00:02:36.910 align:middle line:90% I find useful in PDEs. 00:02:36.910 --> 00:02:40.300 align:middle line:84% But triangle inequality, integration by parts, 00:02:40.300 --> 00:02:47.780 align:middle line:84% change of variables, those three power the analysis machine 00:02:47.780 --> 00:02:49.860 align:middle line:90% that I'm most familiar with. 00:02:49.860 --> 00:02:52.280 align:middle line:84% So how to prove the triangle inequality-- 00:02:52.280 --> 00:02:56.780 align:middle line:90% 00:02:56.780 --> 00:03:05.000 align:middle line:84% if x and y is an R, then clearly, by property number 6, 00:03:05.000 --> 00:03:07.400 align:middle line:84% the x is less than or equal to the absolute value 00:03:07.400 --> 00:03:10.610 align:middle line:84% of x and y is less than or equal to the absolute value of y. 00:03:10.610 --> 00:03:13.040 align:middle line:84% So their sum is less than or equal to the absolute value 00:03:13.040 --> 00:03:14.030 align:middle line:90% of x plus y. 00:03:14.030 --> 00:03:18.060 align:middle line:90% 00:03:18.060 --> 00:03:22.240 align:middle line:84% And not only that, I can replace x by minus x and I 00:03:22.240 --> 00:03:27.090 align:middle line:84% get minus x plus minus y, this is 00:03:27.090 --> 00:03:31.860 align:middle line:84% less than or equal to the absolute value of minus x 00:03:31.860 --> 00:03:33.690 align:middle line:84% plus the absolute value of minus y. 00:03:33.690 --> 00:03:37.185 align:middle line:84% But this is just absolute value of x plus y. 00:03:37.185 --> 00:03:44.040 align:middle line:90% 00:03:44.040 --> 00:03:48.430 align:middle line:84% And both of these therefore imply-- 00:03:48.430 --> 00:04:05.660 align:middle line:84% so if I just take this and multiply through by minus 1, 00:04:05.660 --> 00:04:09.320 align:middle line:84% this inequality, which, by number 5 00:04:09.320 --> 00:04:13.340 align:middle line:84% in the properties that we proved, for the absolute value 00:04:13.340 --> 00:04:16.010 align:middle line:84% implies that the absolute value of x plus y is less than 00:04:16.010 --> 00:04:21.290 align:middle line:84% or equal to the absolute value of x 00:04:21.290 --> 00:04:25.690 align:middle line:90% plus the absolute value of y. 00:04:25.690 --> 00:04:31.750 align:middle line:84% So let me just make a remark about the reverse triangle 00:04:31.750 --> 00:04:41.630 align:middle line:84% Inequality, which will make an appearance in your assignment. 00:04:41.630 --> 00:04:47.750 align:middle line:84% And so this involves the absolute value of x minus y 00:04:47.750 --> 00:04:50.840 align:middle line:84% and the absolute value of the difference 00:04:50.840 --> 00:04:52.700 align:middle line:90% in the absolute values. 00:04:52.700 --> 00:04:53.570 align:middle line:90% Why is it reverse? 00:04:53.570 --> 00:04:54.965 align:middle line:90% Because the inequality reverses. 00:04:54.965 --> 00:05:02.280 align:middle line:90% 00:05:02.280 --> 00:05:06.810 align:middle line:84% The absolute value of x minus the absolute value and y, 00:05:06.810 --> 00:05:08.470 align:middle line:90% take the absolute value. 00:05:08.470 --> 00:05:11.280 align:middle line:84% This is less than or equal to the absolute value 00:05:11.280 --> 00:05:12.090 align:middle line:90% of x minus y. 00:05:12.090 --> 00:05:14.940 align:middle line:84% And you'll prove this in the assignment. 00:05:14.940 --> 00:05:18.750 align:middle line:84% So these two-- so this follows from the triangle inequality. 00:05:18.750 --> 00:05:21.522 align:middle line:84% But these two inequalities get used quite a bit. 00:05:21.522 --> 00:05:23.730 align:middle line:84% And we'll use them quite a bit throughout the course. 00:05:23.730 --> 00:05:34.260 align:middle line:90% 00:05:34.260 --> 00:05:41.710 align:middle line:84% All right, so let me take a minute to reconnect R 00:05:41.710 --> 00:05:46.660 align:middle line:84% with what you know and love from calculus, namely decimals. 00:05:46.660 --> 00:05:50.960 align:middle line:84% And we're going to use this fact about R 00:05:50.960 --> 00:05:54.980 align:middle line:84% to answer a natural question that maybe you're 00:05:54.980 --> 00:05:58.800 align:middle line:84% asking yourself, which is the following. 00:05:58.800 --> 00:06:00.500 align:middle line:84% So let me raise this as a question. 00:06:00.500 --> 00:06:05.390 align:middle line:90% 00:06:05.390 --> 00:06:08.270 align:middle line:84% So this first part is not the question. 00:06:08.270 --> 00:06:12.370 align:middle line:84% We've already addressed this in the first assignment, the fact 00:06:12.370 --> 00:06:14.560 align:middle line:90% that the rationals is countable. 00:06:14.560 --> 00:06:16.540 align:middle line:84% In fact, you did it for the positive rationals. 00:06:16.540 --> 00:06:18.940 align:middle line:84% But if the positive rationals are countable, 00:06:18.940 --> 00:06:24.520 align:middle line:84% you can then prove that all rationals are countable. 00:06:24.520 --> 00:06:33.663 align:middle line:84% And a natural question-- is R countable? 00:06:33.663 --> 00:06:35.330 align:middle line:84% Or is the set of real numbers countable? 00:06:35.330 --> 00:06:41.520 align:middle line:90% 00:06:41.520 --> 00:06:44.850 align:middle line:90% So in the end-- 00:06:44.850 --> 00:06:47.430 align:middle line:84% I'm not going to leave you in suspense-- 00:06:47.430 --> 00:06:50.100 align:middle line:90% the answer is no. 00:06:50.100 --> 00:06:51.532 align:middle line:90% And we're going to prove this. 00:06:51.532 --> 00:06:53.865 align:middle line:84% But let me just make a few remarks about what this says. 00:06:53.865 --> 00:06:56.550 align:middle line:90% 00:06:56.550 --> 00:07:02.040 align:middle line:90% So it's the following. 00:07:02.040 --> 00:07:08.200 align:middle line:84% So if Q-- so this is not a proof of this. 00:07:08.200 --> 00:07:12.260 align:middle line:84% I'm just going to make a remark based on this answer. 00:07:12.260 --> 00:07:32.120 align:middle line:84% So since R is uncountable and Q is countable, 00:07:32.120 --> 00:07:41.810 align:middle line:84% this implies that R takeaway Q, the set of irrational numbers, 00:07:41.810 --> 00:07:47.890 align:middle line:90% is, in fact, uncountable. 00:07:47.890 --> 00:07:48.390 align:middle line:90% Why? 00:07:48.390 --> 00:07:50.910 align:middle line:90% 00:07:50.910 --> 00:07:55.070 align:middle line:84% So I'm just going to say in words why this is true. 00:07:55.070 --> 00:07:58.780 align:middle line:84% So suppose otherwise that both the rational numbers 00:07:58.780 --> 00:08:01.210 align:middle line:84% and the irrationals are countable. 00:08:01.210 --> 00:08:03.700 align:middle line:84% Then you can map in a bijective fashion 00:08:03.700 --> 00:08:10.130 align:middle line:84% the rationals to the natural numbers and the irrationals 00:08:10.130 --> 00:08:11.510 align:middle line:90% also to the natural numbers. 00:08:11.510 --> 00:08:15.050 align:middle line:84% But then you can map those natural numbers to 0, minus 1, 00:08:15.050 --> 00:08:18.120 align:middle line:90% minus 2, minus 3, and so on. 00:08:18.120 --> 00:08:20.390 align:middle line:84% So then you find that there is a bijection 00:08:20.390 --> 00:08:24.320 align:middle line:84% from Q to the natural numbers and the irrationals 00:08:24.320 --> 00:08:27.200 align:middle line:84% to the integers less than or equal to 0. 00:08:27.200 --> 00:08:31.040 align:middle line:84% And therefore, there's bijection from R to the integers. 00:08:31.040 --> 00:08:34.720 align:middle line:84% But the integers are countable, which would contradict the fact 00:08:34.720 --> 00:08:36.760 align:middle line:90% that R is uncountable. 00:08:36.760 --> 00:08:40.630 align:middle line:84% Therefore, the set of irrationals is uncountable. 00:08:40.630 --> 00:08:45.370 align:middle line:84% So in short, if you didn't follow that, 00:08:45.370 --> 00:08:48.850 align:middle line:84% the set of irrationals is uncountable 00:08:48.850 --> 00:08:53.650 align:middle line:84% because otherwise this would force R to be countable. 00:08:53.650 --> 00:08:56.328 align:middle line:84% Now, I haven't proved yet that R is uncountable. 00:08:56.328 --> 00:08:57.620 align:middle line:90% We'll do that in just a minute. 00:08:57.620 --> 00:09:02.290 align:middle line:84% And we'll use a theorem about decimal expansions for-- 00:09:02.290 --> 00:09:04.105 align:middle line:84% or decimal representations of real numbers. 00:09:04.105 --> 00:09:11.560 align:middle line:90% 00:09:11.560 --> 00:09:15.440 align:middle line:84% So now, connecting rationals and irrationals 00:09:15.440 --> 00:09:19.580 align:middle line:84% to what you've seen since you were small, 00:09:19.580 --> 00:09:21.260 align:middle line:90% so to decimals, we simply-- 00:09:21.260 --> 00:09:31.140 align:middle line:84% we typically-- we think of rational numbers 00:09:31.140 --> 00:09:44.080 align:middle line:84% as in terms of decimal representations. 00:09:44.080 --> 00:09:46.640 align:middle line:90% What does that mean? 00:09:46.640 --> 00:09:49.030 align:middle line:90% That means we take-- 00:09:49.030 --> 00:10:01.420 align:middle line:84% if x is in Q, then we'll write x as some number times some digit 00:10:01.420 --> 00:10:10.330 align:middle line:84% times 10 to the k plus 10 to the k minus 1 plus d0. 00:10:10.330 --> 00:10:15.700 align:middle line:84% And then we go to the tenths spot, so d to the-- 00:10:15.700 --> 00:10:19.570 align:middle line:84% d minus 1 10 to the minus 1 plus-- 00:10:19.570 --> 00:10:20.710 align:middle line:90% so this is discussion. 00:10:20.710 --> 00:10:26.640 align:middle line:84% I'm not stating any theorems right now, 00:10:26.640 --> 00:10:35.920 align:middle line:84% until we get down to maybe some last decimal spot. 00:10:35.920 --> 00:10:57.540 align:middle line:84% And with these digits, a subset of 0, 1, 9. 00:10:57.540 --> 00:11:00.190 align:middle line:84% So all I'm saying is typically we think of x as this way. 00:11:00.190 --> 00:11:14.520 align:middle line:84% And then we write x as dk, dk minus 1, d minus 1, d minus 2, 00:11:14.520 --> 00:11:15.150 align:middle line:90% and so on. 00:11:15.150 --> 00:11:19.940 align:middle line:90% 00:11:19.940 --> 00:11:24.380 align:middle line:84% I'm using this notation because I'm going to be-- 00:11:24.380 --> 00:11:26.380 align:middle line:84% this is the notation I need to state the theorem 00:11:26.380 --> 00:11:27.970 align:middle line:90% about the real numbers. 00:11:27.970 --> 00:11:33.010 align:middle line:84% But it's quite silly that I'm about to write this down 00:11:33.010 --> 00:11:36.130 align:middle line:90% in this setting. 00:11:36.130 --> 00:11:38.470 align:middle line:90% Welcome to MIT. 00:11:38.470 --> 00:11:48.610 align:middle line:84% So for example, 1 times 10 plus 1 times 00:11:48.610 --> 00:11:53.185 align:middle line:84% 10 to the 0 plus 1 times 10 to the minus 1. 00:11:53.185 --> 00:11:56.230 align:middle line:90% 00:11:56.230 --> 00:12:04.950 align:middle line:84% This is supposed to represent the number 11.1, which 00:12:04.950 --> 00:12:06.180 align:middle line:90% is the rational number. 00:12:06.180 --> 00:12:23.120 align:middle line:90% 00:12:23.120 --> 00:12:25.640 align:middle line:84% It's quite funny that I had to think about that for a minute 00:12:25.640 --> 00:12:28.400 align:middle line:84% to go from the decimal expansion to the rational numbers. 00:12:28.400 --> 00:12:33.170 align:middle line:84% But don't give me too hard of a time. 00:12:33.170 --> 00:12:37.695 align:middle line:84% In a lot of at least pure math, we typically 00:12:37.695 --> 00:12:39.070 align:middle line:84% write things in rational numbers. 00:12:39.070 --> 00:12:40.720 align:middle line:90% We don't write in decimal-- 00:12:40.720 --> 00:12:43.480 align:middle line:84% I haven't had to write in decimal expansion in quite 00:12:43.480 --> 00:12:44.360 align:middle line:90% some time. 00:12:44.360 --> 00:12:48.670 align:middle line:84% Anyways, now, of course, not all rational numbers 00:12:48.670 --> 00:12:51.280 align:middle line:84% can be written as a finite decimal expansion. 00:12:51.280 --> 00:12:53.680 align:middle line:84% For example, 1/3 you have to write 00:12:53.680 --> 00:13:01.940 align:middle line:90% as 0.33333 and so on and so on. 00:13:01.940 --> 00:13:05.110 align:middle line:84% So that's not necessarily covered under the discussion 00:13:05.110 --> 00:13:07.790 align:middle line:90% that I have right here. 00:13:07.790 --> 00:13:10.540 align:middle line:84% But it will be covered in what I'm 00:13:10.540 --> 00:13:13.760 align:middle line:84% about to say for the real numbers. 00:13:13.760 --> 00:13:16.180 align:middle line:90% So we could cover 1/3. 00:13:16.180 --> 00:13:18.340 align:middle line:84% And then we can also cover every real number 00:13:18.340 --> 00:13:22.630 align:middle line:84% if we allow infinite decimal representations. 00:13:22.630 --> 00:13:24.910 align:middle line:84% So I'll use the word decimal expansions or decimal 00:13:24.910 --> 00:13:27.430 align:middle line:84% representations interchangeably there. 00:13:27.430 --> 00:13:31.538 align:middle line:90% So let me make a definition. 00:13:31.538 --> 00:13:33.580 align:middle line:84% And what I'm going to state is about real numbers 00:13:33.580 --> 00:13:35.380 align:middle line:90% between 0 and 1. 00:13:35.380 --> 00:13:37.660 align:middle line:84% Of course, if you want to talk about real numbers 00:13:37.660 --> 00:13:41.500 align:middle line:84% bigger than 1, then you just tack on 1 to the-- 00:13:41.500 --> 00:13:43.285 align:middle line:90% if you like, to the d0 place. 00:13:43.285 --> 00:13:51.880 align:middle line:90% 00:13:51.880 --> 00:13:54.340 align:middle line:84% So the infinite part is having to deal with what 00:13:54.340 --> 00:13:57.560 align:middle line:90% comes after the decimal point. 00:13:57.560 --> 00:14:00.290 align:middle line:84% So the let me make the following. 00:14:00.290 --> 00:14:03.220 align:middle line:84% So this is a rigorous definition. 00:14:03.220 --> 00:14:06.550 align:middle line:90% Let x in 0, 1 excluding 0. 00:14:06.550 --> 00:14:10.270 align:middle line:90% 00:14:10.270 --> 00:14:16.201 align:middle line:84% And let d minus j be in 0, 1 to 9. 00:14:16.201 --> 00:14:24.767 align:middle line:90% 00:14:24.767 --> 00:14:29.130 align:middle line:90% for j, a natural number. 00:14:29.130 --> 00:14:34.290 align:middle line:90% So we say x is-- 00:14:34.290 --> 00:14:38.590 align:middle line:90% 00:14:38.590 --> 00:14:40.480 align:middle line:84% so what does it mean to say that x 00:14:40.480 --> 00:14:43.990 align:middle line:84% is given by an infinite decimal expansion 00:14:43.990 --> 00:14:48.130 align:middle line:84% or given by an infinite decimal representation? 00:14:48.130 --> 00:14:52.690 align:middle line:84% We say x is represented by the digits. 00:14:52.690 --> 00:15:04.070 align:middle line:90% 00:15:04.070 --> 00:15:13.800 align:middle line:84% And we write x is equal to d minus 1, d minus 2, d minus 3. 00:15:13.800 --> 00:15:17.740 align:middle line:90% 00:15:17.740 --> 00:15:21.110 align:middle line:84% If-- so this is the precise meaning of this. 00:15:21.110 --> 00:15:27.955 align:middle line:84% If x is equal to the supremum of the set-- 00:15:27.955 --> 00:15:31.020 align:middle line:90% 00:15:31.020 --> 00:15:35.010 align:middle line:84% so I take a finite decimal expansion. 00:15:35.010 --> 00:15:41.900 align:middle line:90% 00:15:41.900 --> 00:15:46.760 align:middle line:84% And then so I get a number for each n, a natural number. 00:15:46.760 --> 00:15:51.200 align:middle line:84% So the first one is just d minus 1 10 to the minus 1. 00:15:51.200 --> 00:15:54.260 align:middle line:84% And then for n equals 2, I tack on next hundredths place, 00:15:54.260 --> 00:15:55.920 align:middle line:90% and then the thousandths place. 00:15:55.920 --> 00:16:00.650 align:middle line:84% And so I get a set of real numbers. 00:16:00.650 --> 00:16:02.720 align:middle line:84% If I take its supremum and I get x, 00:16:02.720 --> 00:16:08.810 align:middle line:84% then I say that x is represented by these digits or that x-- 00:16:08.810 --> 00:16:12.200 align:middle line:84% or that this gives a decimal representation of x. 00:16:12.200 --> 00:16:25.075 align:middle line:84% So that you see this lines up with what you know, 00:16:25.075 --> 00:16:32.470 align:middle line:84% if I look at 0.250000, remember this is now the sup of what? 00:16:32.470 --> 00:16:35.350 align:middle line:90% 00:16:35.350 --> 00:16:36.640 align:middle line:90% Of a certain set. 00:16:36.640 --> 00:16:39.400 align:middle line:90% So let me do first this guy. 00:16:39.400 --> 00:16:43.330 align:middle line:90% So this is 2/10. 00:16:43.330 --> 00:16:48.070 align:middle line:84% And then now, I add the hundredths place plus 5/100, 00:16:48.070 --> 00:16:53.980 align:middle line:84% and then 2/10 plus 5/100-- this is a set, remember-- 00:16:53.980 --> 00:16:59.590 align:middle line:90% plus 0/1000 over plus-- 00:16:59.590 --> 00:17:03.280 align:middle line:84% and at each next guy, I'm supposed 00:17:03.280 --> 00:17:06.069 align:middle line:84% to take this and add on another 0. 00:17:06.069 --> 00:17:13.210 align:middle line:84% So as a set, this is just a set containing two elements, 00:17:13.210 --> 00:17:24.930 align:middle line:90% namely 2/10 and 25/100. 00:17:24.930 --> 00:17:28.420 align:middle line:84% And so this is bigger than that guy. 00:17:28.420 --> 00:17:31.770 align:middle line:84% So the supremum is 25/100, which equals 1/4. 00:17:31.770 --> 00:17:33.790 align:middle line:84% OK, so all of that to tell you that, 00:17:33.790 --> 00:17:36.400 align:middle line:90% yes, this definition should-- 00:17:36.400 --> 00:17:39.510 align:middle line:84% or at least does ring true with what 00:17:39.510 --> 00:17:51.040 align:middle line:84% you were taught, that 1/4 is equal to 0.25, all right? 00:17:51.040 --> 00:17:56.715 align:middle line:84% OK, so now that what we're done with the triangle inequality, 00:17:56.715 --> 00:17:57.715 align:middle line:90% I'm going to erase this. 00:17:57.715 --> 00:18:34.220 align:middle line:90% 00:18:34.220 --> 00:18:37.360 align:middle line:84% And so this is the last bit of discussion 00:18:37.360 --> 00:18:40.693 align:middle line:84% about the elementary properties of the real numbers. 00:18:40.693 --> 00:18:42.985 align:middle line:84% Next, we'll be talking about sequences of real numbers. 00:18:42.985 --> 00:18:46.900 align:middle line:90% 00:18:46.900 --> 00:18:54.310 align:middle line:84% OK, so all right, so I have this definition. 00:18:54.310 --> 00:18:57.550 align:middle line:84% The fundamental theorem about the real numbers 00:18:57.550 --> 00:19:00.460 align:middle line:84% is that every real number, at least in this setting between 0 00:19:00.460 --> 00:19:05.690 align:middle line:84% and 1, can be represented by a set of digits 00:19:05.690 --> 00:19:11.070 align:middle line:84% or it has a decimal representation. 00:19:11.070 --> 00:19:14.310 align:middle line:84% And every decimal-- if you like, every set of digits 00:19:14.310 --> 00:19:17.340 align:middle line:84% corresponds to a decimal representation 00:19:17.340 --> 00:19:18.550 align:middle line:90% of a real number. 00:19:18.550 --> 00:19:24.340 align:middle line:84% So let me state the second thing I said first. 00:19:24.340 --> 00:19:56.870 align:middle line:84% So for every set of digits d minus j with each of these d 00:19:56.870 --> 00:20:10.530 align:middle line:84% minus j's in 0 up to 9, then there 00:20:10.530 --> 00:20:23.660 align:middle line:84% exists a unique x in closed interval 0, 1 such 00:20:23.660 --> 00:20:28.700 align:middle line:90% that x is equal to d minus 1-- 00:20:28.700 --> 00:20:30.680 align:middle line:84% with this decimal representation. 00:20:30.680 --> 00:20:33.320 align:middle line:90% 00:20:33.320 --> 00:20:38.150 align:middle line:84% So you give me some digits and I can 00:20:38.150 --> 00:20:40.610 align:middle line:84% find a unique real number that has 00:20:40.610 --> 00:20:44.000 align:middle line:90% that decimal representation. 00:20:44.000 --> 00:20:48.020 align:middle line:84% Now, before I state the second part of the theorem, 00:20:48.020 --> 00:20:49.970 align:middle line:84% this says that I give you digits, 00:20:49.970 --> 00:20:51.600 align:middle line:90% I get a unique real number. 00:20:51.600 --> 00:20:53.900 align:middle line:84% Now, you ask about the other direction. 00:20:53.900 --> 00:20:57.620 align:middle line:84% If I have a real number, do I get a unique set of digits? 00:20:57.620 --> 00:21:01.430 align:middle line:84% And of course, that is not true necessarily 00:21:01.430 --> 00:21:16.960 align:middle line:84% because, for example, 1/2, this is equal to 0.500 and so on. 00:21:16.960 --> 00:21:22.420 align:middle line:84% But this is also equal to 0.4999 and so on. 00:21:22.420 --> 00:21:31.370 align:middle line:84% So yes, every set of digits gives me a unique real number. 00:21:31.370 --> 00:21:32.990 align:middle line:84% So if I gave you these digits, they 00:21:32.990 --> 00:21:34.940 align:middle line:90% would spit out 1/2 and only 1/2. 00:21:34.940 --> 00:21:37.940 align:middle line:90% It wouldn't spit out 1/4. 00:21:37.940 --> 00:21:40.260 align:middle line:84% But if you give me a real number, 00:21:40.260 --> 00:21:46.480 align:middle line:84% there does not necessarily exist a unique set 00:21:46.480 --> 00:21:49.570 align:middle line:84% of digits giving a decimal expansion for that number. 00:21:49.570 --> 00:21:52.060 align:middle line:84% So I can have two different decimal expansions. 00:21:52.060 --> 00:21:54.745 align:middle line:90% I have 0.5 and I have 0.4999. 00:21:54.745 --> 00:21:56.620 align:middle line:84% I will always have at least one, but it's not 00:21:56.620 --> 00:21:58.300 align:middle line:90% necessarily unique. 00:21:58.300 --> 00:22:02.590 align:middle line:84% But we can single out one unique choice 00:22:02.590 --> 00:22:06.730 align:middle line:84% by requiring that, in some sense, if I were to truncate 00:22:06.730 --> 00:22:10.660 align:middle line:84% it, then the decimal expansion would be less than the number 00:22:10.660 --> 00:22:13.870 align:middle line:84% I'm looking at and wanting to expand. 00:22:13.870 --> 00:22:17.190 align:middle line:84% So that's the second part of this theorem, 00:22:17.190 --> 00:22:28.430 align:middle line:84% is that for every x in 0, 1, now excluding 0, 00:22:28.430 --> 00:22:36.070 align:middle line:90% there exists unique digits-- 00:22:36.070 --> 00:22:44.140 align:middle line:90% 00:22:44.140 --> 00:22:47.530 align:middle line:84% and as before, these are in 0 up to 9-- 00:22:47.530 --> 00:22:53.560 align:middle line:90% such that x is-- 00:22:53.560 --> 00:22:56.230 align:middle line:84% such that these give a decimal representation of x. 00:22:56.230 --> 00:22:58.760 align:middle line:90% 00:22:58.760 --> 00:23:04.610 align:middle line:84% And so what singles out a choice if we're ever coming up 00:23:04.610 --> 00:23:05.720 align:middle line:90% against these two things? 00:23:05.720 --> 00:23:08.510 align:middle line:90% 00:23:08.510 --> 00:23:17.128 align:middle line:84% And if I truncate the decimal expansion-- 00:23:17.128 --> 00:23:18.920 align:middle line:84% so I'm not going to write zeros after that. 00:23:18.920 --> 00:23:22.880 align:middle line:84% I'm just going to truncate it-- this always is less than x 00:23:22.880 --> 00:23:31.635 align:middle line:84% and this is less than or equal to plus 10 to the minus n. 00:23:31.635 --> 00:23:37.710 align:middle line:90% 00:23:37.710 --> 00:23:47.520 align:middle line:84% All right, so this inequality will always hold. 00:23:47.520 --> 00:23:53.600 align:middle line:84% And in fact, so this inequality will always hold. 00:23:53.600 --> 00:23:55.280 align:middle line:84% And in fact, x is always bigger than 00:23:55.280 --> 00:23:58.140 align:middle line:84% or equal to this side, no matter what. 00:23:58.140 --> 00:24:03.320 align:middle line:84% If x is given by a decimal representation, 00:24:03.320 --> 00:24:06.950 align:middle line:90% so we always have these two. 00:24:06.950 --> 00:24:08.720 align:middle line:84% This is a less than or equal to and this 00:24:08.720 --> 00:24:10.160 align:middle line:90% is a less than or equal to. 00:24:10.160 --> 00:24:11.960 align:middle line:84% But we single out a unique choice 00:24:11.960 --> 00:24:15.110 align:middle line:84% of decimal representation by choosing 00:24:15.110 --> 00:24:17.630 align:middle line:90% this to be a strict less than. 00:24:17.630 --> 00:24:23.690 align:middle line:84% So then that forces us to choose this guy over this guy. 00:24:23.690 --> 00:24:32.540 align:middle line:84% So for theorem 2, this would say that the unique representation 00:24:32.540 --> 00:24:36.680 align:middle line:84% that satisfies this for 1/2 is given by 0.4999. 00:24:36.680 --> 00:24:42.060 align:middle line:84% And for example, for 1 would be just 0.9999 00:24:42.060 --> 00:24:45.390 align:middle line:84% because if I take this representation and cut it off, 00:24:45.390 --> 00:24:51.990 align:middle line:84% this is actually equal to 1/2 and not less than 1/2. 00:24:51.990 --> 00:24:55.410 align:middle line:84% OK, so that was a lot of discussion about the theorem. 00:24:55.410 --> 00:24:57.510 align:middle line:84% But that was for a reason because I'm not actually 00:24:57.510 --> 00:25:00.450 align:middle line:90% going to prove this theorem. 00:25:00.450 --> 00:25:04.980 align:middle line:84% It's not entirely difficult. It just uses 00:25:04.980 --> 00:25:07.420 align:middle line:84% the least upper bound property of the real numbers. 00:25:07.420 --> 00:25:12.350 align:middle line:84% But it's a little clunky to write down. 00:25:12.350 --> 00:25:16.400 align:middle line:84% And what I'd rather do is just use this theorem 00:25:16.400 --> 00:25:21.110 align:middle line:84% to prove my answer, the answer to my question 00:25:21.110 --> 00:25:25.880 align:middle line:84% up there, on whether R is accountable or not. 00:25:25.880 --> 00:25:27.950 align:middle line:84% In the textbook, there is a proof 00:25:27.950 --> 00:25:29.450 align:middle line:84% of this theorem, which you can read. 00:25:29.450 --> 00:25:35.130 align:middle line:90% 00:25:35.130 --> 00:25:39.270 align:middle line:84% OK, so we're actually going to use this theorem 00:25:39.270 --> 00:25:46.790 align:middle line:84% to prove the following theorem due to Cantor. 00:25:46.790 --> 00:25:50.330 align:middle line:84% So we just went from absolute values, which maybe it was not 00:25:50.330 --> 00:25:53.600 align:middle line:84% too shock and awe, now to this theorem, which to me is shock 00:25:53.600 --> 00:26:12.530 align:middle line:84% and awe, is the following, that the interval 0, 1 is 00:26:12.530 --> 00:26:13.280 align:middle line:90% uncountable. 00:26:13.280 --> 00:26:17.410 align:middle line:90% 00:26:17.410 --> 00:26:22.710 align:middle line:84% All right, so maybe you're asking yourself, 00:26:22.710 --> 00:26:25.700 align:middle line:84% well, that's showing this is uncountable. 00:26:25.700 --> 00:26:32.490 align:middle line:84% And that's saying R is uncountable. 00:26:32.490 --> 00:26:35.700 align:middle line:84% So where do we go from this to that? 00:26:35.700 --> 00:26:37.310 align:middle line:84% Well, I'll let you think about that. 00:26:37.310 --> 00:26:40.280 align:middle line:84% But let's assume this was uncountable 00:26:40.280 --> 00:26:41.640 align:middle line:90% and R was countable. 00:26:41.640 --> 00:26:44.990 align:middle line:84% So I'm just going to say this, say why out loud. 00:26:44.990 --> 00:26:49.700 align:middle line:84% If this was uncountable and R was countable, 00:26:49.700 --> 00:26:55.640 align:middle line:84% then since this is a subset of R, that means this under 00:26:55.640 --> 00:26:57.840 align:middle line:90% would be-- 00:26:57.840 --> 00:26:59.790 align:middle line:90% so let me just write. 00:26:59.790 --> 00:27:08.592 align:middle line:90% 00:27:08.592 --> 00:27:09.810 align:middle line:90% Why does this imply-- 00:27:09.810 --> 00:27:15.090 align:middle line:90% 00:27:15.090 --> 00:27:16.920 align:middle line:84% And I'm going to give you a fake proof, 00:27:16.920 --> 00:27:22.078 align:middle line:84% but you can actually make this completely rigorous-- 00:27:22.078 --> 00:27:23.620 align:middle line:84% fake proof in the sense that I'm just 00:27:23.620 --> 00:27:29.050 align:middle line:84% going to write symbols that imply this. 00:27:29.050 --> 00:27:34.150 align:middle line:84% So why is R uncountable then if this is uncountable? 00:27:34.150 --> 00:27:39.460 align:middle line:84% Well, the function that takes an element in here 00:27:39.460 --> 00:27:43.100 align:middle line:84% just into R, this is clearly bijective. 00:27:43.100 --> 00:27:49.690 align:middle line:84% so just the identity map from 0, 1 into R is an injective map. 00:27:49.690 --> 00:27:57.310 align:middle line:84% So I clearly have that the cardinality of this set 00:27:57.310 --> 00:28:01.900 align:middle line:84% is less than or equal to the cardinality of R. 00:28:01.900 --> 00:28:04.190 align:middle line:84% In fact, maybe you can think about it. 00:28:04.190 --> 00:28:05.690 align:middle line:84% Maybe I'll put it on the assignment. 00:28:05.690 --> 00:28:10.480 align:middle line:84% But in fact, one can prove they have the same cardinality. 00:28:10.480 --> 00:28:14.470 align:middle line:90% But I have that. 00:28:14.470 --> 00:28:15.850 align:middle line:90% And this is uncountable. 00:28:15.850 --> 00:28:19.120 align:middle line:84% Therefore, its cardinality is bigger 00:28:19.120 --> 00:28:22.370 align:middle line:90% than the natural numbers. 00:28:22.370 --> 00:28:24.850 align:middle line:84% So this should imply that cardinality 00:28:24.850 --> 00:28:32.600 align:middle line:84% of the natural numbers is less than the cardinality of R. 00:28:32.600 --> 00:28:37.230 align:middle line:84% In other words, R is uncountable. 00:28:37.230 --> 00:28:41.160 align:middle line:84% All right, so let's prove this theorem due to Cantor. 00:28:41.160 --> 00:28:44.370 align:middle line:84% Now, this other theorem due to Cantor, 00:28:44.370 --> 00:28:48.428 align:middle line:84% that the cardinality of the power set 00:28:48.428 --> 00:28:50.220 align:middle line:84% is bigger than the cardinality of the set-- 00:28:50.220 --> 00:28:51.150 align:middle line:90% I mean, that was-- 00:28:51.150 --> 00:28:54.700 align:middle line:90% 00:28:54.700 --> 00:28:57.420 align:middle line:84% at least the proof, the first time 00:28:57.420 --> 00:28:59.610 align:middle line:84% I saw it, that was some crazy shit. 00:28:59.610 --> 00:29:03.600 align:middle line:84% And this is also going to be some pretty crazy and clever 00:29:03.600 --> 00:29:06.620 align:middle line:90% shit, too. 00:29:06.620 --> 00:29:10.490 align:middle line:84% So this is what's referred to as Cantor's diagonalization 00:29:10.490 --> 00:29:14.432 align:middle line:90% argument or diagonal argument. 00:29:14.432 --> 00:29:15.390 align:middle line:90% And it's the following. 00:29:15.390 --> 00:29:17.307 align:middle line:84% So we're going to prove this by contradiction. 00:29:17.307 --> 00:29:23.730 align:middle line:90% 00:29:23.730 --> 00:29:26.613 align:middle line:84% So we're going to assume that this is countable. 00:29:26.613 --> 00:29:27.530 align:middle line:90% So it can't be finite. 00:29:27.530 --> 00:29:30.690 align:middle line:84% There's infinitely many of the elements in here. 00:29:30.690 --> 00:29:41.082 align:middle line:84% So suppose that, in fact, it has the cardinality 00:29:41.082 --> 00:29:42.040 align:middle line:90% of the natural numbers. 00:29:42.040 --> 00:29:45.120 align:middle line:90% 00:29:45.120 --> 00:29:49.480 align:middle line:84% And we're going to arrive at a contradiction. 00:29:49.480 --> 00:29:53.970 align:middle line:84% So then there exists a function, which 00:29:53.970 --> 00:29:58.890 align:middle line:84% I'm going to call x, from natural numbers 00:29:58.890 --> 00:30:05.000 align:middle line:90% into 0, 1, which is bijective. 00:30:05.000 --> 00:30:20.110 align:middle line:90% 00:30:20.110 --> 00:30:32.670 align:middle line:84% Now, so let me label this inequality star. 00:30:32.670 --> 00:30:38.910 align:middle line:90% For each n, we write x of n. 00:30:38.910 --> 00:30:41.730 align:middle line:90% So this is an element of 0, 1. 00:30:41.730 --> 00:30:42.960 align:middle line:90% So this is bijective. 00:30:42.960 --> 00:30:46.560 align:middle line:84% So every element in 0 gets mapped onto. 00:30:46.560 --> 00:30:54.760 align:middle line:84% So we write x of n to be in its decimal representation. 00:30:54.760 --> 00:31:00.610 align:middle line:84% I have two indices now-- one for the digit place and the one 00:31:00.610 --> 00:31:02.605 align:middle line:90% for which-- 00:31:02.605 --> 00:31:07.940 align:middle line:90% 00:31:07.940 --> 00:31:10.052 align:middle line:90% excuse me. 00:31:10.052 --> 00:31:11.760 align:middle line:84% At least I sheltered you from that sneeze 00:31:11.760 --> 00:31:13.440 align:middle line:84% by putting my hand over the microphone. 00:31:13.440 --> 00:31:14.870 align:middle line:90% Is the microphone still on? 00:31:14.870 --> 00:31:15.870 align:middle line:90% Yeah. 00:31:15.870 --> 00:31:25.260 align:middle line:84% OK, so for each n, we write each x of n in its decimal 00:31:25.260 --> 00:31:36.050 align:middle line:90% representation, satisfying star. 00:31:36.050 --> 00:31:41.460 align:middle line:84% So each x in here can be written uniquely 00:31:41.460 --> 00:31:43.890 align:middle line:84% in a given decimal representation satisfying 00:31:43.890 --> 00:31:45.800 align:middle line:90% this inequality here. 00:31:45.800 --> 00:31:47.670 align:middle line:84% And what we're going to do is we're 00:31:47.670 --> 00:31:57.610 align:middle line:84% going to come up with a real number in the interval 0, 1 00:31:57.610 --> 00:32:01.030 align:middle line:84% which x does not map to, which therefore contradicts 00:32:01.030 --> 00:32:02.770 align:middle line:90% the fact that x is surjective. 00:32:02.770 --> 00:32:07.660 align:middle line:84% And bijective surjective means that everything 00:32:07.660 --> 00:32:08.950 align:middle line:90% has to get mapped onto. 00:32:08.950 --> 00:32:11.350 align:middle line:90% That's the onto part. 00:32:11.350 --> 00:32:13.330 align:middle line:84% Injective means it maps different things 00:32:13.330 --> 00:32:14.770 align:middle line:90% to different things. 00:32:14.770 --> 00:32:22.780 align:middle line:84% So what's the idea of finding this element that 00:32:22.780 --> 00:32:25.680 align:middle line:90% doesn't get mapped onto? 00:32:25.680 --> 00:32:57.990 align:middle line:84% So let me write down just a few of these decimal expansions. 00:32:57.990 --> 00:33:00.240 align:middle line:84% And I may not want to do a fourth one. 00:33:00.240 --> 00:33:05.550 align:middle line:90% 00:33:05.550 --> 00:33:08.970 align:middle line:90% And maybe I do. 00:33:08.970 --> 00:33:10.698 align:middle line:90% So this is, again, discussion. 00:33:10.698 --> 00:33:12.240 align:middle line:84% So I'm going to put this in brackets, 00:33:12.240 --> 00:33:13.823 align:middle line:84% meaning this is not part of the proof, 00:33:13.823 --> 00:33:17.520 align:middle line:84% but this is part of the idea of what's going on. 00:33:17.520 --> 00:33:22.290 align:middle line:90% And where's the diagonal part? 00:33:22.290 --> 00:33:28.420 align:middle line:90% 00:33:28.420 --> 00:33:35.200 align:middle line:84% OK, so and this should go off in this direction . 00:33:35.200 --> 00:33:38.120 align:middle line:84% And the decimal expansion is marked off in that direction. 00:33:38.120 --> 00:33:43.360 align:middle line:84% And so somehow, this list, if it kept going in that direction, 00:33:43.360 --> 00:33:47.890 align:middle line:84% would have to contain every real number between 0 and 1. 00:33:47.890 --> 00:33:52.360 align:middle line:84% And so what's the idea to come up with a real number between 0 00:33:52.360 --> 00:33:58.120 align:middle line:84% and one that is not in this list is we go down 00:33:58.120 --> 00:34:06.850 align:middle line:84% the diagonal of the decimal expansions 00:34:06.850 --> 00:34:14.010 align:middle line:84% and we change each decimal to something else than what it is. 00:34:14.010 --> 00:34:20.810 align:middle line:84% And so therefore-- and then I take my element y. 00:34:20.810 --> 00:34:31.449 align:middle line:84% So remember, we're trying to find a y not in this list. 00:34:31.449 --> 00:34:35.340 align:middle line:84% And then I would take y to be the real number 00:34:35.340 --> 00:34:46.190 align:middle line:84% that I get by changing each of these decimal digits. 00:34:46.190 --> 00:34:49.130 align:middle line:84% So you can imagine if y somehow popped up-- 00:34:49.130 --> 00:34:52.909 align:middle line:84% let's say it popped up, since I don't want to write anymore, 00:34:52.909 --> 00:34:56.389 align:middle line:84% as x4, then somehow this number would 00:34:56.389 --> 00:35:01.820 align:middle line:84% have to be the digit for y, but I changed it. 00:35:01.820 --> 00:35:03.740 align:middle line:84% And therefore, it's not the digit of y. 00:35:03.740 --> 00:35:07.370 align:middle line:90% 00:35:07.370 --> 00:35:10.340 align:middle line:90% So that's it in a nutshell. 00:35:10.340 --> 00:35:12.470 align:middle line:90% 00:35:12.470 --> 00:35:13.970 align:middle line:84% If you didn't get that, that's fine. 00:35:13.970 --> 00:35:19.730 align:middle line:84% Let me now write out the details. 00:35:19.730 --> 00:35:23.330 align:middle line:84% So it's a little bit simpler to imagine if, let's say, 00:35:23.330 --> 00:35:24.740 align:middle line:90% all of these were 0's and 1's. 00:35:24.740 --> 00:35:28.310 align:middle line:84% So instead of being base 10, now it's base 2. 00:35:28.310 --> 00:35:31.340 align:middle line:90% And so instead of-- 00:35:31.340 --> 00:35:33.890 align:middle line:84% if the digit was the 0, I've flipped it to 1. 00:35:33.890 --> 00:35:37.820 align:middle line:84% So you can imagine that if y did-- 00:35:37.820 --> 00:35:42.590 align:middle line:84% so and then I form y by flipping the digits. 00:35:42.590 --> 00:35:51.440 align:middle line:84% So if this was 0, 0, 1, 1, then y would start off 1, 1, 0, 0. 00:35:51.440 --> 00:35:56.840 align:middle line:84% And therefore, if y appeared, say, fourth in line, 00:35:56.840 --> 00:36:02.510 align:middle line:84% it would have to be 1 and 0 in the same spot 00:36:02.510 --> 00:36:05.540 align:middle line:84% just by how I've constructed y by flipping it. 00:36:05.540 --> 00:36:07.730 align:middle line:84% And therefore, y can't be in that list. 00:36:07.730 --> 00:36:12.010 align:middle line:84% And therefore, this map cannot be surjective, 00:36:12.010 --> 00:36:14.690 align:middle line:90% which is our contradiction. 00:36:14.690 --> 00:36:16.900 align:middle line:84% So let me make this more precise. 00:36:16.900 --> 00:36:20.670 align:middle line:90% 00:36:20.670 --> 00:36:27.810 align:middle line:90% Let ej-- this is going to be 1. 00:36:27.810 --> 00:36:39.680 align:middle line:84% If the j-th digit of the j-th x of j 00:36:39.680 --> 00:36:50.630 align:middle line:84% does not equal 1; 2, if the j-th digit 00:36:50.630 --> 00:36:57.240 align:middle line:84% of the j-th guy in that list, or x of j, does equal 1. 00:36:57.240 --> 00:37:03.550 align:middle line:90% 00:37:03.550 --> 00:37:09.370 align:middle line:84% By part 1 of the theorem, previous theorem, 00:37:09.370 --> 00:37:18.320 align:middle line:84% there exists the unique y in 0, 1 without 0-- 00:37:18.320 --> 00:37:18.820 align:middle line:90% why? 00:37:18.820 --> 00:37:22.600 align:middle line:84% Because some of these digits are non-zero. 00:37:22.600 --> 00:37:24.130 align:middle line:90% It's always either 1 or 2-- 00:37:24.130 --> 00:37:28.875 align:middle line:90% 00:37:28.875 --> 00:37:36.650 align:middle line:84% such that y is equal to e minus 1, e minus 2, and so on. 00:37:36.650 --> 00:37:41.620 align:middle line:90% 00:37:41.620 --> 00:37:59.860 align:middle line:84% So moreover, since all of these digits are either 1 or 2, 00:37:59.860 --> 00:38:05.750 align:middle line:90% they are certainly non-zero. 00:38:05.750 --> 00:38:08.600 align:middle line:90% 00:38:08.600 --> 00:38:12.200 align:middle line:84% So y is given by a decimal expansion where all 00:38:12.200 --> 00:38:14.900 align:middle line:90% of these things are non-zero. 00:38:14.900 --> 00:38:21.560 align:middle line:84% So that means that for all n, a natural number, 00:38:21.560 --> 00:38:26.870 align:middle line:84% if I take a finite decimal expansion-- so I cut this 00:38:26.870 --> 00:38:31.070 align:middle line:84% off at e to the minus n-- since e to the e minus n 00:38:31.070 --> 00:38:34.770 align:middle line:84% plus 1 is either 1 or 2, that's going to be positive. 00:38:34.770 --> 00:38:39.470 align:middle line:90% So that's less than y. 00:38:39.470 --> 00:38:41.690 align:middle line:84% And this is always the case, that this 00:38:41.690 --> 00:38:42.980 align:middle line:90% is less than or equal to. 00:38:42.980 --> 00:38:49.530 align:middle line:90% 00:38:49.530 --> 00:38:52.680 align:middle line:84% So because all these digits are positive, 00:38:52.680 --> 00:38:54.870 align:middle line:84% if I cut it off, if I truncate it, 00:38:54.870 --> 00:38:58.410 align:middle line:84% then that number is going to be less than y 00:38:58.410 --> 00:39:01.380 align:middle line:84% because I'm missing stuff out in the millionths place 00:39:01.380 --> 00:39:06.720 align:middle line:84% or whatever each time for each n. 00:39:06.720 --> 00:39:17.570 align:middle line:84% And therefore, this decimal expansion 00:39:17.570 --> 00:39:31.490 align:middle line:84% is the unique decimal representation 00:39:31.490 --> 00:39:35.330 align:middle line:84% from part 2 of the theorem that I stated. 00:39:35.330 --> 00:39:38.090 align:middle line:84% Every element in 0, 1 has a unique representation, 00:39:38.090 --> 00:39:40.070 align:middle line:90% satisfying that inequality. 00:39:40.070 --> 00:39:44.120 align:middle line:84% Because all of these digits are positive, this, in fact, 00:39:44.120 --> 00:39:46.130 align:middle line:84% is a unique representation of this element 00:39:46.130 --> 00:39:51.410 align:middle line:84% y, which I've constructed by flipping digits from this map. 00:39:51.410 --> 00:39:56.580 align:middle line:90% 00:39:56.580 --> 00:40:09.240 align:middle line:84% Now, since x is surjective, there 00:40:09.240 --> 00:40:12.890 align:middle line:90% exists an n, natural number-- 00:40:12.890 --> 00:40:14.750 align:middle line:84% this n has nothing to do with that n. 00:40:14.750 --> 00:40:19.100 align:middle line:84% Maybe I'll use a different letter, m-- 00:40:19.100 --> 00:40:23.000 align:middle line:90% such that y equals x of m. 00:40:23.000 --> 00:40:26.300 align:middle line:90% 00:40:26.300 --> 00:40:34.130 align:middle line:84% Then let's say I look at e to the minus m. 00:40:34.130 --> 00:40:36.976 align:middle line:90% 00:40:36.976 --> 00:40:41.640 align:middle line:84% So then this says that the m-th digit, 00:40:41.640 --> 00:40:52.790 align:middle line:84% which should be giving me e to the minus m because this is 00:40:52.790 --> 00:40:55.280 align:middle line:90% equal to -- 00:40:55.280 --> 00:41:14.580 align:middle line:84% this, remember, is equal to 1 if dm 2 if dm equals 1. 00:41:14.580 --> 00:41:18.915 align:middle line:84% In particular, this does not equal dm. 00:41:18.915 --> 00:41:22.230 align:middle line:90% 00:41:22.230 --> 00:41:27.885 align:middle line:84% So I have proven that this number does not equal itself. 00:41:27.885 --> 00:41:32.050 align:middle line:90% 00:41:32.050 --> 00:41:34.310 align:middle line:90% All right, I can never if-- 00:41:34.310 --> 00:41:37.220 align:middle line:84% I can never have this number, whatever 00:41:37.220 --> 00:41:41.000 align:middle line:84% it is, equal to this thing because I'm always changing it. 00:41:41.000 --> 00:41:47.210 align:middle line:84% If this thing is 1, then this number is 2. 00:41:47.210 --> 00:41:49.917 align:middle line:84% If this number is not 1, then this number is 1. 00:41:49.917 --> 00:41:51.500 align:middle line:84% And therefore, it's not equal to that. 00:41:51.500 --> 00:41:55.276 align:middle line:90% 00:41:55.276 --> 00:41:58.510 align:middle line:90% And this is a contradiction. 00:41:58.510 --> 00:42:11.170 align:middle line:84% Thus, it shows that this set is uncountable. 00:42:11.170 --> 00:42:18.710 align:middle line:90% 00:42:18.710 --> 00:42:20.650 align:middle line:84% So I hope I explained that well enough. 00:42:20.650 --> 00:42:23.390 align:middle line:90% 00:42:23.390 --> 00:42:27.220 align:middle line:84% You can also look in the textbook for an explanation 00:42:27.220 --> 00:42:28.045 align:middle line:90% as well. 00:42:28.045 --> 00:42:29.920 align:middle line:84% And of course, you can ask me in office hours 00:42:29.920 --> 00:42:34.480 align:middle line:90% for more explanation. 00:42:34.480 --> 00:42:37.462 align:middle line:90% 00:42:37.462 --> 00:42:39.070 align:middle line:90% Let me fuel up real quick. 00:42:39.070 --> 00:42:45.880 align:middle line:90% 00:42:45.880 --> 00:42:48.595 align:middle line:84% OK, now, we're moving onto a new chapter. 00:42:48.595 --> 00:42:56.717 align:middle line:90% 00:42:56.717 --> 00:42:57.675 align:middle line:90% We're going to get to-- 00:42:57.675 --> 00:43:02.060 align:middle line:90% 00:43:02.060 --> 00:43:04.500 align:middle line:84% now, we're going to get to the analysis part. 00:43:04.500 --> 00:43:07.550 align:middle line:84% So analysis as I've said before, is the study of limits. 00:43:07.550 --> 00:43:09.620 align:middle line:84% Real analysis is the study of limits that 00:43:09.620 --> 00:43:12.780 align:middle line:90% has to deal with real numbers. 00:43:12.780 --> 00:43:19.620 align:middle line:84% So let's get to our first notion of a limit. 00:43:19.620 --> 00:43:21.915 align:middle line:84% And that is the limit of a sequence of real numbers. 00:43:21.915 --> 00:43:24.510 align:middle line:90% 00:43:24.510 --> 00:43:33.190 align:middle line:84% So now, we're moving onto sequences and series. 00:43:33.190 --> 00:43:37.470 align:middle line:90% 00:43:37.470 --> 00:43:39.450 align:middle line:84% So what is the precise definition 00:43:39.450 --> 00:43:41.145 align:middle line:90% of a sequence of real numbers? 00:43:41.145 --> 00:43:46.040 align:middle line:90% 00:43:46.040 --> 00:44:01.090 align:middle line:84% A sequence of real numbers is precisely 00:44:01.090 --> 00:44:04.990 align:middle line:84% a function x from the natural numbers 00:44:04.990 --> 00:44:08.440 align:middle line:90% into R, just a function. 00:44:08.440 --> 00:44:11.270 align:middle line:84% It doesn't have to be bijective, surjective, injective, 00:44:11.270 --> 00:44:11.770 align:middle line:90% anything. 00:44:11.770 --> 00:44:14.140 align:middle line:84% It's just a function from the natural numbers to R. 00:44:14.140 --> 00:44:18.430 align:middle line:84% So that's precisely what a sequence of real numbers is. 00:44:18.430 --> 00:44:22.060 align:middle line:84% Now, we typically don't think of a sequence as a function. 00:44:22.060 --> 00:44:24.070 align:middle line:84% In fact, we don't even use that notation. 00:44:24.070 --> 00:44:28.820 align:middle line:90% 00:44:28.820 --> 00:44:41.280 align:middle line:84% We denote x of n by x sub n and the associated 00:44:41.280 --> 00:44:51.900 align:middle line:84% sequence by this curly brackets, n equals 1 to infinity. 00:44:51.900 --> 00:44:58.770 align:middle line:84% Or we might not even write the n equals 1 to infinity part 00:44:58.770 --> 00:45:01.470 align:middle line:90% or just start listing them. 00:45:01.470 --> 00:45:05.450 align:middle line:90% 00:45:05.450 --> 00:45:12.870 align:middle line:84% All right, so let me quickly say that although-- 00:45:12.870 --> 00:45:19.170 align:middle line:84% so again, a sequence, unambiguously defined, 00:45:19.170 --> 00:45:22.320 align:middle line:84% is a function from the natural numbers to real. 00:45:22.320 --> 00:45:25.380 align:middle line:84% So don't confuse this with a set. 00:45:25.380 --> 00:45:26.850 align:middle line:90% A sequence is not a set. 00:45:26.850 --> 00:45:29.530 align:middle line:90% 00:45:29.530 --> 00:45:35.650 align:middle line:84% So think of a sequence, even though again it's 00:45:35.650 --> 00:45:40.720 align:middle line:84% defined as a function, as just an infinite listing 00:45:40.720 --> 00:45:43.330 align:middle line:90% of elements of R-- 00:45:43.330 --> 00:45:45.850 align:middle line:84% one element, the first element, the second element, 00:45:45.850 --> 00:45:48.700 align:middle line:90% the third element, and so on. 00:45:48.700 --> 00:45:50.620 align:middle line:84% And they don't have to be different. 00:45:50.620 --> 00:46:02.290 align:middle line:84% So for example, 1, 1, 1, 1, that is a perfectly good sequence. 00:46:02.290 --> 00:46:04.130 align:middle line:84% In terms of what is the function, 00:46:04.130 --> 00:46:08.840 align:middle line:84% this is x of n equals 1 for all n. 00:46:08.840 --> 00:46:11.620 align:middle line:84% But again, I will not think of really a sequence 00:46:11.620 --> 00:46:13.870 align:middle line:90% as a function. 00:46:13.870 --> 00:46:19.450 align:middle line:84% Just as we never thought about functions as subsets of-- 00:46:19.450 --> 00:46:23.350 align:middle line:84% as certain subsets of the Cartesian product of two sets, 00:46:23.350 --> 00:46:30.095 align:middle line:84% you should really think of these as just a list of real numbers. 00:46:30.095 --> 00:46:31.720 align:middle line:84% And when we refer to them, we'll either 00:46:31.720 --> 00:46:34.360 align:middle line:84% write them like that or with curly brackets 00:46:34.360 --> 00:46:37.568 align:middle line:84% around an expression for x sub n. 00:46:37.568 --> 00:46:44.270 align:middle line:90% 00:46:44.270 --> 00:46:48.755 align:middle line:84% So for example, if I were to write this, 00:46:48.755 --> 00:47:00.260 align:middle line:84% this means this is the sequence given by 1, 1/2, 1/3, 1/4, 1/5, 00:47:00.260 --> 00:47:00.920 align:middle line:90% and so on. 00:47:00.920 --> 00:47:05.530 align:middle line:90% 00:47:05.530 --> 00:47:06.030 align:middle line:90% OK. 00:47:06.030 --> 00:47:54.400 align:middle line:90% 00:47:54.400 --> 00:48:00.650 align:middle line:90% So here's another definition. 00:48:00.650 --> 00:48:04.570 align:middle line:84% So a sequence is bounded if, in some sense, 00:48:04.570 --> 00:48:08.180 align:middle line:84% it doesn't run off to arbitrarily large values. 00:48:08.180 --> 00:48:28.330 align:middle line:84% So sequence x sub n is bounded if there exists 00:48:28.330 --> 00:48:30.760 align:middle line:84% some real number b bigger than or equal 00:48:30.760 --> 00:48:39.310 align:middle line:84% to 0 such that for all n in the natural numbers xn 00:48:39.310 --> 00:48:43.900 align:middle line:84% is less than or equal to b in absolute value. 00:48:43.900 --> 00:48:50.140 align:middle line:84% So again, for example, if we take a look back at both 00:48:50.140 --> 00:48:52.780 align:middle line:90% of those two sequences-- 00:48:52.780 --> 00:48:58.400 align:middle line:84% 1, 1, 1, and the sequence 1 over n-- 00:48:58.400 --> 00:49:02.030 align:middle line:90% 00:49:02.030 --> 00:49:02.970 align:middle line:90% these are bounded. 00:49:02.970 --> 00:49:03.470 align:middle line:90% Why? 00:49:03.470 --> 00:49:04.910 align:middle line:90% Because every element in this-- 00:49:04.910 --> 00:49:08.420 align:middle line:90% 00:49:08.420 --> 00:49:14.000 align:middle line:84% I shouldn't say element, but every entry in this sequence 00:49:14.000 --> 00:49:15.140 align:middle line:90% is equal to 1. 00:49:15.140 --> 00:49:16.910 align:middle line:90% So it's bounded by 1. 00:49:16.910 --> 00:49:21.980 align:middle line:84% And every entry in this sequence is less than or equal to 1. 00:49:21.980 --> 00:49:23.900 align:middle line:84% So it's bounded by 1 in absolute value. 00:49:23.900 --> 00:49:24.990 align:middle line:90% It's also positive. 00:49:24.990 --> 00:49:27.280 align:middle line:90% So they're both bounded. 00:49:27.280 --> 00:49:35.680 align:middle line:84% Another example is the sequence minus 1 to the n. 00:49:35.680 --> 00:49:41.530 align:middle line:84% So I'll write equals, although this shouldn't-- this 00:49:41.530 --> 00:49:42.530 align:middle line:90% doesn't mean anything. 00:49:42.530 --> 00:49:45.490 align:middle line:90% I'm just going to write it-- 00:49:45.490 --> 00:49:49.260 align:middle line:84% write it-- or start listing the entries-- minus 1, 00:49:49.260 --> 00:49:54.280 align:middle line:90% 1, minus 1, 1, minus 1, 1. 00:49:54.280 --> 00:50:01.990 align:middle line:84% This is also bounded because the absolute value of minus 1 00:50:01.990 --> 00:50:02.830 align:middle line:90% to the n is 1. 00:50:02.830 --> 00:50:06.470 align:middle line:84% So that's bounded by 1 for all n. 00:50:06.470 --> 00:50:07.820 align:middle line:90% So what's a non-example? 00:50:07.820 --> 00:50:16.430 align:middle line:90% 00:50:16.430 --> 00:50:20.800 align:middle line:84% The sequence n-- in other words, this 00:50:20.800 --> 00:50:26.960 align:middle line:84% is the sequence 1, 2, 3, 4, 5, and so on. 00:50:26.960 --> 00:50:34.240 align:middle line:84% Now, precisely why is it un-- so I should say this is bounded. 00:50:34.240 --> 00:50:38.530 align:middle line:90% 00:50:38.530 --> 00:50:48.390 align:middle line:84% And otherwise, I'll refer to x sub n as unbounded, unbounded. 00:50:48.390 --> 00:50:52.470 align:middle line:90% 00:50:52.470 --> 00:50:53.520 align:middle line:90% So why is it unbounded? 00:50:53.520 --> 00:50:56.670 align:middle line:84% Because the entries are getting larger and larger in size. 00:50:56.670 --> 00:50:59.350 align:middle line:90% 00:50:59.350 --> 00:51:02.650 align:middle line:84% But if I were to ask you to prove this, 00:51:02.650 --> 00:51:05.020 align:middle line:90% you would have to show-- 00:51:05.020 --> 00:51:07.510 align:middle line:84% and this is also a good first exercise-- 00:51:07.510 --> 00:51:10.390 align:middle line:84% what does it mean to be unbounded. 00:51:10.390 --> 00:51:13.870 align:middle line:84% So I said this at one point, that if you 00:51:13.870 --> 00:51:17.290 align:middle line:84% come across a reasonably interesting definition, 00:51:17.290 --> 00:51:21.370 align:middle line:84% you should look up or try to come up with examples. 00:51:21.370 --> 00:51:23.740 align:middle line:84% And typically, when you do come up with examples, 00:51:23.740 --> 00:51:25.600 align:middle line:84% you should come up with non-examples. 00:51:25.600 --> 00:51:30.320 align:middle line:84% And what that requires you to do is negate the definition. 00:51:30.320 --> 00:51:34.420 align:middle line:84% What does it mean for something to not be that? 00:51:34.420 --> 00:51:39.280 align:middle line:84% So even though I'm saying that if it's not bounded 00:51:39.280 --> 00:51:41.050 align:middle line:84% I call it unbounded, I haven't actually 00:51:41.050 --> 00:51:43.750 align:middle line:84% written a mathematical statement to this effect 00:51:43.750 --> 00:51:45.470 align:middle line:90% on what this means. 00:51:45.470 --> 00:51:50.240 align:middle line:84% So let me make this right here as a remark. 00:51:50.240 --> 00:51:58.560 align:middle line:84% So a sequence is unbounded, so if it doesn't 00:51:58.560 --> 00:52:00.070 align:middle line:90% satisfy that definition. 00:52:00.070 --> 00:52:02.460 align:middle line:84% So we need to negate this definition. 00:52:02.460 --> 00:52:05.910 align:middle line:84% The definition says this is bounded 00:52:05.910 --> 00:52:09.450 align:middle line:84% if there exists a b bigger than or equal to 0 such 00:52:09.450 --> 00:52:12.480 align:middle line:90% that this holds for all n. 00:52:12.480 --> 00:52:16.020 align:middle line:84% Now, when I negate a there exists, that becomes a for all. 00:52:16.020 --> 00:52:19.960 align:middle line:84% And when I negate a for all, that becomes a there exists. 00:52:19.960 --> 00:52:23.410 align:middle line:84% And then I negate this condition. 00:52:23.410 --> 00:52:30.420 align:middle line:84% So the sequence is unbounded if for all b bigger than 00:52:30.420 --> 00:52:37.080 align:middle line:84% or equal to 0 there exists a natural number in n 00:52:37.080 --> 00:52:42.840 align:middle line:84% such that xn in absolute value is bigger than b. 00:52:42.840 --> 00:52:50.100 align:middle line:90% 00:52:50.100 --> 00:52:52.800 align:middle line:84% Now, using this-- so this is-- if you like, 00:52:52.800 --> 00:52:54.630 align:middle line:84% you can take that as a second definition. 00:52:54.630 --> 00:52:57.870 align:middle line:84% Or it's not really a definition, since it's just the negation 00:52:57.870 --> 00:52:59.160 align:middle line:90% of the first definition. 00:52:59.160 --> 00:53:02.490 align:middle line:90% 00:53:02.490 --> 00:53:04.710 align:middle line:90% Why is this set unbounded? 00:53:04.710 --> 00:53:07.200 align:middle line:84% Because of the Archimedean property. 00:53:07.200 --> 00:53:12.940 align:middle line:90% 00:53:12.940 --> 00:53:15.820 align:middle line:84% So now, I'm giving you a little short proof on why 00:53:15.820 --> 00:53:18.760 align:middle line:90% this sequence is unbounded. 00:53:18.760 --> 00:53:23.830 align:middle line:84% Let b be bigger than or equal to 0. 00:53:23.830 --> 00:53:27.730 align:middle line:84% By the Archimedean property, there 00:53:27.730 --> 00:53:34.030 align:middle line:84% exists a natural number n such that n is bigger 00:53:34.030 --> 00:53:37.600 align:middle line:84% than b, which is exactly what we wanted 00:53:37.600 --> 00:53:47.270 align:middle line:84% to prove because, in this case, x sub n is n, is unbounded. 00:53:47.270 --> 00:54:01.380 align:middle line:90% 00:54:01.380 --> 00:54:09.640 align:middle line:84% OK, now, what does it mean for us 00:54:09.640 --> 00:54:11.140 align:middle line:90% to have a limit of a sequence? 00:54:11.140 --> 00:54:14.272 align:middle line:84% What does a limit of a sequence mean? 00:54:14.272 --> 00:54:15.730 align:middle line:84% What does it mean for a real number 00:54:15.730 --> 00:54:17.950 align:middle line:90% to be the limit of a sequence? 00:54:17.950 --> 00:54:21.293 align:middle line:90% 00:54:21.293 --> 00:54:23.710 align:middle line:84% What does it mean for a sequence to converge to something? 00:54:23.710 --> 00:54:41.000 align:middle line:84% So the sequence x sub n converges to a real number 00:54:41.000 --> 00:54:46.670 align:middle line:84% x in R if the following condition is satisfied-- 00:54:46.670 --> 00:54:55.760 align:middle line:84% if for every epsilon positive, there exists a natural number 00:54:55.760 --> 00:55:01.940 align:middle line:84% capital M such that for all n bigger 00:55:01.940 --> 00:55:08.270 align:middle line:84% than or equal to capital M xn minus x absolute value 00:55:08.270 --> 00:55:10.112 align:middle line:90% is less than epsilon. 00:55:10.112 --> 00:55:11.570 align:middle line:84% Now, remember the absolute value is 00:55:11.570 --> 00:55:13.153 align:middle line:84% meant to be something like a distance. 00:55:13.153 --> 00:55:14.900 align:middle line:90% So what does this statement say? 00:55:14.900 --> 00:55:18.590 align:middle line:84% This says if I give you a little bit of tolerance, epsilon, 00:55:18.590 --> 00:55:21.890 align:middle line:84% and you go far enough out in the sequence, 00:55:21.890 --> 00:55:29.600 align:middle line:84% then the distance between that entry in the sequence and x 00:55:29.600 --> 00:55:31.490 align:middle line:90% are very close to each other. 00:55:31.490 --> 00:55:36.130 align:middle line:90% 00:55:36.130 --> 00:55:40.030 align:middle line:84% Let me finish stating the rest of this definition 00:55:40.030 --> 00:55:42.610 align:middle line:84% and we'll do a little more discussion. 00:55:42.610 --> 00:55:49.240 align:middle line:90% 00:55:49.240 --> 00:55:57.660 align:middle line:84% If a sequence converges, we say it's convergent. 00:55:57.660 --> 00:56:05.360 align:middle line:90% 00:56:05.360 --> 00:56:10.280 align:middle line:84% Otherwise, we say it's divergent. 00:56:10.280 --> 00:56:16.540 align:middle line:90% 00:56:16.540 --> 00:56:19.750 align:middle line:90% So all right, let me draw-- 00:56:19.750 --> 00:56:24.610 align:middle line:84% let me draw a little picture of what this is meant to be. 00:56:24.610 --> 00:56:31.570 align:middle line:84% So the real numbers, it was this-- 00:56:31.570 --> 00:56:35.510 align:middle line:84% I mean, it is this unique ordered field 00:56:35.510 --> 00:56:37.160 align:middle line:84% with the least upper bound property. 00:56:37.160 --> 00:56:40.130 align:middle line:84% But when you think about the real numbers, 00:56:40.130 --> 00:56:42.410 align:middle line:84% think of it as you've always thought about it, 00:56:42.410 --> 00:56:44.790 align:middle line:90% as just the real number line. 00:56:44.790 --> 00:56:48.870 align:middle line:84% So what does this definition mean? 00:56:48.870 --> 00:56:56.600 align:middle line:84% It means that a sequence converges to some 00:56:56.600 --> 00:57:02.710 align:middle line:84% x if for every epsilon I can do the following. 00:57:02.710 --> 00:57:03.530 align:middle line:90% If I look at-- 00:57:03.530 --> 00:57:12.130 align:middle line:90% 00:57:12.130 --> 00:57:22.250 align:middle line:84% if I give you an epsilon and you go on either side of x epsilon 00:57:22.250 --> 00:57:27.500 align:middle line:84% amount, then you should be able to find a capital M so 00:57:27.500 --> 00:57:34.100 align:middle line:84% that if you look at entries in your sequence x sub n for n 00:57:34.100 --> 00:57:37.980 align:middle line:84% bigger than or equal to M they're in this guy. 00:57:37.980 --> 00:57:43.640 align:middle line:84% So then I should have x sub M there, x sub M plus 1, 00:57:43.640 --> 00:57:45.560 align:middle line:90% x sub M plus 2. 00:57:45.560 --> 00:57:48.560 align:middle line:84% And in general, x sub n should be in there, 00:57:48.560 --> 00:57:50.930 align:middle line:84% as long as n is bigger than or equal to M. 00:57:50.930 --> 00:57:57.300 align:middle line:84% So I should be able to do this for every epsilon. 00:57:57.300 --> 00:57:59.900 align:middle line:84% So for any amount of tolerance epsilon, 00:57:59.900 --> 00:58:03.560 align:middle line:84% you should be able to go far enough out in the sequence 00:58:03.560 --> 00:58:08.330 align:middle line:84% that all entries in that sequence 00:58:08.330 --> 00:58:12.280 align:middle line:90% are within this tolerance of x. 00:58:12.280 --> 00:58:44.740 align:middle line:90% 00:58:44.740 --> 00:58:48.550 align:middle line:84% I mean, although this is a picture of what the definition 00:58:48.550 --> 00:58:50.740 align:middle line:90% means, how should you-- 00:58:50.740 --> 00:58:58.640 align:middle line:84% I mean, you've probably gotten plenty of experience 00:58:58.640 --> 00:59:01.430 align:middle line:84% with sequences when you were doing calculus. 00:59:01.430 --> 00:59:04.490 align:middle line:90% But what does this mean loosely? 00:59:04.490 --> 00:59:09.290 align:middle line:84% It means if I look close to x, some little interval containing 00:59:09.290 --> 00:59:13.730 align:middle line:84% x, then all of the sequence, all of the elements of the sequence 00:59:13.730 --> 00:59:15.755 align:middle line:90% should be eventually in this-- 00:59:15.755 --> 00:59:18.840 align:middle line:90% 00:59:18.840 --> 00:59:19.840 align:middle line:90% in this little interval. 00:59:19.840 --> 00:59:22.510 align:middle line:90% 00:59:22.510 --> 00:59:24.430 align:middle line:84% Or another way to think about it is 00:59:24.430 --> 00:59:33.280 align:middle line:84% that as I go on in the sequence, the entries are getting closer 00:59:33.280 --> 00:59:40.960 align:middle line:84% and closer and closer to x Now, the way one makes 00:59:40.960 --> 00:59:46.420 align:middle line:84% these last two intuitive statements precise 00:59:46.420 --> 00:59:48.850 align:middle line:90% is via this definition. 00:59:48.850 --> 00:59:52.780 align:middle line:84% The closer and closer part is encapsulated 00:59:52.780 --> 00:59:56.050 align:middle line:90% in this for all epsilon part. 00:59:56.050 --> 01:00:00.700 align:middle line:84% And eventually getting closer and closer means for all n 01:00:00.700 --> 01:00:02.260 align:middle line:84% bigger than or equal to capital M. 01:00:02.260 --> 01:00:04.870 align:middle line:84% As long as I go far enough out in the sequence, 01:00:04.870 --> 01:00:07.420 align:middle line:90% they're getting close to x. 01:00:07.420 --> 01:00:09.445 align:middle line:90% So I hope that's pretty clear. 01:00:09.445 --> 01:00:44.080 align:middle line:90% 01:00:44.080 --> 01:00:46.750 align:middle line:84% All right, so again, we have a definition. 01:00:46.750 --> 01:00:50.440 align:middle line:90% 01:00:50.440 --> 01:00:52.790 align:middle line:90% And it's reasonably interesting. 01:00:52.790 --> 01:00:56.840 align:middle line:84% So we should do examples, and then also negate it. 01:00:56.840 --> 01:01:00.280 align:middle line:84% But before I do either of those, I 01:01:00.280 --> 01:01:03.310 align:middle line:84% do want to just prove a very simple fact 01:01:03.310 --> 01:01:07.870 align:middle line:84% about convergent sequences, namely 01:01:07.870 --> 01:01:12.670 align:middle line:84% that if I have a sequence which converges to x, then x 01:01:12.670 --> 01:01:15.640 align:middle line:84% is the only thing that sequence can converge to. 01:01:15.640 --> 01:01:19.750 align:middle line:84% I cannot have a convergent sequence which converges to two 01:01:19.750 --> 01:01:20.470 align:middle line:90% different things. 01:01:20.470 --> 01:01:22.780 align:middle line:84% Like I said, if you go by this intuition 01:01:22.780 --> 01:01:26.080 align:middle line:84% that eventually the x sub n's are getting 01:01:26.080 --> 01:01:28.900 align:middle line:84% closer and closer and closer to x, well, 01:01:28.900 --> 01:01:31.090 align:middle line:84% then there's no way they can be also getting closer 01:01:31.090 --> 01:01:34.733 align:middle line:90% to something other than x. 01:01:34.733 --> 01:01:36.400 align:middle line:84% So I just want to prove that real quick. 01:01:36.400 --> 01:01:40.150 align:middle line:84% And then we'll do examples and a negation of this definition. 01:01:40.150 --> 01:01:43.531 align:middle line:90% 01:01:43.531 --> 01:01:46.200 align:middle line:90% So the following theorem-- 01:01:46.200 --> 01:01:52.170 align:middle line:84% and so I'm going to state first the theorem that 01:01:52.170 --> 01:01:54.030 align:middle line:84% has nothing to do with sequences, 01:01:54.030 --> 01:01:59.640 align:middle line:84% but is a nice way if you want to show two things are equal. 01:01:59.640 --> 01:02:01.587 align:middle line:84% You can show they're smaller than anything. 01:02:01.587 --> 01:02:03.420 align:middle line:84% So the first theorem that I'm going to state 01:02:03.420 --> 01:02:15.120 align:middle line:84% is if x and y are in R and all epsilon positive x minus y 01:02:15.120 --> 01:02:21.420 align:middle line:84% is less than epsilon, then x equals y. 01:02:21.420 --> 01:02:23.010 align:middle line:84% This is not a surprising statement. 01:02:23.010 --> 01:02:27.300 align:middle line:84% If I have x and y and the distance between them 01:02:27.300 --> 01:02:29.970 align:middle line:84% is arbitrarily small, then they have to be the same thing. 01:02:29.970 --> 01:02:33.540 align:middle line:90% 01:02:33.540 --> 01:02:34.710 align:middle line:90% So what's the proof? 01:02:34.710 --> 01:02:37.590 align:middle line:90% 01:02:37.590 --> 01:02:47.700 align:middle line:84% Suppose xy is in R. And for all epsilon positive, x minus y 01:02:47.700 --> 01:02:48.700 align:middle line:90% is less than epsilon. 01:02:48.700 --> 01:02:50.250 align:middle line:84% So now, I want to prove x equals y. 01:02:50.250 --> 01:02:56.460 align:middle line:84% So let's assume that this does not hold and arrive 01:02:56.460 --> 01:02:57.370 align:middle line:90% at a contradiction. 01:02:57.370 --> 01:02:59.920 align:middle line:84% So this is a short proof by contradiction. 01:02:59.920 --> 01:03:03.300 align:middle line:90% Suppose x does not equal y. 01:03:03.300 --> 01:03:13.360 align:middle line:84% Then remember we proved that the absolute value of something 01:03:13.360 --> 01:03:17.800 align:middle line:84% is 0 if and only if that thing is equal to 0 01:03:17.800 --> 01:03:19.510 align:middle line:90% and it's always non-negative. 01:03:19.510 --> 01:03:22.930 align:middle line:84% So x not equal to y means the absolute value of x minus y 01:03:22.930 --> 01:03:24.670 align:middle line:90% is less than-- is bigger than 0. 01:03:24.670 --> 01:03:27.340 align:middle line:90% 01:03:27.340 --> 01:03:32.320 align:middle line:84% Then by this assumption, for epsilon 01:03:32.320 --> 01:03:36.040 align:middle line:84% equal to x minus y over 2, I get-- 01:03:36.040 --> 01:03:41.260 align:middle line:90% 01:03:41.260 --> 01:03:45.730 align:middle line:84% which implies if I just subtract that over, x minus y 01:03:45.730 --> 01:03:46.720 align:middle line:90% is less than 0. 01:03:46.720 --> 01:03:52.150 align:middle line:90% So 1/2 is less than 0 or-- 01:03:52.150 --> 01:03:53.980 align:middle line:84% and that's a very false statement. 01:03:53.980 --> 01:03:56.230 align:middle line:84% We've already proven that the absolute value is always 01:03:56.230 --> 01:03:57.970 align:middle line:90% non-negative. 01:03:57.970 --> 01:04:03.580 align:middle line:84% So if I have two real numbers that are arbitrarily 01:04:03.580 --> 01:04:07.030 align:middle line:84% close to each other, then they have to be the same. 01:04:07.030 --> 01:04:11.330 align:middle line:84% I'm going to use this theorem to prove that-- 01:04:11.330 --> 01:04:14.530 align:middle line:84% I don't know why I put a 1 there-- 01:04:14.530 --> 01:04:22.170 align:middle line:84% why limits-- and I have not said that's what we call x-- 01:04:22.170 --> 01:04:26.490 align:middle line:84% but why a convergent sequence can only converge to one thing. 01:04:26.490 --> 01:04:29.840 align:middle line:90% 01:04:29.840 --> 01:04:33.580 align:middle line:84% So here is the statement of the second theorem. 01:04:33.580 --> 01:04:40.960 align:middle line:84% If x sub n is a sequence and it converges 01:04:40.960 --> 01:04:49.270 align:middle line:84% to x and it also converges to y, then x equals y. 01:04:49.270 --> 01:04:55.625 align:middle line:84% So a convergent sequence can only converge to one thing. 01:04:55.625 --> 01:04:57.500 align:middle line:84% And again, like I said, there should be clear 01:04:57.500 --> 01:05:03.680 align:middle line:84% because what's happening is if I have two things that are not 01:05:03.680 --> 01:05:05.990 align:middle line:84% equal and x sub n converges to x, 01:05:05.990 --> 01:05:10.100 align:middle line:84% that means as long as I go far enough out in the sequence, 01:05:10.100 --> 01:05:16.340 align:middle line:84% they're supposed to be here in this small interval around x. 01:05:16.340 --> 01:05:20.750 align:middle line:84% Well, if they're supposed to be converging to y, 01:05:20.750 --> 01:05:22.340 align:middle line:84% if I go far enough out, they're also 01:05:22.340 --> 01:05:28.270 align:middle line:84% supposed to be over here as long as I go far enough out. 01:05:28.270 --> 01:05:33.200 align:middle line:84% But you can't be in two places at once ever. 01:05:33.200 --> 01:05:35.800 align:middle line:84% And so that's why x must equal y. 01:05:35.800 --> 01:05:43.130 align:middle line:90% 01:05:43.130 --> 01:05:44.780 align:middle line:84% Although, I suppose that now we're 01:05:44.780 --> 01:05:48.530 align:middle line:84% doing all the classes online, some are recorded. 01:05:48.530 --> 01:05:54.040 align:middle line:84% So you can be two places at once, but at least not here. 01:05:54.040 --> 01:05:56.950 align:middle line:84% OK, so we're going to use this theorem 01:05:56.950 --> 01:05:58.360 align:middle line:90% to prove this statement. 01:05:58.360 --> 01:06:02.910 align:middle line:90% 01:06:02.910 --> 01:06:15.110 align:middle line:90% Suppose-- converges to x and y. 01:06:15.110 --> 01:06:17.840 align:middle line:84% And I want to verify that for all epsilon positive 01:06:17.840 --> 01:06:19.857 align:middle line:84% the absolute value of x minus y is less than 01:06:19.857 --> 01:06:20.690 align:middle line:90% or equal to epsilon. 01:06:20.690 --> 01:06:27.340 align:middle line:90% 01:06:27.340 --> 01:06:30.310 align:middle line:84% Now, remember if I'm proving something for all epsilon, 01:06:30.310 --> 01:06:35.860 align:middle line:84% I have to-- that means I do it for arbitrary epsilon. 01:06:35.860 --> 01:06:39.760 align:middle line:84% You get points just for setting let epsilon be positive. 01:06:39.760 --> 01:06:42.640 align:middle line:84% OK, now, I want to show the absolute value of x minus y 01:06:42.640 --> 01:06:44.440 align:middle line:84% is less than or equal to epsilon. 01:06:44.440 --> 01:06:46.900 align:middle line:84% And we're basically going to do the argument that I just 01:06:46.900 --> 01:06:47.400 align:middle line:90% erased. 01:06:47.400 --> 01:06:49.870 align:middle line:90% 01:06:49.870 --> 01:07:03.400 align:middle line:84% So since x sub n converges to x, that 01:07:03.400 --> 01:07:10.860 align:middle line:84% implies that there exists a natural number M1 such that 01:07:10.860 --> 01:07:14.400 align:middle line:84% for all n bigger than or equal to M1, 01:07:14.400 --> 01:07:19.140 align:middle line:84% x sub n minus x is less than epsilon over 2. 01:07:19.140 --> 01:07:20.502 align:middle line:90% Why the epsilon over 2? 01:07:20.502 --> 01:07:21.585 align:middle line:90% You'll see, magic happens. 01:07:21.585 --> 01:07:25.910 align:middle line:90% 01:07:25.910 --> 01:07:37.310 align:middle line:84% And since x sub n converges to y, 01:07:37.310 --> 01:07:42.350 align:middle line:84% thus this implies that there exists an M2 natural number 01:07:42.350 --> 01:07:47.990 align:middle line:84% such that for all n bigger than or equal to M2-- 01:07:47.990 --> 01:07:49.670 align:middle line:84% so again, this is the part that says 01:07:49.670 --> 01:07:52.710 align:middle line:84% as long as I go far enough out s sub n has to be close to x. 01:07:52.710 --> 01:07:54.260 align:middle line:84% Now, this is the part that says as 01:07:54.260 --> 01:07:58.630 align:middle line:84% long as I go far enough out x sub n has to be close to y-- 01:07:58.630 --> 01:08:03.580 align:middle line:84% so that for all n bigger than or equal to M2 x sub n minus y 01:08:03.580 --> 01:08:05.290 align:middle line:90% is less than epsilon over 2. 01:08:05.290 --> 01:08:08.690 align:middle line:90% 01:08:08.690 --> 01:08:20.319 align:middle line:84% Then since the sum M1 plus-- these are both natural numbers. 01:08:20.319 --> 01:08:28.300 align:middle line:84% This is bigger than or equal to M1 and M2. 01:08:28.300 --> 01:08:33.640 align:middle line:84% This implies-- now, I'm going to use these two inequalities 01:08:33.640 --> 01:08:36.550 align:middle line:90% and the triangle inequality. 01:08:36.550 --> 01:08:39.340 align:middle line:84% Then if I look at the absolute value of x minus y, 01:08:39.340 --> 01:08:41.680 align:middle line:84% this is equal to the absolute value 01:08:41.680 --> 01:08:46.404 align:middle line:84% of x minus x sub n plus x sub n minus-- 01:08:46.404 --> 01:08:49.300 align:middle line:90% 01:08:49.300 --> 01:08:58.420 align:middle line:84% so let me-- not x sub n, but x sub M1 plus M2 plus x sub M1 01:08:58.420 --> 01:08:59.139 align:middle line:90% plus M2. 01:08:59.139 --> 01:09:03.319 align:middle line:90% 01:09:03.319 --> 01:09:05.060 align:middle line:84% Now, by the triangle inequality, this 01:09:05.060 --> 01:09:15.460 align:middle line:84% is less than or equal to x M2 plus M1 plus M2 minus y. 01:09:15.460 --> 01:09:19.189 align:middle line:84% Now, this is less than epsilon over 2 01:09:19.189 --> 01:09:22.910 align:middle line:84% because M1 plus M2 is bigger than or equal to M1. 01:09:22.910 --> 01:09:26.439 align:middle line:90% 01:09:26.439 --> 01:09:29.390 align:middle line:84% And for the same reason, since plus M1 plus M2 01:09:29.390 --> 01:09:32.359 align:middle line:84% is bigger than or equal to M2, this is less than epsilon 01:09:32.359 --> 01:09:33.514 align:middle line:90% over 2. 01:09:33.514 --> 01:09:35.750 align:middle line:84% Ah, now you see why I divided both of those 01:09:35.750 --> 01:09:39.330 align:middle line:84% by 2 because when I add them up, I get epsilon. 01:09:39.330 --> 01:09:42.439 align:middle line:84% So I've shown that for every epsilon positive, 01:09:42.439 --> 01:09:45.169 align:middle line:84% the absolute value of x minus y is less than epsilon. 01:09:45.169 --> 01:09:58.600 align:middle line:90% 01:09:58.600 --> 01:10:03.570 align:middle line:84% And therefore, by the theorem that I proved a minute ago, 01:10:03.570 --> 01:10:04.380 align:middle line:90% x equals y. 01:10:04.380 --> 01:10:11.120 align:middle line:90% 01:10:11.120 --> 01:10:14.000 align:middle line:90% Why did I state that theorem? 01:10:14.000 --> 01:10:15.320 align:middle line:90% Because I want to use notation. 01:10:15.320 --> 01:10:18.440 align:middle line:84% And I want this notation to be already apparent 01:10:18.440 --> 01:10:19.550 align:middle line:90% that it's consistent. 01:10:19.550 --> 01:10:22.710 align:middle line:90% 01:10:22.710 --> 01:10:26.500 align:middle line:84% And I'm going to use terminology. 01:10:26.500 --> 01:10:37.080 align:middle line:84% So I mean, this is not really a new definition. 01:10:37.080 --> 01:10:38.430 align:middle line:90% It's just some terminology. 01:10:38.430 --> 01:10:49.040 align:middle line:90% 01:10:49.040 --> 01:10:57.745 align:middle line:84% We call x the limit of the sequence. 01:10:57.745 --> 01:11:01.760 align:middle line:90% 01:11:01.760 --> 01:11:07.090 align:middle line:84% Now, the fact that I've shown only that there's only one 01:11:07.090 --> 01:11:10.360 align:middle line:84% guy that a sequence can converge to 01:11:10.360 --> 01:11:12.520 align:middle line:84% is why I get to use the word "the--" 01:11:12.520 --> 01:11:15.700 align:middle line:84% so not we call x a limit of x sub n. 01:11:15.700 --> 01:11:19.720 align:middle line:84% It's the limit because there's only one, if one exists-- 01:11:19.720 --> 01:11:26.050 align:middle line:90% 01:11:26.050 --> 01:11:32.560 align:middle line:84% and write x equals limit as n goes to infinity of x sub n. 01:11:32.560 --> 01:11:39.650 align:middle line:90% 01:11:39.650 --> 01:11:46.582 align:middle line:84% OK, so let's pause on negating the definition of what 01:11:46.582 --> 01:11:48.290 align:middle line:84% it means for a sequence to be convergent. 01:11:48.290 --> 01:11:50.870 align:middle line:84% And let's do a couple of examples 01:11:50.870 --> 01:11:54.650 align:middle line:84% of convergent sequences, proving that they 01:11:54.650 --> 01:11:57.350 align:middle line:84% are convergent sequences and computing what they-- 01:11:57.350 --> 01:12:00.040 align:middle line:90% 01:12:00.040 --> 01:12:02.740 align:middle line:84% and at least proving that these sequences 01:12:02.740 --> 01:12:06.970 align:middle line:84% do converge to the limit that I'm telling you. 01:12:06.970 --> 01:12:11.860 align:middle line:84% So for example, let's look at the constant sequence 01:12:11.860 --> 01:12:19.225 align:middle line:84% x sub n equals 1 for all N. So that's the sequence 1, 1, 1, 01:12:19.225 --> 01:12:20.680 align:middle line:90% and so on. 01:12:20.680 --> 01:12:26.305 align:middle line:84% Then I'm just going to write 1, not x sub n. 01:12:26.305 --> 01:12:30.310 align:middle line:84% The limit as n goes to infinity of 1 equals 1. 01:12:30.310 --> 01:12:36.490 align:middle line:84% This proof is not very enlightening, 01:12:36.490 --> 01:12:39.880 align:middle line:90% but I'll show you how it goes. 01:12:39.880 --> 01:12:41.680 align:middle line:90% So you're supposed to show-- 01:12:41.680 --> 01:12:42.950 align:middle line:90% what does this mean? 01:12:42.950 --> 01:12:44.750 align:middle line:84% This means for every epsilon positive, 01:12:44.750 --> 01:12:46.750 align:middle line:84% there exist a natural number capital M such 01:12:46.750 --> 01:12:50.650 align:middle line:84% that if I'm bigger than capital M, this minus the limit 01:12:50.650 --> 01:12:52.310 align:middle line:90% should be less than epsilon. 01:12:52.310 --> 01:12:53.930 align:middle line:90% So let epsilon be positive. 01:12:53.930 --> 01:12:58.120 align:middle line:84% I have to now come up with a capital M 01:12:58.120 --> 01:13:00.730 align:middle line:84% so that x sub n minus x is less than epsilon 01:13:00.730 --> 01:13:03.550 align:middle line:84% whenever little n is bigger than or equal to capital M. 01:13:03.550 --> 01:13:05.712 align:middle line:84% So I'm going to choose capital M to be 1. 01:13:05.712 --> 01:13:07.045 align:middle line:90% I can do that for this sequence. 01:13:07.045 --> 01:13:13.820 align:middle line:90% 01:13:13.820 --> 01:13:18.620 align:middle line:84% Then if n is bigger than or equal to M 01:13:18.620 --> 01:13:24.120 align:middle line:84% and I look at the n-th entry in the sequence, 01:13:24.120 --> 01:13:28.280 align:middle line:84% well, this is just 1 minus 1 equals 0. 01:13:28.280 --> 01:13:32.270 align:middle line:90% And that's less than epsilon. 01:13:32.270 --> 01:13:38.300 align:middle line:84% So not very enlightening because capital M equals 1 01:13:38.300 --> 01:13:40.110 align:middle line:90% is good enough. 01:13:40.110 --> 01:13:43.070 align:middle line:84% So let's do something a little more. 01:13:43.070 --> 01:13:47.480 align:middle line:84% So how about the limit as n goes to infinity of 1 01:13:47.480 --> 01:13:49.040 align:middle line:90% over n equals 0? 01:13:49.040 --> 01:13:54.320 align:middle line:90% 01:13:54.320 --> 01:13:55.595 align:middle line:90% So let's do a proof of this. 01:13:55.595 --> 01:13:59.730 align:middle line:90% 01:13:59.730 --> 01:14:04.050 align:middle line:84% Again, I'm supposed to be proving this epsilon statement 01:14:04.050 --> 01:14:12.210 align:middle line:84% in the definition in order to be able to state this, which 01:14:12.210 --> 01:14:13.990 align:middle line:90% is for every epsilon statement. 01:14:13.990 --> 01:14:16.990 align:middle line:84% So I start the proof off with let epsilon be positive. 01:14:16.990 --> 01:14:18.480 align:middle line:90% Now, I have to find-- 01:14:18.480 --> 01:14:22.110 align:middle line:84% I have to tell you how to choose capital M to ensure 01:14:22.110 --> 01:14:24.120 align:middle line:84% that little n bigger than or equal to capital M 01:14:24.120 --> 01:14:29.310 align:middle line:84% implies x sub n minus 0, in this case, is less than epsilon. 01:14:29.310 --> 01:14:37.050 align:middle line:84% So I will say choose capital M natural number such 01:14:37.050 --> 01:14:43.050 align:middle line:84% that m is bigger than 1 over epsilon. 01:14:43.050 --> 01:14:45.827 align:middle line:84% In fact, I'm not going to write it that way. 01:14:45.827 --> 01:14:47.160 align:middle line:90% I'm going to write it this way-- 01:14:47.160 --> 01:14:50.160 align:middle line:90% 01:14:50.160 --> 01:14:52.180 align:middle line:90% 1 over M is less than epsilon. 01:14:52.180 --> 01:14:53.730 align:middle line:90% And Why Can I do this? 01:14:53.730 --> 01:15:00.270 align:middle line:84% Why can I find such a natural number that satisfies this? 01:15:00.270 --> 01:15:03.420 align:middle line:84% And you can't just insert statements into your proof-- 01:15:03.420 --> 01:15:08.590 align:middle line:84% choose capital M to satisfy the unicorn property-- 01:15:08.590 --> 01:15:14.930 align:middle line:84% and not clarify why you can choose 01:15:14.930 --> 01:15:19.850 align:middle line:84% such a natural number that satisfies the unicorn property. 01:15:19.850 --> 01:15:22.160 align:middle line:84% I can do this by the Archimedean property. 01:15:22.160 --> 01:15:24.260 align:middle line:84% This is equivalently stating that I 01:15:24.260 --> 01:15:27.200 align:middle line:84% can find a natural number M so that n 01:15:27.200 --> 01:15:28.730 align:middle line:84% is bigger than 1 over epsilon, what 01:15:28.730 --> 01:15:31.670 align:middle line:90% I had written there originally. 01:15:31.670 --> 01:15:32.960 align:middle line:90% So let me write it this way-- 01:15:32.960 --> 01:15:36.420 align:middle line:90% 01:15:36.420 --> 01:15:41.760 align:middle line:84% such an M exists by the Archimedean property 01:15:41.760 --> 01:15:43.650 align:middle line:90% of the natural numbers. 01:15:43.650 --> 01:15:45.960 align:middle line:84% I'll say that a few times if I have to 01:15:45.960 --> 01:15:51.890 align:middle line:84% or to clarify why certain natural numbers satisfying 01:15:51.890 --> 01:15:53.318 align:middle line:90% certain properties exist. 01:15:53.318 --> 01:15:54.860 align:middle line:84% And then at one point, I'll just stop 01:15:54.860 --> 01:15:57.235 align:middle line:84% because it should be clear that I've done it enough times 01:15:57.235 --> 01:16:00.920 align:middle line:84% you know how or why something does exist. 01:16:00.920 --> 01:16:09.300 align:middle line:84% OK, so I choose a natural number and a natural number 01:16:09.300 --> 01:16:10.950 align:middle line:90% M so that I have this. 01:16:10.950 --> 01:16:17.600 align:middle line:84% Now, I need to show you that this number works. 01:16:17.600 --> 01:16:25.100 align:middle line:84% So let n be bigger than capital M. Then 01:16:25.100 --> 01:16:29.150 align:middle line:84% if I look at x sub n minus my proposed limit, which is 0-- 01:16:29.150 --> 01:16:31.760 align:middle line:90% so I look at 1 minus n 0-- 01:16:31.760 --> 01:16:34.130 align:middle line:90% this is equal to just 1 over n. 01:16:34.130 --> 01:16:37.040 align:middle line:84% Since little n is bigger than or equal to capital M, 01:16:37.040 --> 01:16:42.830 align:middle line:84% this is less than or equal to capital M, which 01:16:42.830 --> 01:16:43.715 align:middle line:90% is less than epsilon. 01:16:43.715 --> 01:16:47.550 align:middle line:90% 01:16:47.550 --> 01:16:52.440 align:middle line:84% And therefore, I've proven what I wanted to prove. 01:16:52.440 --> 01:16:54.150 align:middle line:90% So how do these-- 01:16:54.150 --> 01:16:58.330 align:middle line:84% if I just give you a sequence and I ask you to prove that-- 01:16:58.330 --> 01:17:01.380 align:middle line:90% so this is some discussion. 01:17:01.380 --> 01:17:03.310 align:middle line:84% How do you come up with these proofs? 01:17:03.310 --> 01:17:06.727 align:middle line:84% How should the proof typically look? 01:17:06.727 --> 01:17:07.560 align:middle line:90% If I want to prove-- 01:17:07.560 --> 01:17:10.860 align:middle line:90% 01:17:10.860 --> 01:17:18.840 align:middle line:84% equals L or x, what's the proof of that look like? 01:17:18.840 --> 01:17:22.530 align:middle line:84% It should always-- so the proof should be let epsilon 01:17:22.530 --> 01:17:23.650 align:middle line:90% be positive. 01:17:23.650 --> 01:17:27.030 align:middle line:84% And then maybe you need to do some explaining. 01:17:27.030 --> 01:17:34.350 align:middle line:84% And then you'll say choose M so that something. 01:17:34.350 --> 01:17:38.430 align:middle line:90% 01:17:38.430 --> 01:17:44.150 align:middle line:84% And that such an M exists probably 01:17:44.150 --> 01:17:51.050 align:middle line:84% will be explained there or in the same sentence. 01:17:51.050 --> 01:17:52.640 align:middle line:84% And then the next part of the proof 01:17:52.640 --> 01:17:56.840 align:middle line:84% should be showing that this capital M works. 01:17:56.840 --> 01:18:02.570 align:middle line:84% So then if n is bigger than or equal to capital M, 01:18:02.570 --> 01:18:04.850 align:middle line:84% then you should show that it works 01:18:04.850 --> 01:18:08.240 align:middle line:84% by looking at x sub n minus x and doing 01:18:08.240 --> 01:18:12.920 align:middle line:84% a calculation or inequalities and arriving at this being less 01:18:12.920 --> 01:18:15.320 align:middle line:84% than epsilon, which is what you're wanting to prove. 01:18:15.320 --> 01:18:19.670 align:middle line:84% You're wanting, in the end, to verify this definition, which 01:18:19.670 --> 01:18:22.370 align:middle line:84% says for every epsilon, you can find a capital M 01:18:22.370 --> 01:18:24.470 align:middle line:90% so that I get this. 01:18:24.470 --> 01:18:27.390 align:middle line:84% So in a proof, you are carrying that out. 01:18:27.390 --> 01:18:29.990 align:middle line:84% You're saying for epsilon positive, 01:18:29.990 --> 01:18:32.450 align:middle line:90% I choose M so that this happens. 01:18:32.450 --> 01:18:41.520 align:middle line:90% 01:18:41.520 --> 01:18:47.840 align:middle line:90% Now, how do you-- 01:18:47.840 --> 01:18:50.520 align:middle line:84% how do you come up with such a capital M? 01:18:50.520 --> 01:18:52.100 align:middle line:84% So again, this is just discussion. 01:18:52.100 --> 01:18:54.560 align:middle line:84% And maybe we won't get to any more examples. 01:18:54.560 --> 01:18:59.890 align:middle line:84% I'll just finish this discussion, 01:18:59.890 --> 01:19:01.360 align:middle line:90% and then we'll call it a day. 01:19:01.360 --> 01:19:03.070 align:middle line:84% How do you come up with such a capital M? 01:19:03.070 --> 01:19:06.790 align:middle line:90% How did I come up with M? 01:19:06.790 --> 01:19:08.800 align:middle line:90% Why did I choose M in this way? 01:19:08.800 --> 01:19:12.280 align:middle line:84% In the end, you can see it here, is 01:19:12.280 --> 01:19:16.390 align:middle line:84% that I chose it because if I fiddle around 01:19:16.390 --> 01:19:21.710 align:middle line:84% with the thing I want to bound, I ended up with 1 over n, which 01:19:21.710 --> 01:19:27.140 align:middle line:84% I can control and make it smaller than epsilon. 01:19:27.140 --> 01:19:29.810 align:middle line:84% So typically, how do you find capital M? 01:19:29.810 --> 01:19:37.070 align:middle line:90% 01:19:37.070 --> 01:19:40.550 align:middle line:84% So this is a discussion, I guess, within a discussion. 01:19:40.550 --> 01:19:43.930 align:middle line:90% 01:19:43.930 --> 01:19:47.560 align:middle line:84% Typically, you'll take x sub n minus x, your proposed limit. 01:19:47.560 --> 01:19:53.560 align:middle line:84% Maybe I give you a sequence explicitly. 01:19:53.560 --> 01:19:58.450 align:middle line:84% Or maybe it's a expression involving some other sequences. 01:19:58.450 --> 01:20:00.100 align:middle line:90% And you fiddle around with it. 01:20:00.100 --> 01:20:04.360 align:middle line:84% Maybe you write it differently, or add and subtract something, 01:20:04.360 --> 01:20:08.290 align:middle line:84% or multiply it by 1 in a fancy way. 01:20:08.290 --> 01:20:12.520 align:middle line:84% And you'll get some expression typically 01:20:12.520 --> 01:20:15.280 align:middle line:90% involving something involving n. 01:20:15.280 --> 01:20:23.270 align:middle line:90% 01:20:23.270 --> 01:20:29.270 align:middle line:84% And as long as this expression involving n is simple enough, 01:20:29.270 --> 01:20:32.390 align:middle line:84% then you choose capital M so that this thing 01:20:32.390 --> 01:20:34.080 align:middle line:90% is less than epsilon. 01:20:34.080 --> 01:20:37.310 align:middle line:84% Now, simple enough, for example, 1 over n, 01:20:37.310 --> 01:20:40.430 align:middle line:84% I can choose capital M so that this is less than epsilon 01:20:40.430 --> 01:20:42.970 align:middle line:90% by the Archimedean property. 01:20:42.970 --> 01:20:47.990 align:middle line:84% If, for example-- and that's basically 1 over n, 01:20:47.990 --> 01:20:50.120 align:middle line:84% or 5 over n, or something like that, 01:20:50.120 --> 01:20:55.460 align:middle line:84% that exhausts all the simplest ones for now. 01:20:55.460 --> 01:20:57.920 align:middle line:84% But if, for example, you ended up with 1 01:20:57.920 --> 01:21:03.350 align:middle line:84% over n squared minus 3n plus 100, 01:21:03.350 --> 01:21:06.890 align:middle line:84% I would not say this is a simple enough expression 01:21:06.890 --> 01:21:10.430 align:middle line:84% to be able to say choose capital M so that this is less 01:21:10.430 --> 01:21:14.030 align:middle line:84% than epsilon without more explanation than what I just 01:21:14.030 --> 01:21:14.720 align:middle line:90% gave here. 01:21:14.720 --> 01:21:18.540 align:middle line:90% 01:21:18.540 --> 01:21:25.900 align:middle line:84% And we'll do some more examples next time 01:21:25.900 --> 01:21:30.010 align:middle line:84% to flesh this out a little bit more. 01:21:30.010 --> 01:21:32.340 align:middle line:90% All right, so we'll stop there. 01:21:32.340 --> 01:21:41.000 align:middle line:90%