WEBVTT 00:00:00.000 --> 00:00:01.449 align:middle line:90% [SQUEAKING] 00:00:01.449 --> 00:00:04.347 align:middle line:90% [RUSTLING] 00:00:04.347 --> 00:00:05.796 align:middle line:90% [CLICKING] 00:00:05.796 --> 00:00:11.420 align:middle line:90% 00:00:11.420 --> 00:00:15.700 align:middle line:84% TOBIAS COLDING: So one of the things I want to focus on today, 00:00:15.700 --> 00:00:20.840 align:middle line:84% and we'll get to this in just a minute, is how to write a proof. 00:00:20.840 --> 00:00:30.410 align:middle line:84% So this is going to be how to write a mathematical proof. 00:00:30.410 --> 00:00:40.360 align:middle line:90% 00:00:40.360 --> 00:00:44.400 align:middle line:84% So how do you write a proof, as opposed to, 00:00:44.400 --> 00:00:49.570 align:middle line:84% how do you think about why something should be true? 00:00:49.570 --> 00:00:52.760 align:middle line:84% And so this is one of the important topics 00:00:52.760 --> 00:00:54.800 align:middle line:90% for this course. 00:00:54.800 --> 00:01:00.160 align:middle line:84% So overall-- I mean, there's a lot of important concepts 00:01:00.160 --> 00:01:02.180 align:middle line:90% that we will talk about. 00:01:02.180 --> 00:01:04.760 align:middle line:84% But some of the general things that we will learn 00:01:04.760 --> 00:01:06.960 align:middle line:84% is how to prove something, but also 00:01:06.960 --> 00:01:10.170 align:middle line:84% how to write a proper mathematical proof. 00:01:10.170 --> 00:01:12.587 align:middle line:84% And those things are a little bit different. 00:01:12.587 --> 00:01:14.170 align:middle line:84% And we'll talk about this in a minute. 00:01:14.170 --> 00:01:16.960 align:middle line:90% 00:01:16.960 --> 00:01:19.580 align:middle line:90% I also want to say that-- 00:01:19.580 --> 00:01:23.520 align:middle line:84% so on the course information, the last page of the course-- 00:01:23.520 --> 00:01:25.480 align:middle line:84% several people were asking about this. 00:01:25.480 --> 00:01:31.720 align:middle line:84% On the last page of course information, there is-- 00:01:31.720 --> 00:01:33.140 align:middle line:90% so there's the notes. 00:01:33.140 --> 00:01:35.760 align:middle line:84% There's the notes for the class that I write. 00:01:35.760 --> 00:01:38.280 align:middle line:84% And I typically-- I try to write them in advance, 00:01:38.280 --> 00:01:39.620 align:middle line:90% a couple of days in advance. 00:01:39.620 --> 00:01:42.960 align:middle line:84% And then I try to go over it the day, also, 00:01:42.960 --> 00:01:46.260 align:middle line:84% to make sure to find typos and things like that, 00:01:46.260 --> 00:01:48.800 align:middle line:90% so that it's more readable. 00:01:48.800 --> 00:01:53.160 align:middle line:84% So if I were you, I would first read the lecture notes 00:01:53.160 --> 00:01:55.880 align:middle line:90% that I post after the lectures. 00:01:55.880 --> 00:01:59.030 align:middle line:84% And then there's also-- on the last page of the course 00:01:59.030 --> 00:02:01.470 align:middle line:90% information, there is-- 00:02:01.470 --> 00:02:03.730 align:middle line:90% and I try to do this in advance. 00:02:03.730 --> 00:02:06.150 align:middle line:84% But it's a little bit difficult to do it in advance. 00:02:06.150 --> 00:02:10.830 align:middle line:90% So it says the date. 00:02:10.830 --> 00:02:16.210 align:middle line:84% And it says some key words from the lecture on that date. 00:02:16.210 --> 00:02:20.510 align:middle line:84% So over the weekend, I posted it for the two lectures 00:02:20.510 --> 00:02:22.950 align:middle line:90% for this week. 00:02:22.950 --> 00:02:24.450 align:middle line:90% But then I go back-- 00:02:24.450 --> 00:02:25.890 align:middle line:90% I will try to go back. 00:02:25.890 --> 00:02:29.190 align:middle line:84% So I actually went back this morning and changed some of it 00:02:29.190 --> 00:02:32.510 align:middle line:84% from the second lecture so that it fits more. 00:02:32.510 --> 00:02:34.550 align:middle line:90% And I think I will only-- 00:02:34.550 --> 00:02:38.350 align:middle line:84% I give references there to the book. 00:02:38.350 --> 00:02:42.310 align:middle line:84% And initially, I was thinking of giving both references to Rudin 00:02:42.310 --> 00:02:45.830 align:middle line:90% and to the main textbook. 00:02:45.830 --> 00:02:47.550 align:middle line:90% I think it's probably just-- 00:02:47.550 --> 00:02:51.910 align:middle line:84% maybe it's-- I think it's probably smarter just to give it 00:02:51.910 --> 00:02:53.040 align:middle line:90% from the main textbook. 00:02:53.040 --> 00:02:56.070 align:middle line:90% 00:02:56.070 --> 00:03:00.190 align:middle line:90% And Rudin is a great book. 00:03:00.190 --> 00:03:09.310 align:middle line:84% This was a book that was created here a long time ago. 00:03:09.310 --> 00:03:12.610 align:middle line:84% And it is a classic in the subject. 00:03:12.610 --> 00:03:13.990 align:middle line:90% It's a great book. 00:03:13.990 --> 00:03:16.250 align:middle line:84% It's more formal than the other book. 00:03:16.250 --> 00:03:22.390 align:middle line:84% So the other book is a little bit more accessible. 00:03:22.390 --> 00:03:24.390 align:middle line:84% And so that's why we're using that. 00:03:24.390 --> 00:03:27.750 align:middle line:84% But actually, Rudin is a fantastic book. 00:03:27.750 --> 00:03:32.070 align:middle line:84% If it was just myself, I would just-- 00:03:32.070 --> 00:03:34.170 align:middle line:90% I think Rudin is hard to beat. 00:03:34.170 --> 00:03:37.410 align:middle line:84% But the other one is more accessible. 00:03:37.410 --> 00:03:40.030 align:middle line:84% And so it might actually be better the first time 00:03:40.030 --> 00:03:40.850 align:middle line:90% you learn it. 00:03:40.850 --> 00:03:44.150 align:middle line:90% 00:03:44.150 --> 00:03:47.230 align:middle line:84% So I think, from here on out-- so I deleted 00:03:47.230 --> 00:03:49.490 align:middle line:90% the references I had to Rudin. 00:03:49.490 --> 00:03:52.150 align:middle line:84% I think I will just give references to the main textbook. 00:03:52.150 --> 00:03:55.150 align:middle line:90% 00:03:55.150 --> 00:03:57.990 align:middle line:84% But I certainly encourage you to look at Rudin also. 00:03:57.990 --> 00:04:02.910 align:middle line:84% I also encourage you to look at chapters that we will skip. 00:04:02.910 --> 00:04:06.030 align:middle line:84% But there's also things that we initially skip, 00:04:06.030 --> 00:04:09.270 align:middle line:90% and then we come back to. 00:04:09.270 --> 00:04:12.030 align:middle line:90% But again, you should-- 00:04:12.030 --> 00:04:15.550 align:middle line:84% it's like the-- when I list the things that we 00:04:15.550 --> 00:04:17.950 align:middle line:84% will cover in the lecture in advance, 00:04:17.950 --> 00:04:21.730 align:middle line:84% it's only a sort of approximation. 00:04:21.730 --> 00:04:27.670 align:middle line:84% So once I write the lecture, it might be slightly different. 00:04:27.670 --> 00:04:29.510 align:middle line:90% OK. 00:04:29.510 --> 00:04:33.490 align:middle line:84% So we had-- so last time, we talked about the theorem-- 00:04:33.490 --> 00:04:35.030 align:middle line:90% we didn't prove it-- 00:04:35.030 --> 00:04:43.120 align:middle line:84% that the reals is a complete ordered field. 00:04:43.120 --> 00:04:48.590 align:middle line:90% 00:04:48.590 --> 00:04:52.580 align:middle line:84% And so, in particular, it was an ordered set 00:04:52.580 --> 00:04:55.780 align:middle line:90% that was also a field. 00:04:55.780 --> 00:04:57.120 align:middle line:90% And it was complete. 00:04:57.120 --> 00:04:59.120 align:middle line:84% And so, again, "complete" meant-- 00:04:59.120 --> 00:05:06.340 align:middle line:84% so if you had a subset A here, which was a subset of R, 00:05:06.340 --> 00:05:08.250 align:middle line:90% then A is bounded-- 00:05:08.250 --> 00:05:10.940 align:middle line:90% 00:05:10.940 --> 00:05:29.660 align:middle line:84% is bounded from above if there exists M in R 00:05:29.660 --> 00:05:43.510 align:middle line:84% such that for all a in A, a here is less than or equal to M. 00:05:43.510 --> 00:05:46.220 align:middle line:84% And then, if that's the case, then we 00:05:46.220 --> 00:06:08.920 align:middle line:84% say that an M here is said to be an upper bound for A. 00:06:08.920 --> 00:06:21.380 align:middle line:84% Likewise, if A here is a subset of R, 00:06:21.380 --> 00:06:39.660 align:middle line:84% then we say that A is bounded from below 00:06:39.660 --> 00:06:59.620 align:middle line:84% if there exists little m in R such that m here is less than a 00:06:59.620 --> 00:07:00.520 align:middle line:90% for all a. 00:07:00.520 --> 00:07:05.930 align:middle line:90% 00:07:05.930 --> 00:07:09.160 align:middle line:84% So this is upper and lower bounds, OK? 00:07:09.160 --> 00:07:10.820 align:middle line:90% And several people ask-- 00:07:10.820 --> 00:07:14.420 align:middle line:90% on Piazza, they ask-- 00:07:14.420 --> 00:07:21.820 align:middle line:84% they were a little unclear about, 00:07:21.820 --> 00:07:26.320 align:middle line:84% where does this M, capital M, where does this have to be? 00:07:26.320 --> 00:07:28.380 align:middle line:90% In what set does it have to be? 00:07:28.380 --> 00:07:31.980 align:middle line:84% And likewise for this little m, right? 00:07:31.980 --> 00:07:34.560 align:middle line:84% And so let me try to give you some example. 00:07:34.560 --> 00:07:36.160 align:middle line:84% And I tried to do that on Piazza. 00:07:36.160 --> 00:07:38.020 align:middle line:90% But let me just-- 00:07:38.020 --> 00:07:40.080 align:middle line:84% because several people were asking about it, 00:07:40.080 --> 00:07:42.060 align:middle line:90% let me just say again. 00:07:42.060 --> 00:07:46.320 align:middle line:84% So this capital M, when you have it like this, 00:07:46.320 --> 00:07:51.250 align:middle line:84% this capital M does not need to be in the set, right? 00:07:51.250 --> 00:07:52.270 align:middle line:90% A is bounded. 00:07:52.270 --> 00:07:53.180 align:middle line:90% So here's an example. 00:07:53.180 --> 00:07:56.690 align:middle line:90% 00:07:56.690 --> 00:08:01.690 align:middle line:84% If you're looking at the set, 0, 1-- so this is the interval. 00:08:01.690 --> 00:08:02.770 align:middle line:90% This is the interval. 00:08:02.770 --> 00:08:05.890 align:middle line:90% But it does not contain 0. 00:08:05.890 --> 00:08:07.530 align:middle line:90% And it does not contain 1. 00:08:07.530 --> 00:08:12.610 align:middle line:84% But it contains everything in between 0 and 1. 00:08:12.610 --> 00:08:14.670 align:middle line:90% This one here is bounded. 00:08:14.670 --> 00:08:21.930 align:middle line:84% So if this has A, and we think about A here as a subset of R, 00:08:21.930 --> 00:08:23.700 align:middle line:90% then A is bounded. 00:08:23.700 --> 00:08:27.010 align:middle line:90% 00:08:27.010 --> 00:08:31.530 align:middle line:84% And so one usually says it's bounded if it's 00:08:31.530 --> 00:08:32.980 align:middle line:90% both from above and below. 00:08:32.980 --> 00:08:43.250 align:middle line:90% 00:08:43.250 --> 00:08:45.630 align:middle line:84% So if one just says that it's bounded, 00:08:45.630 --> 00:08:49.930 align:middle line:84% then one means that it's bounded from above and below. 00:08:49.930 --> 00:08:50.590 align:middle line:90% It's bounded. 00:08:50.590 --> 00:08:53.470 align:middle line:90% 00:08:53.470 --> 00:08:56.240 align:middle line:84% And so, because I bounded from above-- 00:08:56.240 --> 00:09:01.170 align:middle line:90% 00:09:01.170 --> 00:09:04.690 align:middle line:84% above, because you can just-- you have that if x here 00:09:04.690 --> 00:09:07.410 align:middle line:84% is in this interval, then, of course, 00:09:07.410 --> 00:09:09.990 align:middle line:84% x is less than or equal to 1, right? 00:09:09.990 --> 00:09:15.410 align:middle line:84% In fact, it's strictly less than 1, right? 00:09:15.410 --> 00:09:17.970 align:middle line:84% But you see, this set here, there 00:09:17.970 --> 00:09:24.370 align:middle line:84% is no element in this set that is an upper bound for this set, 00:09:24.370 --> 00:09:25.250 align:middle line:90% right? 00:09:25.250 --> 00:09:27.990 align:middle line:84% The upper bound is outside the set. 00:09:27.990 --> 00:09:30.970 align:middle line:84% But it is in all of R. And that's fine. 00:09:30.970 --> 00:09:37.430 align:middle line:84% An upper bound will typically just be in the larger set. 00:09:37.430 --> 00:09:43.570 align:middle line:90% And the same with lower bound. 00:09:43.570 --> 00:09:48.250 align:middle line:84% Now, this-- everything on this board except this theorem-- 00:09:48.250 --> 00:09:50.942 align:middle line:84% but this is in terms of ordered sets. 00:09:50.942 --> 00:09:52.810 align:middle line:90% It has nothing to do-- 00:09:52.810 --> 00:09:56.930 align:middle line:84% it's just much more general than R. So if you take-- 00:09:56.930 --> 00:10:06.050 align:middle line:84% if S here is an ordered set, and A here is a subset-- 00:10:06.050 --> 00:10:09.178 align:middle line:90% 00:10:09.178 --> 00:10:14.530 align:middle line:84% is a subset-- so it's an ordered set. 00:10:14.530 --> 00:10:18.970 align:middle line:90% So it's a set with an ordering. 00:10:18.970 --> 00:10:25.050 align:middle line:84% And if A here is a subset of S, then, again, we 00:10:25.050 --> 00:10:45.170 align:middle line:84% can say that A is bounded from above if there exists M-- 00:10:45.170 --> 00:10:50.170 align:middle line:84% again, M does not have to be in A. 00:10:50.170 --> 00:10:52.390 align:middle line:84% But it can be just in a larger set, 00:10:52.390 --> 00:11:06.390 align:middle line:84% so that-- such that for all a in A, it's less or equal to M. 00:11:06.390 --> 00:11:12.810 align:middle line:84% So this is like the definition in general, being bounded above, 00:11:12.810 --> 00:11:17.010 align:middle line:84% if you have a ordered set, and you have a subset. 00:11:17.010 --> 00:11:19.170 align:middle line:90% And now, just to illustrate it. 00:11:19.170 --> 00:11:23.970 align:middle line:84% So typically, we're going to think about where the set is 00:11:23.970 --> 00:11:26.170 align:middle line:90% R with the usual ordering. 00:11:26.170 --> 00:11:28.030 align:middle line:90% And you have a subset of it. 00:11:28.030 --> 00:11:32.410 align:middle line:84% And you talk about whether it's bounded from above or below. 00:11:32.410 --> 00:11:34.170 align:middle line:84% But of course, you could also do it 00:11:34.170 --> 00:11:37.390 align:middle line:84% in a more general setting like this. 00:11:37.390 --> 00:11:43.600 align:middle line:84% And so here's an example, that if you look at-- which maybe is 00:11:43.600 --> 00:11:45.360 align:middle line:90% counterintuitive-- 00:11:45.360 --> 00:11:53.640 align:middle line:84% but if you take-- if S here is the half line, 00:11:53.640 --> 00:11:57.120 align:middle line:90% so it's all the real numbers-- 00:11:57.120 --> 00:12:02.140 align:middle line:84% it's between, say, 0 and infinity. 00:12:02.140 --> 00:12:05.720 align:middle line:90% 00:12:05.720 --> 00:12:12.120 align:middle line:90% And then you'll have A here. 00:12:12.120 --> 00:12:16.050 align:middle line:90% It's a subset, 0 to 1. 00:12:16.050 --> 00:12:21.936 align:middle line:90% 00:12:21.936 --> 00:12:27.500 align:middle line:84% Now, A here, and the ordering-- so this is, of course, 00:12:27.500 --> 00:12:30.480 align:middle line:84% a subset of R. But you should forget 00:12:30.480 --> 00:12:33.840 align:middle line:84% about R. I mean, you know the ordering from R. 00:12:33.840 --> 00:12:36.600 align:middle line:84% But forget about it, because this is now what you 00:12:36.600 --> 00:12:40.600 align:middle line:90% think about the set S, right? 00:12:40.600 --> 00:12:45.840 align:middle line:84% And so, if you're looking A as a subset of this, then A here-- 00:12:45.840 --> 00:12:47.410 align:middle line:90% A is bounded from above. 00:12:47.410 --> 00:12:55.920 align:middle line:90% 00:12:55.920 --> 00:12:56.420 align:middle line:90% Why? 00:12:56.420 --> 00:13:02.040 align:middle line:84% This is just because, for all A in A, you have 00:13:02.040 --> 00:13:05.280 align:middle line:84% that A here is strictly less than 1. 00:13:05.280 --> 00:13:10.120 align:middle line:84% So 1 here is an upper bound for A, right? 00:13:10.120 --> 00:13:18.040 align:middle line:84% But now, think about, does this set here have a lower bound? 00:13:18.040 --> 00:13:23.740 align:middle line:84% Well, a lower bound have to be inside this set, S, 00:13:23.740 --> 00:13:25.920 align:middle line:84% because we're thinking about this here 00:13:25.920 --> 00:13:28.540 align:middle line:90% as a subset of this ordered set. 00:13:28.540 --> 00:13:32.000 align:middle line:84% We don't think about it as a subset of the real line. 00:13:32.000 --> 00:13:36.840 align:middle line:84% But if you think about it as a subset of this set here, 00:13:36.840 --> 00:13:40.760 align:middle line:84% then there is no lower bound, because 0 is not 00:13:40.760 --> 00:13:44.540 align:middle line:84% in this set, and no negative numbers in this set. 00:13:44.540 --> 00:13:47.320 align:middle line:84% So this here-- and this seems maybe 00:13:47.320 --> 00:13:49.040 align:middle line:90% a little bit counterintuitive. 00:13:49.040 --> 00:14:08.790 align:middle line:84% But A is not bounded from below as a subset of S. Obviously, 00:14:08.790 --> 00:14:11.040 align:middle line:84% it is bounded from below if you were thinking about it 00:14:11.040 --> 00:14:13.500 align:middle line:90% as a subset of a real line. 00:14:13.500 --> 00:14:16.040 align:middle line:84% But if you think about it as a subset of S, 00:14:16.040 --> 00:14:19.600 align:middle line:90% it's not bounded from below. 00:14:19.600 --> 00:14:20.390 align:middle line:90% OK. 00:14:20.390 --> 00:14:31.760 align:middle line:84% So another person was asking the infinite and minus infinity. 00:14:31.760 --> 00:14:35.830 align:middle line:90% Those are not real numbers, OK? 00:14:35.830 --> 00:14:36.565 align:middle line:90% Yeah? 00:14:36.565 --> 00:14:38.190 align:middle line:84% AUDIENCE: So on the homework, should we 00:14:38.190 --> 00:14:41.593 align:middle line:84% consider all of the things as subsets of R? 00:14:41.593 --> 00:14:42.510 align:middle line:90% TOBIAS COLDING: Sorry? 00:14:42.510 --> 00:14:47.270 align:middle line:84% AUDIENCE: For question 3 on the homework [INAUDIBLE] 00:14:47.270 --> 00:14:52.470 align:middle line:90% they're all subsets of R. 00:14:52.470 --> 00:14:56.050 align:middle line:84% TOBIAS COLDING: If it's not directly stated that-- 00:14:56.050 --> 00:15:01.550 align:middle line:84% if it's not directly stated that there's a set, S, and then 00:15:01.550 --> 00:15:04.830 align:middle line:84% it has to be a subset of that set, 00:15:04.830 --> 00:15:09.270 align:middle line:84% then it would be assumed that it's a-- yeah. 00:15:09.270 --> 00:15:13.870 align:middle line:84% Of course, you are welcome to ask on Piazza. 00:15:13.870 --> 00:15:15.590 align:middle line:90% I don't have the homework here. 00:15:15.590 --> 00:15:17.670 align:middle line:84% But I would say that you're safe-- 00:15:17.670 --> 00:15:22.510 align:middle line:84% if it's not stated directly what the larger set is, then 00:15:22.510 --> 00:15:24.350 align:middle line:84% you can assume it's R, because we're going 00:15:24.350 --> 00:15:25.890 align:middle line:90% to concentrate mostly on R. 00:15:25.890 --> 00:15:29.350 align:middle line:84% This is another distinction between Rudin 00:15:29.350 --> 00:15:32.090 align:middle line:84% and this book that is the main textbook. 00:15:32.090 --> 00:15:37.590 align:middle line:84% Rudin is more abstract than the main textbook we are using. 00:15:37.590 --> 00:15:42.890 align:middle line:84% In particular, the main textbook here, 00:15:42.890 --> 00:15:44.890 align:middle line:84% and I'll do the same in my lectures, 00:15:44.890 --> 00:15:47.910 align:middle line:84% is that a lot of things are motivated from R. 00:15:47.910 --> 00:15:50.550 align:middle line:84% And then you abstract afterwards, 00:15:50.550 --> 00:15:53.990 align:middle line:84% so whereas Rudin starts with the abstraction, 00:15:53.990 --> 00:15:58.390 align:middle line:84% and then it specializes to R. And of course, there's 00:15:58.390 --> 00:16:01.880 align:middle line:84% an argument for maybe starting with the familiar and then 00:16:01.880 --> 00:16:03.880 align:middle line:84% abstract, instead of doing the other way around. 00:16:03.880 --> 00:16:10.510 align:middle line:90% 00:16:10.510 --> 00:16:13.830 align:middle line:84% And so for a-- whether it's just an ordered set-- 00:16:13.830 --> 00:16:19.250 align:middle line:84% I mean, just in general, so remember that completeness-- 00:16:19.250 --> 00:16:23.950 align:middle line:90% 00:16:23.950 --> 00:16:28.910 align:middle line:84% so completeness-- so this is when you have an ordered set. 00:16:28.910 --> 00:16:33.110 align:middle line:84% So if you have an ordered set, and A here 00:16:33.110 --> 00:16:40.400 align:middle line:84% is a subset of S that is bounded from above-- 00:16:40.400 --> 00:16:47.230 align:middle line:90% 00:16:47.230 --> 00:16:55.530 align:middle line:84% from above-- then, it is complete if S is complete. 00:16:55.530 --> 00:17:03.630 align:middle line:84% So if S is complete, then any subset 00:17:03.630 --> 00:17:11.829 align:middle line:84% that is bounded from above has a least upper bound. 00:17:11.829 --> 00:17:18.840 align:middle line:90% Then A has a least upper bound. 00:17:18.840 --> 00:17:24.069 align:middle line:90% 00:17:24.069 --> 00:17:31.050 align:middle line:84% And that least upper bound is denoted by the sup of A, right? 00:17:31.050 --> 00:17:36.180 align:middle line:84% So this thing here, it has the property 00:17:36.180 --> 00:17:48.340 align:middle line:84% that a is less than or equal to M for all a in A. And also, 00:17:48.340 --> 00:17:54.430 align:middle line:90% if M1 is another upper bound-- 00:17:54.430 --> 00:17:57.060 align:middle line:90% 00:17:57.060 --> 00:18:03.820 align:middle line:90% upper bound for A-- 00:18:03.820 --> 00:18:08.560 align:middle line:84% then M is less than or equal to M1. 00:18:08.560 --> 00:18:11.300 align:middle line:90% 00:18:11.300 --> 00:18:15.020 align:middle line:90% So that's a least upper bound. 00:18:15.020 --> 00:18:17.820 align:middle line:90% And the same thing is-- 00:18:17.820 --> 00:18:22.860 align:middle line:84% so, similarly, when we have an ordered set, 00:18:22.860 --> 00:18:26.770 align:middle line:84% and we have a set, if it's complete, and if we have-- 00:18:26.770 --> 00:18:33.140 align:middle line:90% 00:18:33.140 --> 00:18:56.460 align:middle line:84% if A here is bounded from below, then inf of A 00:18:56.460 --> 00:19:06.460 align:middle line:84% is the least upper-- is the largest, the greatest, 00:19:06.460 --> 00:19:14.400 align:middle line:90% lower bound for A. 00:19:14.400 --> 00:19:19.020 align:middle line:84% So this, again, means that this number here, little m-- 00:19:19.020 --> 00:19:28.620 align:middle line:84% so m here is less or equal to a for all a in A. 00:19:28.620 --> 00:19:42.820 align:middle line:84% And if m1 here is another lower bound, 00:19:42.820 --> 00:19:48.940 align:middle line:84% then m1 here is less or equal to m. 00:19:48.940 --> 00:19:51.780 align:middle line:90% So this is a symmetric property. 00:19:51.780 --> 00:19:56.860 align:middle line:84% And so, again, R, the reals, is complete. 00:19:56.860 --> 00:20:01.440 align:middle line:84% So whenever you take a bounded subset of the real-- 00:20:01.440 --> 00:20:04.180 align:middle line:84% so if it's bounded, that means that it's bounded from above 00:20:04.180 --> 00:20:05.660 align:middle line:90% and below-- 00:20:05.660 --> 00:20:09.880 align:middle line:84% then it has a least upper bound and a greatest lower bound. 00:20:09.880 --> 00:20:15.220 align:middle line:84% And those are denoted by sup and inf. 00:20:15.220 --> 00:20:19.620 align:middle line:84% And in general, sup and inf are not 00:20:19.620 --> 00:20:24.660 align:middle line:84% in the set itself, in the set A. I mean, they could be. 00:20:24.660 --> 00:20:27.627 align:middle line:84% But in general, they wouldn't be. 00:20:27.627 --> 00:20:29.710 align:middle line:84% AUDIENCE: The example that we talked about earlier 00:20:29.710 --> 00:20:32.860 align:middle line:84% [INAUDIBLE] open set [INAUDIBLE] but that 00:20:32.860 --> 00:20:34.610 align:middle line:90% is not complete [INAUDIBLE]. 00:20:34.610 --> 00:20:36.170 align:middle line:90% TOBIAS COLDING: That's right. 00:20:36.170 --> 00:20:38.430 align:middle line:90% So no. 00:20:38.430 --> 00:20:42.666 align:middle line:84% So if you're looking at this set here, 00:20:42.666 --> 00:20:56.490 align:middle line:84% so suppose that this here is the set S. 00:20:56.490 --> 00:21:01.610 align:middle line:84% And if you take A here is a subset of S, 00:21:01.610 --> 00:21:06.250 align:middle line:84% and A here is bounded from-- let's say that A here 00:21:06.250 --> 00:21:07.580 align:middle line:90% is bounded from below. 00:21:07.580 --> 00:21:16.010 align:middle line:90% 00:21:16.010 --> 00:21:20.930 align:middle line:84% If it's bounded from below, then it couldn't be-- 00:21:20.930 --> 00:21:26.530 align:middle line:84% this set here, this here is not bounded from below. 00:21:26.530 --> 00:21:29.770 align:middle line:84% So it means that if it's bounded from below, 00:21:29.770 --> 00:21:35.970 align:middle line:84% then A here could not have numbers that was approaching 0. 00:21:35.970 --> 00:21:39.930 align:middle line:84% And so actually, if it is bounded from below, 00:21:39.930 --> 00:21:42.890 align:middle line:84% if it's a subset of this, and it's bounded from below, 00:21:42.890 --> 00:21:45.850 align:middle line:84% then actually, it would have a least lower bound-- 00:21:45.850 --> 00:21:47.590 align:middle line:90% a greatest lower bound, right? 00:21:47.590 --> 00:21:49.370 align:middle line:90% It should mean-- 00:21:49.370 --> 00:21:51.130 align:middle line:90% Yeah. 00:21:51.130 --> 00:21:52.190 align:middle line:90% So it is-- 00:21:52.190 --> 00:21:53.970 align:middle line:84% I mean, I'm not going to prove anything 00:21:53.970 --> 00:21:55.810 align:middle line:90% is complete or not complete. 00:21:55.810 --> 00:22:07.330 align:middle line:84% But it's pretty convincing that it is actually complete, 00:22:07.330 --> 00:22:08.010 align:middle line:90% in this sense. 00:22:08.010 --> 00:22:10.150 align:middle line:84% There's another sense that is not complete. 00:22:10.150 --> 00:22:12.610 align:middle line:84% But anyway, in this sense, it is complete. 00:22:12.610 --> 00:22:13.130 align:middle line:90% Yeah. 00:22:13.130 --> 00:22:15.730 align:middle line:90% No, it's a little bit-- 00:22:15.730 --> 00:22:17.250 align:middle line:90% it can be confusing. 00:22:17.250 --> 00:22:19.730 align:middle line:84% Later on, we'll talk about maybe some other notion 00:22:19.730 --> 00:22:20.670 align:middle line:90% of completeness. 00:22:20.670 --> 00:22:21.910 align:middle line:90% And then it's not complete. 00:22:21.910 --> 00:22:27.090 align:middle line:84% But in this notion, it is complete. 00:22:27.090 --> 00:22:33.090 align:middle line:84% OK, so the next thing I want to get to 00:22:33.090 --> 00:22:35.910 align:middle line:84% is this with how to write a mathematical proof. 00:22:35.910 --> 00:22:39.138 align:middle line:90% 00:22:39.138 --> 00:22:44.570 align:middle line:84% And again, that's a super important thing 00:22:44.570 --> 00:22:48.630 align:middle line:90% to learn in this class. 00:22:48.630 --> 00:22:52.850 align:middle line:90% And it's not just a-- 00:22:52.850 --> 00:22:59.610 align:middle line:90% it's not just a formal thing. 00:22:59.610 --> 00:23:01.550 align:middle line:90% I mean, it's-- yeah. 00:23:01.550 --> 00:23:03.690 align:middle line:90% It may seem like a formal thing. 00:23:03.690 --> 00:23:07.210 align:middle line:90% Why bother with that? 00:23:07.210 --> 00:23:09.970 align:middle line:90% But it's really, really useful. 00:23:09.970 --> 00:23:16.850 align:middle line:84% And the more complicated things become, the more useful it is. 00:23:16.850 --> 00:23:21.290 align:middle line:84% And so I want to illustrate this with how 00:23:21.290 --> 00:23:23.200 align:middle line:90% to write a mathematical proof. 00:23:23.200 --> 00:23:25.420 align:middle line:84% So I'm going to illustrate this-- 00:23:25.420 --> 00:23:38.420 align:middle line:84% so we will illustrate it by proving again-- 00:23:38.420 --> 00:23:41.110 align:middle line:90% 00:23:41.110 --> 00:23:58.840 align:middle line:84% by writing a formal proof of why square root of 2 00:23:58.840 --> 00:23:59.750 align:middle line:90% is a real number. 00:23:59.750 --> 00:24:02.320 align:middle line:90% 00:24:02.320 --> 00:24:04.880 align:middle line:90% So I want to write-- 00:24:04.880 --> 00:24:09.440 align:middle line:84% so last time, we talked about that square root of 2 00:24:09.440 --> 00:24:12.120 align:middle line:90% was a real number. 00:24:12.120 --> 00:24:14.840 align:middle line:84% But when we proved it, it was more 00:24:14.840 --> 00:24:18.360 align:middle line:90% like we talked about the idea. 00:24:18.360 --> 00:24:21.500 align:middle line:84% And so then the question is-- so if you 00:24:21.500 --> 00:24:24.440 align:middle line:84% have-- imagine you have a piece of scrap paper. 00:24:24.440 --> 00:24:27.000 align:middle line:84% What we did last time was what you would 00:24:27.000 --> 00:24:28.500 align:middle line:90% be writing on the scrap paper. 00:24:28.500 --> 00:24:33.360 align:middle line:84% So I want to talk about, if you were handing it in or showing it 00:24:33.360 --> 00:24:37.920 align:middle line:84% to somebody, then you wouldn't do that. 00:24:37.920 --> 00:24:40.160 align:middle line:90% You would do what we do now. 00:24:40.160 --> 00:24:44.280 align:middle line:84% But it doesn't necessarily make for a great-- a formal proof 00:24:44.280 --> 00:24:47.200 align:middle line:84% doesn't necessarily make for a great lecture, 00:24:47.200 --> 00:24:48.810 align:middle line:90% because it's kind of formal. 00:24:48.810 --> 00:24:52.424 align:middle line:90% 00:24:52.424 --> 00:24:58.120 align:middle line:84% So let me try to explain how you should write this. 00:24:58.120 --> 00:25:02.040 align:middle line:84% And again, this is a very simple example. 00:25:02.040 --> 00:25:07.420 align:middle line:84% But hopefully, it will give you an idea of what to do. 00:25:07.420 --> 00:25:11.320 align:middle line:84% And again, this is a really important part of this class 00:25:11.320 --> 00:25:18.560 align:middle line:84% that you're bound to use in many other classes later on. 00:25:18.560 --> 00:25:22.600 align:middle line:84% OK, so let's first talk about what was square root of 2. 00:25:22.600 --> 00:25:24.960 align:middle line:90% So square root of 2-- 00:25:24.960 --> 00:25:27.240 align:middle line:90% this here was a number. 00:25:27.240 --> 00:25:32.720 align:middle line:84% And it was given by that it has a property-- that it's positive, 00:25:32.720 --> 00:25:37.140 align:middle line:84% and when you square it, then you get 2. 00:25:37.140 --> 00:25:39.120 align:middle line:84% That was what square root of 2 was. 00:25:39.120 --> 00:25:41.840 align:middle line:90% 00:25:41.840 --> 00:25:51.480 align:middle line:84% Now, so we again-- so we want to prove-- so this is the theorem. 00:25:51.480 --> 00:26:00.640 align:middle line:84% So theorem-- square root of 2 is a real number. 00:26:00.640 --> 00:26:03.680 align:middle line:84% So I know-- maybe I should write it more formally. 00:26:03.680 --> 00:26:14.215 align:middle line:84% There exists alpha greater than 0-- 00:26:14.215 --> 00:26:22.950 align:middle line:84% this is alpha in R with alpha greater than 0 and alpha 00:26:22.950 --> 00:26:24.730 align:middle line:90% squared is equal to 2. 00:26:24.730 --> 00:26:29.350 align:middle line:90% 00:26:29.350 --> 00:26:41.590 align:middle line:84% Now, we look at the set A. The set A is the numbers-- 00:26:41.590 --> 00:26:51.830 align:middle line:84% it's the x, such that x is positive, 00:26:51.830 --> 00:27:00.645 align:middle line:84% and x squared is less than or equal to 2. 00:27:00.645 --> 00:27:03.355 align:middle line:84% AUDIENCE: [INAUDIBLE] see what x is, like where-- 00:27:03.355 --> 00:27:04.230 align:middle line:90% TOBIAS COLDING: Yeah. 00:27:04.230 --> 00:27:10.430 align:middle line:84% So x here-- so now I will specialize to the real numbers. 00:27:10.430 --> 00:27:12.250 align:middle line:84% So that was why I didn't actually write it. 00:27:12.250 --> 00:27:13.950 align:middle line:90% But you're right. 00:27:13.950 --> 00:27:15.520 align:middle line:90% It certainly can't hurt. 00:27:15.520 --> 00:27:17.590 align:middle line:90% OK, so that's the theorem. 00:27:17.590 --> 00:27:19.720 align:middle line:84% And so now we try to write a formal proof. 00:27:19.720 --> 00:27:26.030 align:middle line:90% 00:27:26.030 --> 00:27:43.150 align:middle line:84% Observe first that A here is nonempty, 00:27:43.150 --> 00:27:47.670 align:middle line:90% since 1 is in A, right? 00:27:47.670 --> 00:27:49.470 align:middle line:90% 1 is positive. 00:27:49.470 --> 00:27:52.370 align:middle line:84% And when you square 1, you get 1. 00:27:52.370 --> 00:27:54.670 align:middle line:84% So this is definitely less than or equal to 2. 00:27:54.670 --> 00:27:56.480 align:middle line:90% So A here is nonempty. 00:27:56.480 --> 00:27:59.910 align:middle line:90% 00:27:59.910 --> 00:28:14.870 align:middle line:84% Next, observe that A here is bounded-- 00:28:14.870 --> 00:28:15.860 align:middle line:90% is bounded from above. 00:28:15.860 --> 00:28:21.870 align:middle line:90% 00:28:21.870 --> 00:28:25.350 align:middle line:84% In general, by the way, when you write a proof, 00:28:25.350 --> 00:28:28.950 align:middle line:84% then you really only try to establish the properties 00:28:28.950 --> 00:28:31.810 align:middle line:84% that you need, because otherwise, it's confusing. 00:28:31.810 --> 00:28:33.950 align:middle line:84% If you establish all kinds of other properties 00:28:33.950 --> 00:28:37.390 align:middle line:84% that is not needed, that just makes it confusing. 00:28:37.390 --> 00:28:39.180 align:middle line:90% So it's bounded from above. 00:28:39.180 --> 00:28:41.870 align:middle line:90% 00:28:41.870 --> 00:28:43.010 align:middle line:90% And this is just because-- 00:28:43.010 --> 00:28:48.590 align:middle line:90% 00:28:48.590 --> 00:29:05.662 align:middle line:90% since 2 is an upper bound for A. 00:29:05.662 --> 00:29:08.070 align:middle line:84% Now, the next thing is-- so now we have 00:29:08.070 --> 00:29:10.730 align:middle line:90% a non-empty bounded subset. 00:29:10.730 --> 00:29:18.700 align:middle line:84% So we set-- so set alpha equal to the sup of a. 00:29:18.700 --> 00:29:21.020 align:middle line:90% And we want to prove-- 00:29:21.020 --> 00:29:31.420 align:middle line:84% want to show that alpha is positive, and alpha squared 00:29:31.420 --> 00:29:32.480 align:middle line:90% is equal to 2. 00:29:32.480 --> 00:29:39.380 align:middle line:90% 00:29:39.380 --> 00:29:49.960 align:middle line:90% Well-- all right. 00:29:49.960 --> 00:29:51.840 align:middle line:84% By definition, the way we defined-- 00:29:51.840 --> 00:29:55.100 align:middle line:90% 00:29:55.100 --> 00:30:01.020 align:middle line:84% the way we defined A, all the elements in A were positive. 00:30:01.020 --> 00:30:12.090 align:middle line:84% So by definition, 0 is a lower bound. 00:30:12.090 --> 00:30:17.060 align:middle line:90% 00:30:17.060 --> 00:30:22.100 align:middle line:84% So you for sure have that 0 here is less or equal to-- 00:30:22.100 --> 00:30:26.100 align:middle line:84% well, it's actually-- I mean, even 1 is a lower bound. 00:30:26.100 --> 00:30:31.300 align:middle line:84% So you can also just say that since 1 is a lower bound-- 00:30:31.300 --> 00:30:37.340 align:middle line:84% since 1 here is in A, you also have that this alpha here 00:30:37.340 --> 00:30:41.940 align:middle line:84% must be bigger or equal to 1, because it 00:30:41.940 --> 00:30:46.940 align:middle line:84% was supposed to be an upper bound for A, so in particular. 00:30:46.940 --> 00:30:55.220 align:middle line:84% Now-- and so we have that alpha is less than 1. 00:30:55.220 --> 00:30:57.440 align:middle line:90% And it's the least upper bound. 00:30:57.440 --> 00:30:59.040 align:middle line:90% And 2 was an upper bound. 00:30:59.040 --> 00:31:02.980 align:middle line:90% So you have this, since-- 00:31:02.980 --> 00:31:07.820 align:middle line:84% maybe I should write it with this thing. 00:31:07.820 --> 00:31:14.780 align:middle line:84% Since 1 is in A, we have that 1 here, 00:31:14.780 --> 00:31:15.950 align:middle line:90% less than or equal to alpha. 00:31:15.950 --> 00:31:19.700 align:middle line:90% 00:31:19.700 --> 00:31:35.900 align:middle line:84% And since alpha is the least upper bound, 00:31:35.900 --> 00:31:49.140 align:middle line:84% and 2 is an upper bound, we have that alpha 00:31:49.140 --> 00:31:50.480 align:middle line:90% is less than or equal to 2. 00:31:50.480 --> 00:31:53.020 align:middle line:90% 00:31:53.020 --> 00:31:55.580 align:middle line:84% Now, remember, we were going to do it exactly 00:31:55.580 --> 00:31:56.840 align:middle line:90% like we did last time. 00:31:56.840 --> 00:32:01.860 align:middle line:84% But we're just going to write it down in a proper way as a proof. 00:32:01.860 --> 00:32:04.820 align:middle line:84% So remember that last time, we then divided the proof 00:32:04.820 --> 00:32:08.300 align:middle line:90% into two steps. 00:32:08.300 --> 00:32:10.620 align:middle line:90% The first step was-- 00:32:10.620 --> 00:32:15.900 align:middle line:84% so part 1-- so now, so all we need to prove-- maybe 00:32:15.900 --> 00:32:17.140 align:middle line:90% I'll just say that. 00:32:17.140 --> 00:32:26.980 align:middle line:84% All we need to prove, to show, is 00:32:26.980 --> 00:32:31.020 align:middle line:84% that alpha squared is equal to 2, 00:32:31.020 --> 00:32:33.100 align:middle line:84% because we already have it as positive now, 00:32:33.100 --> 00:32:36.016 align:middle line:90% because it's bigger than 1. 00:32:36.016 --> 00:32:40.260 align:middle line:84% And so there was two parts to this, step 1 and step 2, 00:32:40.260 --> 00:32:43.140 align:middle line:90% or part 1 and part 2. 00:32:43.140 --> 00:32:46.180 align:middle line:84% The first part was-- so part 1, we 00:32:46.180 --> 00:33:00.930 align:middle line:84% will show that alpha squared is less than or equal to 2. 00:33:00.930 --> 00:33:07.106 align:middle line:90% 00:33:07.106 --> 00:33:09.510 align:middle line:90% And so we prove this-- 00:33:09.510 --> 00:33:13.320 align:middle line:90% so assume not. 00:33:13.320 --> 00:33:19.250 align:middle line:90% 00:33:19.250 --> 00:33:20.770 align:middle line:90% We will show that that leads-- 00:33:20.770 --> 00:33:29.955 align:middle line:90% 00:33:29.955 --> 00:33:50.290 align:middle line:84% that this leads to a contradiction to that alpha 00:33:50.290 --> 00:34:03.270 align:middle line:90% is the least upper bound for A. 00:34:03.270 --> 00:34:07.310 align:middle line:84% I'm writing everything out here, just to illustrate it. 00:34:07.310 --> 00:34:11.170 align:middle line:84% As you get more and more familiar with this, 00:34:11.170 --> 00:34:13.370 align:middle line:90% then the structure is the same. 00:34:13.370 --> 00:34:16.949 align:middle line:84% But some of these steps will seem almost too trivial. 00:34:16.949 --> 00:34:18.610 align:middle line:90% And that's probably fine. 00:34:18.610 --> 00:34:21.850 align:middle line:84% But of course, it's always good to err on more 00:34:21.850 --> 00:34:23.759 align:middle line:90% details than fewer details. 00:34:23.759 --> 00:34:27.596 align:middle line:90% 00:34:27.596 --> 00:34:29.929 align:middle line:90% So this is what we want to do. 00:34:29.929 --> 00:34:35.730 align:middle line:90% So we assume not. 00:34:35.730 --> 00:34:44.050 align:middle line:84% So assume that alpha squared is strictly bigger than 2. 00:34:44.050 --> 00:34:47.330 align:middle line:90% 00:34:47.330 --> 00:34:50.409 align:middle line:84% Then-- and this is where it deviates more 00:34:50.409 --> 00:34:54.090 align:middle line:90% from how we did it last time. 00:34:54.090 --> 00:34:56.090 align:middle line:84% And that's why what we did last time 00:34:56.090 --> 00:35:02.970 align:middle line:84% is really what you want to have on the scrap paper. 00:35:02.970 --> 00:35:06.770 align:middle line:84% What we did last time was like the idea. 00:35:06.770 --> 00:35:10.090 align:middle line:84% And then, once you see the idea, then 00:35:10.090 --> 00:35:16.450 align:middle line:84% you try to write it as a formal proof, like what we do now. 00:35:16.450 --> 00:35:21.530 align:middle line:90% And so, so assume not. 00:35:21.530 --> 00:35:25.410 align:middle line:90% Set h equal to-- 00:35:25.410 --> 00:35:30.930 align:middle line:84% so I'm going to set h equal to alpha squared minus 2 00:35:30.930 --> 00:35:32.760 align:middle line:90% over 4 alpha. 00:35:32.760 --> 00:35:35.530 align:middle line:90% 00:35:35.530 --> 00:35:39.010 align:middle line:84% Obviously, if this was the first time you would do it, 00:35:39.010 --> 00:35:41.910 align:middle line:84% you wouldn't just come up with this sort of thing, right? 00:35:41.910 --> 00:35:45.090 align:middle line:84% This is coming from the scrap paper that you had. 00:35:45.090 --> 00:35:47.610 align:middle line:84% So you're setting h equal to that. 00:35:47.610 --> 00:35:48.740 align:middle line:90% And then you observe-- 00:35:48.740 --> 00:35:51.970 align:middle line:90% 00:35:51.970 --> 00:36:04.000 align:middle line:84% you observe that since alpha here is bigger than 1, and alpha 00:36:04.000 --> 00:36:09.920 align:middle line:84% squared is bigger than 2, since that, 00:36:09.920 --> 00:36:13.600 align:middle line:84% then the nominator here is positive. 00:36:13.600 --> 00:36:15.800 align:middle line:90% And the denominator is positive. 00:36:15.800 --> 00:36:17.960 align:middle line:90% So h here is positive. 00:36:17.960 --> 00:36:21.280 align:middle line:90% 00:36:21.280 --> 00:36:23.420 align:middle line:90% You also have that-- 00:36:23.420 --> 00:36:25.670 align:middle line:90% you also have that h is-- 00:36:25.670 --> 00:36:30.460 align:middle line:90% 00:36:30.460 --> 00:36:30.960 align:middle line:90% right. 00:36:30.960 --> 00:36:36.680 align:middle line:84% You also have automatically that h is-- 00:36:36.680 --> 00:36:37.780 align:middle line:90% maybe we don't need that. 00:36:37.780 --> 00:36:38.940 align:middle line:90% Let's see if we need that. 00:36:38.940 --> 00:36:42.120 align:middle line:90% So OK. 00:36:42.120 --> 00:36:55.520 align:middle line:84% So now you set alpha 1 equal to alpha minus h. 00:36:55.520 --> 00:37:10.300 align:middle line:84% And what I want to prove is-- so we want to show that alpha here, 00:37:10.300 --> 00:37:21.520 align:middle line:84% alpha 1, is an upper bound for A. 00:37:21.520 --> 00:37:25.160 align:middle line:84% So I want to prove that this is an upper bound for A. 00:37:25.160 --> 00:37:28.840 align:middle line:84% The first thing I want to do-- if it's an upper bound, it does 00:37:28.840 --> 00:37:30.620 align:middle line:90% need to be positive. 00:37:30.620 --> 00:37:31.630 align:middle line:90% So let's check that. 00:37:31.630 --> 00:37:40.120 align:middle line:90% 00:37:40.120 --> 00:37:46.360 align:middle line:84% So since we already have that h was positive-- 00:37:46.360 --> 00:37:51.440 align:middle line:84% so this here was h was positive, alpha squared minus 2 00:37:51.440 --> 00:37:53.800 align:middle line:90% over 4 alpha. 00:37:53.800 --> 00:37:57.360 align:middle line:84% This thing here is less or equal to-- 00:37:57.360 --> 00:37:59.020 align:middle line:90% I can even forget about the 2. 00:37:59.020 --> 00:38:02.760 align:middle line:84% So I can write it as alpha squared over 4 alpha. 00:38:02.760 --> 00:38:04.360 align:middle line:90% And so this here is-- 00:38:04.360 --> 00:38:06.400 align:middle line:90% I can just divide by alpha. 00:38:06.400 --> 00:38:09.080 align:middle line:90% So I get alpha over 4. 00:38:09.080 --> 00:38:14.200 align:middle line:84% So I have-- since we have that h here is less than alpha over 4, 00:38:14.200 --> 00:38:22.020 align:middle line:90% then alpha 1 is bigger than-- 00:38:22.020 --> 00:38:30.208 align:middle line:84% is bigger or equal to 3/4 of alpha, 00:38:30.208 --> 00:38:34.400 align:middle line:84% because I'm subtracting at most alpha over 4. 00:38:34.400 --> 00:38:37.840 align:middle line:84% And so there's still 3/4 of alpha left, right? 00:38:37.840 --> 00:38:40.920 align:middle line:84% So I have this-- in particular, this thing here. 00:38:40.920 --> 00:38:43.720 align:middle line:84% It's actually bigger than 3 over 4. 00:38:43.720 --> 00:38:45.380 align:middle line:84% But I just need it to be positive. 00:38:45.380 --> 00:38:49.640 align:middle line:90% 00:38:49.640 --> 00:38:51.240 align:middle line:90% So that's positive. 00:38:51.240 --> 00:38:58.670 align:middle line:84% So alpha over-- alpha 1 is positive. 00:38:58.670 --> 00:39:04.990 align:middle line:84% Now I'm looking at alpha 1 squared, right? 00:39:04.990 --> 00:39:13.260 align:middle line:84% And so alpha 1 squared is equal to alpha minus h squared. 00:39:13.260 --> 00:39:15.950 align:middle line:90% 00:39:15.950 --> 00:39:21.950 align:middle line:84% So this here is equal to alpha squared plus h 00:39:21.950 --> 00:39:24.930 align:middle line:90% squared minus 2 alpha h. 00:39:24.930 --> 00:39:28.940 align:middle line:90% 00:39:28.940 --> 00:39:36.830 align:middle line:84% And this thing here is less or equal to alpha squared. 00:39:36.830 --> 00:39:40.470 align:middle line:90% Here, h here is-- 00:39:40.470 --> 00:39:47.950 align:middle line:84% h is-- so now I'd like to say that h is less or equal to 1. 00:39:47.950 --> 00:39:51.030 align:middle line:84% Actually, I'm going to use that it's less than 1/2. 00:39:51.030 --> 00:39:52.890 align:middle line:90% And why is that the case? 00:39:52.890 --> 00:39:56.330 align:middle line:90% It's actually, alpha itself-- 00:39:56.330 --> 00:39:59.430 align:middle line:84% we already know that alpha was less than 2. 00:39:59.430 --> 00:40:06.310 align:middle line:84% So you have that this thing here is less or equal to 2 over 4. 00:40:06.310 --> 00:40:09.290 align:middle line:90% So this is equal to 1/2, right? 00:40:09.290 --> 00:40:14.830 align:middle line:90% 00:40:14.830 --> 00:40:20.230 align:middle line:90% So now I have that-- 00:40:20.230 --> 00:40:27.750 align:middle line:84% now I have that this thing here is less than plus-- 00:40:27.750 --> 00:40:30.510 align:middle line:90% 00:40:30.510 --> 00:40:36.250 align:middle line:84% here, I just use that it's less than this thing here. 00:40:36.250 --> 00:40:37.190 align:middle line:90% This here, I just-- 00:40:37.190 --> 00:40:40.850 align:middle line:84% I think it's fine to just use h here. 00:40:40.850 --> 00:40:42.130 align:middle line:90% I could use h over 2. 00:40:42.130 --> 00:40:43.450 align:middle line:90% Maybe I'll use h over 2. 00:40:43.450 --> 00:40:46.190 align:middle line:84% So one of the h, I just replaced by 1/2. 00:40:46.190 --> 00:40:47.710 align:middle line:90% That is an upper bound. 00:40:47.710 --> 00:40:52.710 align:middle line:84% And the other one, I just keep, right? 00:40:52.710 --> 00:40:57.990 align:middle line:84% So I have that and then minus 2 alpha h. 00:40:57.990 --> 00:41:03.510 align:middle line:84% But if you multiply alpha by h, then it's 00:41:03.510 --> 00:41:05.830 align:middle line:90% disappeared in the denominator. 00:41:05.830 --> 00:41:09.630 align:middle line:90% So you're getting minus 2. 00:41:09.630 --> 00:41:18.500 align:middle line:84% And then you're getting alpha squared minus 2 over 4. 00:41:18.500 --> 00:41:22.510 align:middle line:90% 00:41:22.510 --> 00:41:25.630 align:middle line:90% And so here, you get-- 00:41:25.630 --> 00:41:30.870 align:middle line:84% this here becomes alpha squared plus-- 00:41:30.870 --> 00:41:32.950 align:middle line:90% and here, I'm just writing in. 00:41:32.950 --> 00:41:34.430 align:middle line:84% And I think it's-- at the moment, 00:41:34.430 --> 00:41:37.510 align:middle line:90% we're overkilling it here. 00:41:37.510 --> 00:41:40.750 align:middle line:84% And so I can replace this one here. 00:41:40.750 --> 00:41:43.750 align:middle line:90% So let me just write it in-- 00:41:43.750 --> 00:41:51.440 align:middle line:84% alpha squared minus 2 over 2 times 4 times alpha. 00:41:51.440 --> 00:41:59.000 align:middle line:84% And this thing here is alpha squared minus 2 over 2. 00:41:59.000 --> 00:42:01.420 align:middle line:84% This 2 cancels with one of those 2. 00:42:01.420 --> 00:42:02.720 align:middle line:90% So this is what I have. 00:42:02.720 --> 00:42:06.180 align:middle line:90% 00:42:06.180 --> 00:42:14.900 align:middle line:84% And so now I get that this thing here is-- 00:42:14.900 --> 00:42:17.330 align:middle line:84% so I get that this thing here is-- 00:42:17.330 --> 00:42:24.980 align:middle line:90% 00:42:24.980 --> 00:42:26.440 align:middle line:90% we're adding it here. 00:42:26.440 --> 00:42:31.500 align:middle line:84% So this thing here is alpha squared. 00:42:31.500 --> 00:42:35.820 align:middle line:84% So let me just bring these two together. 00:42:35.820 --> 00:42:39.680 align:middle line:84% Let me just-- maybe I'll just combine all of these. 00:42:39.680 --> 00:42:42.330 align:middle line:84% I don't think I need it this well. 00:42:42.330 --> 00:42:48.216 align:middle line:90% 00:42:48.216 --> 00:42:48.716 align:middle line:90% Right. 00:42:48.716 --> 00:42:52.620 align:middle line:84% So if I combine these two here, I 00:42:52.620 --> 00:42:56.920 align:middle line:84% think I'm fine with just having a 2 here-- 00:42:56.920 --> 00:43:00.300 align:middle line:90% a 4, sorry, instead of the 2. 00:43:00.300 --> 00:43:07.660 align:middle line:84% So if I do that, then I get alpha squared. 00:43:07.660 --> 00:43:10.040 align:middle line:84% Maybe I should just bring it on the common denominator. 00:43:10.040 --> 00:43:13.340 align:middle line:84% So this here, this alpha, sorry, I'm getting rid of. 00:43:13.340 --> 00:43:18.180 align:middle line:84% I'm replacing-- I'm bounding this from above by alpha squared 00:43:18.180 --> 00:43:20.300 align:middle line:90% plus-- 00:43:20.300 --> 00:43:23.120 align:middle line:84% and I'm going to just make it easier for myself. 00:43:23.120 --> 00:43:27.160 align:middle line:84% I think it's fine to just use that alpha is bounded by 1. 00:43:27.160 --> 00:43:32.930 align:middle line:84% And then you can just bound it by alpha squared minus 2 over 4. 00:43:32.930 --> 00:43:35.660 align:middle line:90% 00:43:35.660 --> 00:43:49.740 align:middle line:84% And this thing here is alpha squared minus 2 over 2, right? 00:43:49.740 --> 00:43:54.820 align:middle line:84% Here, again, I'm just using that alpha was bigger or equal to 1, 00:43:54.820 --> 00:43:57.060 align:middle line:90% and-- 00:43:57.060 --> 00:43:58.386 align:middle line:90% right. 00:43:58.386 --> 00:44:02.420 align:middle line:84% And then I just actually ignore this 2, right? 00:44:02.420 --> 00:44:04.600 align:middle line:84% So this is definitely an upper bound. 00:44:04.600 --> 00:44:07.380 align:middle line:90% 00:44:07.380 --> 00:44:09.980 align:middle line:84% And so now you see that this thing here 00:44:09.980 --> 00:44:13.780 align:middle line:90% is equal to alpha squared. 00:44:13.780 --> 00:44:16.400 align:middle line:84% This thing here, you can think about this thing here-- 00:44:16.400 --> 00:44:19.580 align:middle line:84% you can write this thing here, if you will, as alpha squared, 00:44:19.580 --> 00:44:26.700 align:middle line:84% plus alpha squared minus 2 over 4, minus 2 alpha squared, 00:44:26.700 --> 00:44:32.020 align:middle line:90% minus 2 over 4. 00:44:32.020 --> 00:44:36.020 align:middle line:84% And so what you get is that it's alpha squared, 00:44:36.020 --> 00:44:40.440 align:middle line:84% minus alpha squared, minus 2 over 4. 00:44:40.440 --> 00:44:43.020 align:middle line:90% 00:44:43.020 --> 00:44:46.570 align:middle line:84% And this thing here is now-- this thing here 00:44:46.570 --> 00:44:50.010 align:middle line:84% is positive, because alpha squared was strictly bigger 00:44:50.010 --> 00:44:50.890 align:middle line:90% than 2. 00:44:50.890 --> 00:44:52.310 align:middle line:90% This is positive. 00:44:52.310 --> 00:44:54.350 align:middle line:84% So this is strictly less than alpha squared. 00:44:54.350 --> 00:44:57.170 align:middle line:90% 00:44:57.170 --> 00:44:59.410 align:middle line:90% And so this here would also-- 00:44:59.410 --> 00:45:07.820 align:middle line:84% which is still-- so this thing here is also an upper bound. 00:45:07.820 --> 00:45:09.850 align:middle line:90% I'm sorry. 00:45:09.850 --> 00:45:12.450 align:middle line:90% I need to-- no, sorry. 00:45:12.450 --> 00:45:12.950 align:middle line:90% OK. 00:45:12.950 --> 00:45:13.680 align:middle line:90% So I have this. 00:45:13.680 --> 00:45:21.850 align:middle line:90% 00:45:21.850 --> 00:45:24.970 align:middle line:90% No, no, sorry about that. 00:45:24.970 --> 00:45:27.730 align:middle line:90% OK, so-- yeah, no, I-- sorry. 00:45:27.730 --> 00:45:31.650 align:middle line:90% 00:45:31.650 --> 00:45:33.750 align:middle line:84% OK, let's just go over this again. 00:45:33.750 --> 00:45:34.250 align:middle line:90% Sorry. 00:45:34.250 --> 00:45:38.050 align:middle line:90% 00:45:38.050 --> 00:45:39.950 align:middle line:84% So what am I trying to prove here? 00:45:39.950 --> 00:45:42.110 align:middle line:84% I'm trying to write down the formal-- 00:45:42.110 --> 00:45:46.770 align:middle line:84% I mean, that's the problem, that it got so formal that I was now 00:45:46.770 --> 00:45:52.010 align:middle line:84% in-- so what I'm trying to prove here is that I want to say-- 00:45:52.010 --> 00:45:58.850 align:middle line:84% this part 1 is that alpha squared is less or equal to 1. 00:45:58.850 --> 00:46:01.730 align:middle line:84% This is what-- less or equal to 2. 00:46:01.730 --> 00:46:03.970 align:middle line:90% That's what I want to prove. 00:46:03.970 --> 00:46:05.870 align:middle line:90% This is part 1. 00:46:05.870 --> 00:46:07.470 align:middle line:90% And so I assume not. 00:46:07.470 --> 00:46:13.130 align:middle line:84% So I assume that alpha squared was strictly bigger than 2. 00:46:13.130 --> 00:46:18.770 align:middle line:84% If alpha squared was strictly bigger than 2, here, you have 0. 00:46:18.770 --> 00:46:20.770 align:middle line:90% Here, you have alpha. 00:46:20.770 --> 00:46:24.650 align:middle line:90% And now we define this alpha 0. 00:46:24.650 --> 00:46:29.730 align:middle line:84% Alpha 0 was to the left, so to speak, of alpha, 00:46:29.730 --> 00:46:33.490 align:middle line:84% because it was subtracting a little bit. 00:46:33.490 --> 00:46:37.370 align:middle line:84% And what I want to show is that this alpha 0-- 00:46:37.370 --> 00:46:41.270 align:middle line:84% I want to show that this alpha 0 is also an upper bound. 00:46:41.270 --> 00:46:44.010 align:middle line:84% So I want to prove that this square here 00:46:44.010 --> 00:46:46.190 align:middle line:90% is also bigger than 2. 00:46:46.190 --> 00:46:49.343 align:middle line:90% That's what I want to prove. 00:46:49.343 --> 00:46:51.290 align:middle line:90% So I want to find-- 00:46:51.290 --> 00:46:54.670 align:middle line:90% I want to find a lower bound. 00:46:54.670 --> 00:46:56.850 align:middle line:84% And that lower bound should be true. 00:46:56.850 --> 00:47:00.736 align:middle line:84% So this is not exactly what I wanted to do. 00:47:00.736 --> 00:47:03.010 align:middle line:84% So let's just go through it here. 00:47:03.010 --> 00:47:03.810 align:middle line:90% Sorry. 00:47:03.810 --> 00:47:05.390 align:middle line:90% So this here was correct. 00:47:05.390 --> 00:47:07.700 align:middle line:84% And this here is really for the second part. 00:47:07.700 --> 00:47:11.770 align:middle line:90% 00:47:11.770 --> 00:47:13.970 align:middle line:90% OK. 00:47:13.970 --> 00:47:17.290 align:middle line:90% So we define it like this. 00:47:17.290 --> 00:47:25.090 align:middle line:84% And now I say that this thing here is bigger or equal to-- 00:47:25.090 --> 00:47:27.570 align:middle line:90% I want to bound it from below. 00:47:27.570 --> 00:47:30.430 align:middle line:84% And so then I can ignore the h squared. 00:47:30.430 --> 00:47:33.070 align:middle line:84% So it's bigger than-- because that's positive. 00:47:33.070 --> 00:47:36.590 align:middle line:84% So it's bigger than alpha squared minus 2 alpha h. 00:47:36.590 --> 00:47:39.340 align:middle line:90% 00:47:39.340 --> 00:47:41.620 align:middle line:90% And this thing here is-- 00:47:41.620 --> 00:47:46.660 align:middle line:84% so this is equal to alpha squared minus 2 alpha. 00:47:46.660 --> 00:47:49.000 align:middle line:90% I fill in what h was. 00:47:49.000 --> 00:47:50.170 align:middle line:90% This here was h. 00:47:50.170 --> 00:47:56.239 align:middle line:90% 00:47:56.239 --> 00:48:00.880 align:middle line:84% And then you see that this here cancels with that. 00:48:00.880 --> 00:48:04.020 align:middle line:84% And this here-- and you're getting that-- 00:48:04.020 --> 00:48:05.310 align:middle line:90% you're getting alpha squared-- 00:48:05.310 --> 00:48:07.850 align:middle line:90% 00:48:07.850 --> 00:48:18.720 align:middle line:84% this is this alpha squared minus alpha squared minus 2 over 2. 00:48:18.720 --> 00:48:21.840 align:middle line:84% And this thing here is equal to-- 00:48:21.840 --> 00:48:26.400 align:middle line:84% this is equal to alpha squared over 2. 00:48:26.400 --> 00:48:28.520 align:middle line:90% This is subtracting that. 00:48:28.520 --> 00:48:30.700 align:middle line:90% And then you're adding, right? 00:48:30.700 --> 00:48:32.440 align:middle line:90% You're adding 1. 00:48:32.440 --> 00:48:35.225 align:middle line:90% So this is what it is. 00:48:35.225 --> 00:48:38.280 align:middle line:84% Alpha-- we have already proven that alpha 00:48:38.280 --> 00:48:41.120 align:middle line:90% here was bigger or equal to 1. 00:48:41.120 --> 00:48:45.360 align:middle line:84% So using-- so this here is bigger or equal to-- 00:48:45.360 --> 00:48:48.840 align:middle line:84% I claim that this here is bigger or equal to 2 00:48:48.840 --> 00:48:57.220 align:middle line:84% using that alpha is bigger or equal to 1. 00:48:57.220 --> 00:48:59.824 align:middle line:90% 00:48:59.824 --> 00:49:03.360 align:middle line:84% And so you see that alpha 1, that 00:49:03.360 --> 00:49:06.280 align:middle line:84% is a smaller number, because you're subtracting 00:49:06.280 --> 00:49:09.280 align:middle line:90% a positive number from alpha. 00:49:09.280 --> 00:49:12.480 align:middle line:90% That is also an upper bound. 00:49:12.480 --> 00:49:16.200 align:middle line:84% So this is contradicting-- so this contradicts. 00:49:16.200 --> 00:49:18.920 align:middle line:90% And then one often writes-- 00:49:18.920 --> 00:49:21.880 align:middle line:84% one often writes this symbol here. 00:49:21.880 --> 00:49:26.720 align:middle line:84% This is a standard symbol that means this is a contradiction. 00:49:26.720 --> 00:49:27.460 align:middle line:90% Yeah? 00:49:27.460 --> 00:49:29.600 align:middle line:84% AUDIENCE: Did you say using that alpha 00:49:29.600 --> 00:49:31.600 align:middle line:90% is greater than or equal to 1? 00:49:31.600 --> 00:49:32.520 align:middle line:90% TOBIAS COLDING: Yeah. 00:49:32.520 --> 00:49:34.658 align:middle line:90% AUDIENCE: What if alpha is 1? 00:49:34.658 --> 00:49:36.200 align:middle line:84% TOBIAS COLDING: Yeah, then you get 2. 00:49:36.200 --> 00:49:36.910 align:middle line:90% But that's fine. 00:49:36.910 --> 00:49:39.440 align:middle line:90% 00:49:39.440 --> 00:49:41.640 align:middle line:90% I just needed to show-- 00:49:41.640 --> 00:49:47.880 align:middle line:84% all I needed to do was I needed to prove that-- 00:49:47.880 --> 00:49:51.780 align:middle line:84% I have that-- so you see that I have that-- 00:49:51.780 --> 00:49:55.800 align:middle line:84% I'm assuming that this thing here is strictly positive. 00:49:55.800 --> 00:49:58.280 align:middle line:90% And then I define alpha 1. 00:49:58.280 --> 00:50:02.960 align:middle line:84% But alpha 1 is strictly less than alpha. 00:50:02.960 --> 00:50:06.160 align:middle line:84% And then I have that alpha squared-- 00:50:06.160 --> 00:50:09.360 align:middle line:84% alpha 1 squared is bigger or equal to 2. 00:50:09.360 --> 00:50:14.480 align:middle line:84% This here means that alpha 1 is also an upper bound. 00:50:14.480 --> 00:50:17.200 align:middle line:84% This is strictly less than alpha. 00:50:17.200 --> 00:50:20.220 align:middle line:84% This was supposed to be the least upper bound. 00:50:20.220 --> 00:50:22.960 align:middle line:90% That's a contradiction, right? 00:50:22.960 --> 00:50:24.680 align:middle line:84% And so, you see, I fell into the trap. 00:50:24.680 --> 00:50:26.280 align:middle line:84% I was doing it so formally that I 00:50:26.280 --> 00:50:27.820 align:middle line:90% lost track of what I was doing. 00:50:27.820 --> 00:50:30.000 align:middle line:84% So I was actually in the process of proving the-- 00:50:30.000 --> 00:50:33.960 align:middle line:84% I mean, that's a danger with-- but the formal proof is 00:50:33.960 --> 00:50:36.420 align:middle line:90% much easier to read afterwards. 00:50:36.420 --> 00:50:41.640 align:middle line:84% And so, you see, the difference here with the formal proof was, 00:50:41.640 --> 00:50:46.220 align:middle line:84% last time, we did a little calculation. 00:50:46.220 --> 00:50:50.480 align:middle line:84% And then we tried to figure out, what should this h be? 00:50:50.480 --> 00:50:52.620 align:middle line:90% And then it came at the end. 00:50:52.620 --> 00:50:56.020 align:middle line:84% But when you write down the formal proof, 00:50:56.020 --> 00:50:58.760 align:middle line:84% then you would, up front-- you would already 00:50:58.760 --> 00:51:00.440 align:middle line:84% have done that little calculation. 00:51:00.440 --> 00:51:05.160 align:middle line:84% And you would, up front, say what h was supposed to be. 00:51:05.160 --> 00:51:08.920 align:middle line:84% And then you run through the argument. 00:51:08.920 --> 00:51:09.720 align:middle line:90% Yeah? 00:51:09.720 --> 00:51:11.840 align:middle line:84% AUDIENCE: Should it say that alpha-- 00:51:11.840 --> 00:51:15.723 align:middle line:90% oh, I was looking [INAUDIBLE]. 00:51:15.723 --> 00:51:16.640 align:middle line:90% TOBIAS COLDING: Sorry? 00:51:16.640 --> 00:51:18.598 align:middle line:84% AUDIENCE: I was just confused because I thought 00:51:18.598 --> 00:51:21.391 align:middle line:90% that alpha was [INAUDIBLE]. 00:51:21.391 --> 00:51:22.266 align:middle line:90% TOBIAS COLDING: Here? 00:51:22.266 --> 00:51:29.320 align:middle line:84% So if-- I mean, of course, alpha is much bigger than 1. 00:51:29.320 --> 00:51:31.970 align:middle line:84% But I'm just using-- we already-- 00:51:31.970 --> 00:51:35.810 align:middle line:84% I only need here that alpha is bigger or equal to 1 00:51:35.810 --> 00:51:38.150 align:middle line:84% to say that this thing here is bigger or equal to 2. 00:51:38.150 --> 00:51:40.010 align:middle line:84% AUDIENCE: If you get greater than 2, right? 00:51:40.010 --> 00:51:40.490 align:middle line:90% TOBIAS COLDING: Sorry? 00:51:40.490 --> 00:51:41.510 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:51:41.510 --> 00:51:42.830 align:middle line:90% AUDIENCE: I think [INAUDIBLE]. 00:51:42.830 --> 00:51:43.710 align:middle line:90% TOBIAS COLDING: This is-- 00:51:43.710 --> 00:51:44.210 align:middle line:90% right. 00:51:44.210 --> 00:51:45.390 align:middle line:90% This is alpha. 00:51:45.390 --> 00:51:49.390 align:middle line:84% So the inequality here is that alpha 1 squared 00:51:49.390 --> 00:51:51.430 align:middle line:90% is bigger or equal to 2. 00:51:51.430 --> 00:51:55.670 align:middle line:84% And I'm using that alpha at this stage 00:51:55.670 --> 00:51:58.030 align:middle line:84% here to conclude this inequality. 00:51:58.030 --> 00:52:01.830 align:middle line:90% I'm using-- sorry. 00:52:01.830 --> 00:52:03.010 align:middle line:90% No, no, you're right. 00:52:03.010 --> 00:52:04.090 align:middle line:90% Yeah, sorry about that. 00:52:04.090 --> 00:52:04.632 align:middle line:90% You're right. 00:52:04.632 --> 00:52:06.650 align:middle line:84% I'm using that alpha-- no, I'm not using that. 00:52:06.650 --> 00:52:10.030 align:middle line:84% I'm using that alpha squared is bigger or equal to 2. 00:52:10.030 --> 00:52:10.910 align:middle line:90% Yeah, sorry. 00:52:10.910 --> 00:52:11.970 align:middle line:90% That was what you meant. 00:52:11.970 --> 00:52:12.590 align:middle line:90% Yeah? 00:52:12.590 --> 00:52:14.710 align:middle line:90% Thank you. 00:52:14.710 --> 00:52:21.990 align:middle line:84% OK, so that was a formal proof of part 1. 00:52:21.990 --> 00:52:27.810 align:middle line:84% And then there's part 2, so the other inequality. 00:52:27.810 --> 00:52:30.830 align:middle line:90% 00:52:30.830 --> 00:52:34.590 align:middle line:90% So how do we do that? 00:52:34.590 --> 00:52:36.970 align:middle line:90% And when you write this-- 00:52:36.970 --> 00:52:39.910 align:middle line:90% 00:52:39.910 --> 00:52:43.790 align:middle line:84% when you write this, it's always good to try 00:52:43.790 --> 00:52:52.450 align:middle line:84% to explain what you're going to do, and in this case, 00:52:52.450 --> 00:52:54.370 align:middle line:84% what is going to be the contradiction. 00:52:54.370 --> 00:52:56.670 align:middle line:90% So let's try to prove part 2. 00:52:56.670 --> 00:52:59.750 align:middle line:84% And hopefully, we get it straight this time. 00:52:59.750 --> 00:53:04.470 align:middle line:84% So part 2, so I want to show that-- 00:53:04.470 --> 00:53:09.230 align:middle line:84% so we have-- part 1 was establishing that alpha 00:53:09.230 --> 00:53:13.990 align:middle line:90% squared was less or equal to 2. 00:53:13.990 --> 00:53:17.790 align:middle line:84% Now we want to prove-- so part 2, 00:53:17.790 --> 00:53:27.270 align:middle line:84% we want to show that alpha squared is bigger or equal to 2. 00:53:27.270 --> 00:53:29.990 align:middle line:90% 00:53:29.990 --> 00:53:36.390 align:middle line:84% And so, again, we're proving this-- 00:53:36.390 --> 00:53:38.330 align:middle line:84% we will show this by contradiction. 00:53:38.330 --> 00:53:51.710 align:middle line:90% 00:53:51.710 --> 00:54:03.570 align:middle line:84% So assume that alpha squared is strictly less than 2. 00:54:03.570 --> 00:54:06.235 align:middle line:90% 00:54:06.235 --> 00:54:10.114 align:middle line:90% And now it's-- 00:54:10.114 --> 00:54:11.530 align:middle line:90% I think it's helpful. 00:54:11.530 --> 00:54:13.210 align:middle line:84% You don't necessarily have to do it. 00:54:13.210 --> 00:54:17.030 align:middle line:84% But I think it's helpful if you say-- 00:54:17.030 --> 00:54:19.450 align:middle line:90% maybe it's too long. 00:54:19.450 --> 00:54:22.750 align:middle line:84% But it's at least helpful to think about, 00:54:22.750 --> 00:54:25.150 align:middle line:84% where is the contradiction coming from? 00:54:25.150 --> 00:54:29.780 align:middle line:84% So if alpha squared is less than 2-- 00:54:29.780 --> 00:54:31.375 align:middle line:90% so I will say it. 00:54:31.375 --> 00:54:33.000 align:middle line:84% You don't necessarily have to write it. 00:54:33.000 --> 00:54:56.248 align:middle line:84% But indeed, we will show that if this is the case, then 00:54:56.248 --> 00:54:56.790 align:middle line:90% there exist-- 00:54:56.790 --> 00:55:14.340 align:middle line:90% 00:55:14.340 --> 00:55:20.420 align:middle line:84% there exist some numbers, alpha 2, bigger than alpha, 00:55:20.420 --> 00:55:30.700 align:middle line:84% such that alpha 2 squared is also less than 2, 00:55:30.700 --> 00:55:49.100 align:middle line:84% contradicting that alpha is an upper bound for the set A 00:55:49.100 --> 00:55:50.140 align:middle line:90% we defined. 00:55:50.140 --> 00:55:51.860 align:middle line:90% That's what we will prove. 00:55:51.860 --> 00:55:55.570 align:middle line:90% 00:55:55.570 --> 00:56:02.290 align:middle line:84% And so we want to find a real number that is larger than this. 00:56:02.290 --> 00:56:05.500 align:middle line:90% 00:56:05.500 --> 00:56:12.020 align:middle line:84% But still, the square is still less than 2, OK? 00:56:12.020 --> 00:56:15.160 align:middle line:84% And so, in this case, I'm going to-- 00:56:15.160 --> 00:56:19.624 align:middle line:90% 00:56:19.624 --> 00:56:21.900 align:middle line:90% so now we do this-- 00:56:21.900 --> 00:56:26.140 align:middle line:84% the structure is going to be the same. 00:56:26.140 --> 00:56:31.460 align:middle line:84% We had this little scrap paper where we outlined the argument 00:56:31.460 --> 00:56:34.540 align:middle line:90% and figure out what h had to be. 00:56:34.540 --> 00:56:37.180 align:middle line:90% And so I'm going to use this h. 00:56:37.180 --> 00:56:39.220 align:middle line:90% So we're going to, again-- 00:56:39.220 --> 00:56:41.100 align:middle line:84% maybe I'll just erase this here so we 00:56:41.100 --> 00:56:44.280 align:middle line:84% don't get confused with the first part and the second part. 00:56:44.280 --> 00:56:54.320 align:middle line:90% 00:56:54.320 --> 00:56:54.820 align:middle line:90% OK. 00:56:54.820 --> 00:57:15.220 align:middle line:90% 00:57:15.220 --> 00:57:21.640 align:middle line:90% OK, so I'm going to set h again. 00:57:21.640 --> 00:57:26.330 align:middle line:84% I'm going to define it very similar to what I did before. 00:57:26.330 --> 00:57:28.570 align:middle line:84% But now I have to remember that I'm 00:57:28.570 --> 00:57:32.210 align:middle line:84% assuming that alpha squared is less than 2. 00:57:32.210 --> 00:57:38.620 align:middle line:84% So I define h to be 2 minus alpha squared over 4 alpha. 00:57:38.620 --> 00:57:42.570 align:middle line:90% 00:57:42.570 --> 00:57:45.890 align:middle line:90% This is how I define it. 00:57:45.890 --> 00:57:51.410 align:middle line:84% And then what I'm interested in is-- so I have here 0. 00:57:51.410 --> 00:57:54.770 align:middle line:84% I have this that is supposed to be the upper bound. 00:57:54.770 --> 00:57:58.110 align:middle line:84% But I want to prove that that really isn't an upper bound. 00:57:58.110 --> 00:58:04.830 align:middle line:84% That's an element in my set A. And so this is this alpha 2. 00:58:04.830 --> 00:58:08.170 align:middle line:84% So I'm defining-- so I'm setting-- 00:58:08.170 --> 00:58:16.890 align:middle line:84% set this and alpha 2 equal to alpha plus h. 00:58:16.890 --> 00:58:18.270 align:middle line:84% This here was just to illustrate. 00:58:18.270 --> 00:58:21.150 align:middle line:84% You don't necessarily have to include that. 00:58:21.150 --> 00:58:23.950 align:middle line:90% 00:58:23.950 --> 00:58:26.170 align:middle line:90% And I want to show-- 00:58:26.170 --> 00:58:27.350 align:middle line:90% we want to show-- 00:58:27.350 --> 00:58:30.170 align:middle line:90% 00:58:30.170 --> 00:58:33.210 align:middle line:84% I think it's a good idea to write what you want to show. 00:58:33.210 --> 00:58:36.890 align:middle line:84% In particular, if I really had written that before, 00:58:36.890 --> 00:58:38.490 align:middle line:84% then I wouldn't have been in trouble, 00:58:38.490 --> 00:58:42.050 align:middle line:84% because I was trying to prove something else. 00:58:42.050 --> 00:58:48.570 align:middle line:84% So you want to prove that alpha squared here 00:58:48.570 --> 00:58:52.410 align:middle line:90% is also less than 2. 00:58:52.410 --> 00:59:03.190 align:middle line:84% So this is-- so we already have that alpha here is positive. 00:59:03.190 --> 00:59:10.530 align:middle line:90% 00:59:10.530 --> 00:59:13.170 align:middle line:90% It's bigger than 1. 00:59:13.170 --> 00:59:17.850 align:middle line:84% So this number here, this here, we already know, 00:59:17.850 --> 00:59:19.090 align:middle line:90% is bigger than alpha. 00:59:19.090 --> 00:59:23.410 align:middle line:90% 00:59:23.410 --> 00:59:26.450 align:middle line:90% And so, if I can prove that-- 00:59:26.450 --> 00:59:27.890 align:middle line:90% so this here is-- 00:59:27.890 --> 00:59:29.510 align:middle line:90% so it's bigger than alpha. 00:59:29.510 --> 00:59:32.690 align:middle line:84% If I can prove that the square is less than 2, 00:59:32.690 --> 00:59:34.650 align:middle line:90% then alpha here-- 00:59:34.650 --> 00:59:37.630 align:middle line:84% alpha here, sorry-- would not be an upper bound. 00:59:37.630 --> 00:59:44.030 align:middle line:84% There's another element in the set A that is bigger. 00:59:44.030 --> 00:59:45.170 align:middle line:90% So I want to prove that. 00:59:45.170 --> 00:59:49.910 align:middle line:84% So now I'm looking at alpha 2 squared. 00:59:49.910 --> 00:59:53.750 align:middle line:90% So that is alpha plus h, right? 00:59:53.750 --> 00:59:55.930 align:middle line:90% That's how I define alpha 2. 00:59:55.930 --> 00:59:57.110 align:middle line:90% And I square it. 00:59:57.110 --> 01:00:00.090 align:middle line:90% 01:00:00.090 --> 01:00:02.290 align:middle line:84% And if I do that, then I get alpha 01:00:02.290 --> 01:00:12.660 align:middle line:84% squared plus h squared plus 2 h alpha. 01:00:12.660 --> 01:00:16.180 align:middle line:90% 01:00:16.180 --> 01:00:20.260 align:middle line:84% And I now-- so now I want an upper bound. 01:00:20.260 --> 01:00:26.480 align:middle line:84% And so we already have that alpha is this one here. 01:00:26.480 --> 01:00:31.560 align:middle line:84% h is-- you see that h is, in this case here-- 01:00:31.560 --> 01:00:38.600 align:middle line:84% so h is equal to 2 minus alpha squared over 4 alpha. 01:00:38.600 --> 01:00:42.240 align:middle line:90% Alpha is bigger or equal to 1. 01:00:42.240 --> 01:00:45.640 align:middle line:84% This here, the denominator, is bounded by 2. 01:00:45.640 --> 01:00:47.180 align:middle line:90% So again, it's 2 over 4. 01:00:47.180 --> 01:00:50.920 align:middle line:90% So this is bounded by 1/2. 01:00:50.920 --> 01:00:54.340 align:middle line:90% So h is positive. 01:00:54.340 --> 01:01:01.000 align:middle line:84% And we have that h is positive and h is less than 1/2, right? 01:01:01.000 --> 01:01:06.040 align:middle line:84% And so now, this here, I can bound. 01:01:06.040 --> 01:01:10.520 align:middle line:84% I can bound trivially by just using that h is bounded by 1. 01:01:10.520 --> 01:01:15.920 align:middle line:90% So I get this plus h plus 2. 01:01:15.920 --> 01:01:22.920 align:middle line:84% And then h times alpha, again, or 2h alpha, this is-- 01:01:22.920 --> 01:01:27.720 align:middle line:84% when we multiply this by 2 alpha, then the alpha 01:01:27.720 --> 01:01:28.840 align:middle line:90% cancel out. 01:01:28.840 --> 01:01:34.080 align:middle line:84% And the 2 cancel with one of these-- 01:01:34.080 --> 01:01:35.360 align:middle line:90% 2 and this 4. 01:01:35.360 --> 01:01:39.140 align:middle line:84% So you're getting 2 minus alpha squared over 2. 01:01:39.140 --> 01:01:43.010 align:middle line:90% 01:01:43.010 --> 01:01:49.280 align:middle line:84% And so now I get that this thing here is bounded by-- 01:01:49.280 --> 01:01:56.600 align:middle line:90% so h here, h is also at-- 01:01:56.600 --> 01:02:01.920 align:middle line:84% a very rough bound for h is using-- sorry-- 01:02:01.920 --> 01:02:04.760 align:middle line:84% so using-- so I have that, and now 01:02:04.760 --> 01:02:16.920 align:middle line:84% using that h here is equal to alpha 2 minus alpha squared 01:02:16.920 --> 01:02:24.440 align:middle line:84% over 2 over 4 alpha, which is less than 2 01:02:24.440 --> 01:02:31.520 align:middle line:84% minus alpha squared over 2, since alpha here 01:02:31.520 --> 01:02:34.180 align:middle line:90% is bigger than 1, right? 01:02:34.180 --> 01:02:35.880 align:middle line:90% So I have that. 01:02:35.880 --> 01:02:46.120 align:middle line:84% So using that, we have that alpha 2 squared 01:02:46.120 --> 01:02:49.120 align:middle line:84% is less or equal to-- let me just write that line. 01:02:49.120 --> 01:02:50.880 align:middle line:90% I'm just now filling in. 01:02:50.880 --> 01:02:53.080 align:middle line:90% Alpha squared, the H here-- 01:02:53.080 --> 01:02:55.760 align:middle line:90% I'm using this inequality here-- 01:02:55.760 --> 01:02:59.120 align:middle line:90% 2 minus alpha squared over 2. 01:02:59.120 --> 01:03:03.160 align:middle line:84% And then I have the same thing again. 01:03:03.160 --> 01:03:07.560 align:middle line:84% And so you see that this here, it's this here times 2. 01:03:07.560 --> 01:03:15.400 align:middle line:84% So this is just alpha squared minus 2 plus 2 minus alpha 01:03:15.400 --> 01:03:15.900 align:middle line:90% squared. 01:03:15.900 --> 01:03:19.230 align:middle line:90% 01:03:19.230 --> 01:03:24.030 align:middle line:84% And this inequality here, one of these inequalities, I 01:03:24.030 --> 01:03:26.690 align:middle line:84% should have replaced by a strict inequality. 01:03:26.690 --> 01:03:30.370 align:middle line:84% This is because h is positive and is strictly less than 1. 01:03:30.370 --> 01:03:32.630 align:middle line:84% So this here is actually a strict inequality. 01:03:32.630 --> 01:03:34.890 align:middle line:84% So this here is a strict inequality. 01:03:34.890 --> 01:03:37.650 align:middle line:90% 01:03:37.650 --> 01:03:42.750 align:middle line:84% But now you see that this thing here, the alpha cancel out. 01:03:42.750 --> 01:03:44.750 align:middle line:90% And you're getting 2. 01:03:44.750 --> 01:03:50.350 align:middle line:84% So you have that alpha 2 squared is strictly less than 2. 01:03:50.350 --> 01:03:52.630 align:middle line:90% That's what we wanted to prove. 01:03:52.630 --> 01:04:07.150 align:middle line:84% And this gives that alpha 2 is also in A. And yet, alpha 2 01:04:07.150 --> 01:04:10.750 align:middle line:90% was bigger than alpha. 01:04:10.750 --> 01:04:13.130 align:middle line:84% And this is the desired contradiction. 01:04:13.130 --> 01:04:25.096 align:middle line:90% 01:04:25.096 --> 01:04:28.530 align:middle line:84% So again, we already did this last time. 01:04:28.530 --> 01:04:32.230 align:middle line:84% But the point here is that, how do you 01:04:32.230 --> 01:04:34.070 align:middle line:90% write down a formal proof? 01:04:34.070 --> 01:04:36.030 align:middle line:84% And a formal proof, in this case, 01:04:36.030 --> 01:04:38.850 align:middle line:84% is that you try-- very clearly, in this case, 01:04:38.850 --> 01:04:41.390 align:middle line:90% it was divided into two steps. 01:04:41.390 --> 01:04:45.330 align:middle line:84% So you try to divide it into the two steps. 01:04:45.330 --> 01:04:50.790 align:middle line:84% And then maybe you have a last line to say that part 1-- 01:04:50.790 --> 01:05:06.670 align:middle line:84% part 1 and part 2 together show that alpha squared is equal 01:05:06.670 --> 01:05:09.210 align:middle line:90% to 2, completing the proof. 01:05:09.210 --> 01:05:17.630 align:middle line:90% 01:05:17.630 --> 01:05:20.390 align:middle line:84% And again, the point here was that-- so we already 01:05:20.390 --> 01:05:21.890 align:middle line:90% knew how to prove this. 01:05:21.890 --> 01:05:25.396 align:middle line:84% But how do you write down a formal proof? 01:05:25.396 --> 01:05:30.850 align:middle line:84% And so, for this proof, it was divided into these two steps. 01:05:30.850 --> 01:05:34.430 align:middle line:84% So you try to write each step separately. 01:05:34.430 --> 01:05:36.910 align:middle line:84% They were both-- both of the two steps 01:05:36.910 --> 01:05:39.790 align:middle line:90% was proven by contradiction. 01:05:39.790 --> 01:05:41.830 align:middle line:84% And so you would say up front-- you 01:05:41.830 --> 01:05:45.310 align:middle line:84% would say that you're going to prove it by contradiction. 01:05:45.310 --> 01:05:48.870 align:middle line:84% And then, of course, there would be-- 01:05:48.870 --> 01:05:50.730 align:middle line:84% the statement that you want to prove-- 01:05:50.730 --> 01:05:53.090 align:middle line:84% since you're proving it by contradiction, 01:05:53.090 --> 01:05:56.030 align:middle line:90% you would assume the negation. 01:05:56.030 --> 01:05:59.930 align:middle line:84% And then I think, if you can, and it doesn't feel too much, 01:05:59.930 --> 01:06:03.925 align:middle line:84% I think it's very helpful to write what you are trying, 01:06:03.925 --> 01:06:06.050 align:middle line:84% how you are trying to get the contradiction, right? 01:06:06.050 --> 01:06:07.120 align:middle line:90% That's what I tried here. 01:06:07.120 --> 01:06:09.650 align:middle line:90% 01:06:09.650 --> 01:06:11.900 align:middle line:84% And then, you see, then the big difference 01:06:11.900 --> 01:06:14.620 align:middle line:84% from when we were trying to figure out how to prove it 01:06:14.620 --> 01:06:19.820 align:middle line:84% is that you now go ahead, and you just define-- 01:06:19.820 --> 01:06:21.220 align:middle line:90% you define those things. 01:06:21.220 --> 01:06:24.440 align:middle line:90% You define h and alpha 2, right? 01:06:24.440 --> 01:06:29.460 align:middle line:84% That seems to come out of nowhere. 01:06:29.460 --> 01:06:32.780 align:middle line:84% And then you establish the property 01:06:32.780 --> 01:06:35.740 align:middle line:84% that you want to prove that leads to the contradiction. 01:06:35.740 --> 01:06:39.500 align:middle line:90% That's the structure. 01:06:39.500 --> 01:06:41.330 align:middle line:90% Are there any questions to this? 01:06:41.330 --> 01:06:43.800 align:middle line:90% 01:06:43.800 --> 01:06:44.300 align:middle line:90% Yeah? 01:06:44.300 --> 01:06:50.670 align:middle line:84% AUDIENCE: [INAUDIBLE] property of the real numbers [INAUDIBLE] 01:06:50.670 --> 01:06:52.500 align:middle line:90% least upper bound [INAUDIBLE]? 01:06:52.500 --> 01:06:53.417 align:middle line:90% TOBIAS COLDING: Right. 01:06:53.417 --> 01:06:55.340 align:middle line:90% So we're using a few things. 01:06:55.340 --> 01:07:01.580 align:middle line:84% We're using-- and we're using a few things. 01:07:01.580 --> 01:07:05.100 align:middle line:90% We're using that it's ordered. 01:07:05.100 --> 01:07:09.900 align:middle line:84% We're also using something about that it's an ordered field. 01:07:09.900 --> 01:07:10.608 align:middle line:90% And why? 01:07:10.608 --> 01:07:12.150 align:middle line:84% Because we're using something-- well, 01:07:12.150 --> 01:07:14.260 align:middle line:84% we're using something about multiplication 01:07:14.260 --> 01:07:16.360 align:middle line:90% to at all define square root. 01:07:16.360 --> 01:07:21.220 align:middle line:84% And also, we're using something about that 1 is bigger than 0. 01:07:21.220 --> 01:07:30.940 align:middle line:90% 01:07:30.940 --> 01:07:33.940 align:middle line:84% And then-- so we're using-- and then we're also using, 01:07:33.940 --> 01:07:37.100 align:middle line:84% of course, completeness in order to define this least 01:07:37.100 --> 01:07:38.140 align:middle line:90% upper bound. 01:07:38.140 --> 01:07:41.420 align:middle line:84% So we're using a bunch of these things. 01:07:41.420 --> 01:07:44.740 align:middle line:90% But of course, if you have-- 01:07:44.740 --> 01:07:47.100 align:middle line:90% yeah. 01:07:47.100 --> 01:07:49.860 align:middle line:90% I mean, if you're asking when-- 01:07:49.860 --> 01:07:52.900 align:middle line:90% and that's not really-- 01:07:52.900 --> 01:07:59.020 align:middle line:84% at this stage, very, very soon, in one of the next few lectures, 01:07:59.020 --> 01:08:01.700 align:middle line:84% things will be in the real numbers. 01:08:01.700 --> 01:08:04.020 align:middle line:84% We're not going to work on general fields. 01:08:04.020 --> 01:08:06.020 align:middle line:90% And we're not going to-- yeah. 01:08:06.020 --> 01:08:08.500 align:middle line:84% General fields-- this is not part of this topic. 01:08:08.500 --> 01:08:13.220 align:middle line:84% But of course, it's interesting to ask. 01:08:13.220 --> 01:08:15.100 align:middle line:84% And I think that's your point, is 01:08:15.100 --> 01:08:17.340 align:middle line:84% like, when can you define something 01:08:17.340 --> 01:08:20.260 align:middle line:84% like a square root of 2 in a field? 01:08:20.260 --> 01:08:21.340 align:middle line:90% Or something like that. 01:08:21.340 --> 01:08:22.600 align:middle line:90% And there's a lot of things. 01:08:22.600 --> 01:08:24.100 align:middle line:84% You're absolutely right that there's 01:08:24.100 --> 01:08:27.979 align:middle line:84% a lot of these things that was just general property. 01:08:27.979 --> 01:08:30.279 align:middle line:84% But maybe we shouldn't get too much into that. 01:08:30.279 --> 01:08:33.340 align:middle line:90% 01:08:33.340 --> 01:08:35.620 align:middle line:90% So I wanted to-- 01:08:35.620 --> 01:08:39.100 align:middle line:84% there's a few more things I was hoping to do. 01:08:39.100 --> 01:08:44.859 align:middle line:84% So I also wanted to just talk about the Archimedean property 01:08:44.859 --> 01:08:45.859 align:middle line:90% again. 01:08:45.859 --> 01:08:56.399 align:middle line:84% And so I wanted to just outline how to write a formal proof. 01:08:56.399 --> 01:09:01.620 align:middle line:84% And the reason, of course, why I'm illustrating how to 01:09:01.620 --> 01:09:03.620 align:middle line:90% write a formal-- 01:09:03.620 --> 01:09:07.170 align:middle line:84% writing a formal proof is an important part of this class. 01:09:07.170 --> 01:09:12.010 align:middle line:84% And I'm obviously using the things 01:09:12.010 --> 01:09:15.490 align:middle line:84% that we already proved last time, just so 01:09:15.490 --> 01:09:17.676 align:middle line:90% that you can focus on how-- 01:09:17.676 --> 01:09:20.729 align:middle line:84% not to prove-- not the statements, 01:09:20.729 --> 01:09:25.689 align:middle line:84% why the statements should be true, but just how to write it. 01:09:25.689 --> 01:09:31.660 align:middle line:84% And so let's talk about the Archimedean property. 01:09:31.660 --> 01:09:37.850 align:middle line:90% 01:09:37.850 --> 01:09:41.770 align:middle line:84% And so the Archimedean property, just saying that-- 01:09:41.770 --> 01:09:49.010 align:middle line:84% so this is the theorem that the natural numbers is not 01:09:49.010 --> 01:09:50.019 align:middle line:90% bounded from above. 01:09:50.019 --> 01:09:59.050 align:middle line:90% 01:09:59.050 --> 01:10:01.090 align:middle line:84% That's one way of formulating it. 01:10:01.090 --> 01:10:03.670 align:middle line:84% You could also say, like we did last time, 01:10:03.670 --> 01:10:05.970 align:middle line:84% that whenever you take a real, there's 01:10:05.970 --> 01:10:10.050 align:middle line:84% a larger natural number than that real. 01:10:10.050 --> 01:10:12.950 align:middle line:90% But that's the same. 01:10:12.950 --> 01:10:13.450 align:middle line:90% OK. 01:10:13.450 --> 01:10:15.310 align:middle line:90% So now, how do we prove this? 01:10:15.310 --> 01:10:19.370 align:middle line:90% 01:10:19.370 --> 01:10:21.450 align:middle line:84% So again, here, we're thinking about things 01:10:21.450 --> 01:10:24.850 align:middle line:84% as being subset of R. We always-- 01:10:24.850 --> 01:10:28.370 align:middle line:84% when we don't really mention it, then we always 01:10:28.370 --> 01:10:35.530 align:middle line:84% think about it as being subset of R. And so we will prove-- 01:10:35.530 --> 01:10:37.430 align:middle line:84% we will show this by contradiction-- 01:10:37.430 --> 01:10:44.370 align:middle line:90% 01:10:44.370 --> 01:10:45.510 align:middle line:90% by contradiction. 01:10:45.510 --> 01:10:49.210 align:middle line:90% 01:10:49.210 --> 01:11:02.920 align:middle line:84% So suppose alpha is an upper bound for the natural numbers. 01:11:02.920 --> 01:11:05.730 align:middle line:90% 01:11:05.730 --> 01:11:22.290 align:middle line:84% Well, we can assume, after replacing alpha by sup of A, 01:11:22.290 --> 01:11:33.900 align:middle line:84% we can assume that alpha is the least upper bound. 01:11:33.900 --> 01:11:43.650 align:middle line:90% 01:11:43.650 --> 01:11:48.730 align:middle line:84% And now-- and then I think it's nice, again, 01:11:48.730 --> 01:11:54.860 align:middle line:84% to write where the contradiction is coming from. 01:11:54.860 --> 01:12:02.010 align:middle line:90% 01:12:02.010 --> 01:12:33.890 align:middle line:84% So we'll show-- we will get the desired contradiction by showing 01:12:33.890 --> 01:12:50.370 align:middle line:84% that alpha minus 1 is also an upper bound, contradicting 01:12:50.370 --> 01:12:53.910 align:middle line:84% that alpha was the least upper bound. 01:12:53.910 --> 01:13:03.860 align:middle line:90% 01:13:03.860 --> 01:13:11.180 align:middle line:90% And so now, since alpha-- 01:13:11.180 --> 01:13:14.440 align:middle line:84% so this is what we want to prove. 01:13:14.440 --> 01:13:32.880 align:middle line:84% Since alpha is an upper bound for all natural numbers, 01:13:32.880 --> 01:13:35.000 align:middle line:84% if you add 1 to the natural number, 01:13:35.000 --> 01:13:39.200 align:middle line:84% this is also-- this here is also a natural number, right? 01:13:39.200 --> 01:13:41.300 align:middle line:84% n plus 1 is also a natural number. 01:13:41.300 --> 01:13:46.820 align:middle line:84% So n plus 1 must be less or equal to this upper bound. 01:13:46.820 --> 01:13:50.000 align:middle line:90% 01:13:50.000 --> 01:14:01.120 align:middle line:90% But this implies that n-- 01:14:01.120 --> 01:14:05.280 align:middle line:84% subtracting 1 on both sides implies that n is less or equal 01:14:05.280 --> 01:14:09.120 align:middle line:90% to alpha minus 1, and this-- 01:14:09.120 --> 01:14:20.260 align:middle line:84% which holds for all n, all natural numbers. 01:14:20.260 --> 01:14:26.620 align:middle line:84% So alpha minus 1 is also an upper bound. 01:14:26.620 --> 01:14:40.960 align:middle line:90% 01:14:40.960 --> 01:14:41.640 align:middle line:90% OK. 01:14:41.640 --> 01:14:48.720 align:middle line:84% And this proof gives you the desired contradiction. 01:14:48.720 --> 01:14:52.720 align:middle line:84% So that's how you would write that. 01:14:52.720 --> 01:14:57.360 align:middle line:84% Let me talk a little bit about sequences. 01:14:57.360 --> 01:15:00.440 align:middle line:90% So this will be a big topic. 01:15:00.440 --> 01:15:01.660 align:middle line:90% So sequences. 01:15:01.660 --> 01:15:06.600 align:middle line:90% 01:15:06.600 --> 01:15:09.180 align:middle line:84% Well, first-- so we talked about that a little bit. 01:15:09.180 --> 01:15:22.640 align:middle line:84% So if you look at the-- so how do we think about square root 01:15:22.640 --> 01:15:26.340 align:middle line:90% 2 on the real line? 01:15:26.340 --> 01:15:30.040 align:middle line:84% So the way to think about square root of 2 is what we did here-- 01:15:30.040 --> 01:15:34.340 align:middle line:84% that is, this positive thing so that the square is 2. 01:15:34.340 --> 01:15:39.960 align:middle line:84% But if you want to place square root of 2 on the real line, 01:15:39.960 --> 01:15:46.060 align:middle line:84% then you could think about it as 1 as a limit of a sequence. 01:15:46.060 --> 01:15:51.280 align:middle line:84% If you take 1, and then the 1 is less than 1.4, 01:15:51.280 --> 01:16:02.990 align:middle line:84% which is less than 1.41, which is less than 1.414, et cetera. 01:16:02.990 --> 01:16:07.590 align:middle line:90% And so this here is a sequence. 01:16:07.590 --> 01:16:10.050 align:middle line:84% I'll make that precise in just a minute, 01:16:10.050 --> 01:16:13.110 align:middle line:84% but an increasing sequence of numbers. 01:16:13.110 --> 01:16:17.210 align:middle line:84% And the limit is square root of 2. 01:16:17.210 --> 01:16:21.470 align:middle line:84% So if you plug square root of 2 in on the calculator, 01:16:21.470 --> 01:16:23.370 align:middle line:84% it will bring out a lot of digits. 01:16:23.370 --> 01:16:24.970 align:middle line:90% And these are the first digits. 01:16:24.970 --> 01:16:27.070 align:middle line:90% And it will continue on. 01:16:27.070 --> 01:16:30.270 align:middle line:84% And that's how you think about square root of 2 01:16:30.270 --> 01:16:34.670 align:middle line:84% if you want to think about it on the real line. 01:16:34.670 --> 01:16:36.750 align:middle line:90% And so we want to-- 01:16:36.750 --> 01:16:40.870 align:middle line:84% and so square root of 2 is going to be the limit of these. 01:16:40.870 --> 01:16:42.570 align:middle line:84% And we need to make this precise. 01:16:42.570 --> 01:16:48.990 align:middle line:84% And we also need to make precise what, really, a sequence is. 01:16:48.990 --> 01:17:05.740 align:middle line:84% So a sequence-- so a sequence of real numbers-- 01:17:05.740 --> 01:17:13.750 align:middle line:90% 01:17:13.750 --> 01:17:14.720 align:middle line:90% this is a map. 01:17:14.720 --> 01:17:17.310 align:middle line:90% 01:17:17.310 --> 01:17:24.630 align:middle line:84% It's a function from the natural numbers into R. 01:17:24.630 --> 01:17:29.910 align:middle line:84% So for each natural number, you get a real number. 01:17:29.910 --> 01:17:35.150 align:middle line:90% And so a function-- 01:17:35.150 --> 01:17:39.990 align:middle line:84% a function is defined by the value at different points. 01:17:39.990 --> 01:17:42.690 align:middle line:90% So this thing here-- 01:17:42.690 --> 01:17:46.610 align:middle line:84% but if you want to think about it as a sequence, 01:17:46.610 --> 01:17:51.870 align:middle line:84% then it's common to denote it-- to denote this here like that. 01:17:51.870 --> 01:17:55.430 align:middle line:84% So this is how one typically writes a sequence, where, 01:17:55.430 --> 01:18:03.570 align:middle line:84% instead of what the argument is, what this is, inside a bracket, 01:18:03.570 --> 01:18:07.000 align:middle line:90% you write it as a subscript. 01:18:07.000 --> 01:18:10.150 align:middle line:90% So this is a sequence. 01:18:10.150 --> 01:18:15.550 align:middle line:84% And so, in the case over there, if you're 01:18:15.550 --> 01:18:20.470 align:middle line:84% thinking about a1, say, to be equal to 1, 01:18:20.470 --> 01:18:25.830 align:middle line:90% a2 to be equal to 1.4, a3-- 01:18:25.830 --> 01:18:28.830 align:middle line:84% and now I'm thinking about the sequence here. 01:18:28.830 --> 01:18:31.790 align:middle line:84% As I'm thinking about square root of 2, 01:18:31.790 --> 01:18:36.470 align:middle line:84% and I take the number of the subscript here, 01:18:36.470 --> 01:18:39.870 align:middle line:84% I'm just taking the number of digits in square root of 2. 01:18:39.870 --> 01:18:41.630 align:middle line:90% So now I need three digits. 01:18:41.630 --> 01:18:46.310 align:middle line:90% Here, I need a 2, 1, 4, 1. 01:18:46.310 --> 01:18:53.920 align:middle line:90% a4 is 1.414, et cetera. 01:18:53.920 --> 01:18:58.540 align:middle line:84% And so then you have-- so then a n here 01:18:58.540 --> 01:19:05.780 align:middle line:90% is a sequence of real numbers. 01:19:05.780 --> 01:19:07.700 align:middle line:90% And it's actually increasing-- 01:19:07.700 --> 01:19:12.300 align:middle line:84% that the sequence a n is less than a n plus 1. 01:19:12.300 --> 01:19:14.540 align:middle line:90% We have this property. 01:19:14.540 --> 01:19:20.620 align:middle line:84% And we want to talk about-- next time, we want to talk about, 01:19:20.620 --> 01:19:23.100 align:middle line:90% what does a limit mean? 01:19:23.100 --> 01:19:24.670 align:middle line:90% What is a limit? 01:19:24.670 --> 01:19:27.380 align:middle line:90% 01:19:27.380 --> 01:19:31.780 align:middle line:84% And so square root of 2, indeed, will 01:19:31.780 --> 01:19:33.880 align:middle line:90% be the limit of this sequence. 01:19:33.880 --> 01:19:36.500 align:middle line:84% And the limit, you write like this, sometimes, 01:19:36.500 --> 01:19:40.940 align:middle line:90% as n goes to infinity of a n. 01:19:40.940 --> 01:19:43.810 align:middle line:84% But we'll talk about this next time. 01:19:43.810 --> 01:19:47.000 align:middle line:90%