WEBVTT 00:00:00.000 --> 00:00:01.976 align:middle line:90% [SQUEAKING] 00:00:01.976 --> 00:00:04.446 align:middle line:90% [RUSTLING] 00:00:04.446 --> 00:00:05.434 align:middle line:90% [CLICKING] 00:00:05.434 --> 00:00:12.350 align:middle line:90% 00:00:12.350 --> 00:00:15.440 align:middle line:84% CASEY RODRIGUEZ: So, all right, so let's start 00:00:15.440 --> 00:00:16.490 align:middle line:90% talking about some math. 00:00:16.490 --> 00:00:19.640 align:middle line:90% 00:00:19.640 --> 00:00:22.540 align:middle line:84% So there was a last result I was going 00:00:22.540 --> 00:00:26.410 align:middle line:84% to prove for series, which is that if you have an absolutely 00:00:26.410 --> 00:00:29.350 align:middle line:84% convergent series, and you rearrange it, 00:00:29.350 --> 00:00:32.380 align:middle line:84% then that series is also absolutely 00:00:32.380 --> 00:00:34.480 align:middle line:84% convergent and converges to the same thing 00:00:34.480 --> 00:00:36.760 align:middle line:90% as the unarranged series. 00:00:36.760 --> 00:00:39.340 align:middle line:84% But I'm going to leave that to the lecture notes, 00:00:39.340 --> 00:00:41.350 align:middle line:84% and we're just going to start on a new topic. 00:00:41.350 --> 00:00:44.210 align:middle line:90% 00:00:44.210 --> 00:00:53.350 align:middle line:84% So the topic that we're now on is continuous functions. 00:00:53.350 --> 00:00:57.890 align:middle line:90% 00:00:57.890 --> 00:01:04.158 align:middle line:84% And so continuous functions, as we'll see, 00:01:04.158 --> 00:01:06.960 align:middle line:84% is a statement about how a function behaves 00:01:06.960 --> 00:01:13.990 align:middle line:84% near a point in relation to how it behaves at the point. 00:01:13.990 --> 00:01:15.790 align:middle line:84% Now, what it means for a function 00:01:15.790 --> 00:01:18.220 align:middle line:84% to behave near a point-- that's the notion 00:01:18.220 --> 00:01:22.238 align:middle line:84% of the limit of the function, which 00:01:22.238 --> 00:01:23.405 align:middle line:90% is what we'll first discuss. 00:01:23.405 --> 00:01:33.370 align:middle line:90% 00:01:33.370 --> 00:01:34.960 align:middle line:90% OK? 00:01:34.960 --> 00:01:40.280 align:middle line:84% Now, at what points are we going to be looking at? 00:01:40.280 --> 00:01:43.840 align:middle line:84% So this is where limits take place. 00:01:43.840 --> 00:01:48.550 align:middle line:84% And somehow, we want to look at a function near points 00:01:48.550 --> 00:01:53.650 align:middle line:84% where there's a set nearby to actually look at f at. 00:01:53.650 --> 00:01:57.310 align:middle line:84% So f is usually defined on some set S, 00:01:57.310 --> 00:02:00.898 align:middle line:84% and we want to look at points so that there's a lot of S nearby. 00:02:00.898 --> 00:02:03.190 align:middle line:84% And we kind of already dealt with these types of points 00:02:03.190 --> 00:02:07.450 align:middle line:84% when we encountered cluster points in the assignment. 00:02:07.450 --> 00:02:10.639 align:middle line:84% So this is exactly where we start looking at limits. 00:02:10.639 --> 00:02:15.950 align:middle line:84% So this is a definition that I'll 00:02:15.950 --> 00:02:19.030 align:middle line:90% recall from the assignment. 00:02:19.030 --> 00:02:30.580 align:middle line:84% Let S be a subset of R and x and R. 00:02:30.580 --> 00:02:46.510 align:middle line:84% We say x is a cluster point of S if, for all delta positive, 00:02:46.510 --> 00:02:51.300 align:middle line:84% the interval x minus delta, x plus delta intersect 00:02:51.300 --> 00:02:59.030 align:middle line:90% S take away x is non-empty, OK? 00:02:59.030 --> 00:03:02.500 align:middle line:84% So the new terminology-- not so new 00:03:02.500 --> 00:03:04.250 align:middle line:84% since we dealt with it in the assignment-- 00:03:04.250 --> 00:03:07.310 align:middle line:90% is that of a cluster point. 00:03:07.310 --> 00:03:09.680 align:middle line:84% And another way to say this is that-- 00:03:09.680 --> 00:03:11.420 align:middle line:84% so an equivalent way of stating this 00:03:11.420 --> 00:03:16.850 align:middle line:84% is that, for all delta positive, there exists, let's say, 00:03:16.850 --> 00:03:26.400 align:middle line:84% a y in S such that 0 is bigger than x minus y 00:03:26.400 --> 00:03:31.150 align:middle line:90% is less than delta, OK? 00:03:31.150 --> 00:03:37.870 align:middle line:84% So what this means is that, as far as the picture goes, 00:03:37.870 --> 00:03:40.270 align:middle line:84% x is a cluster point if, when I ever 00:03:40.270 --> 00:03:42.850 align:middle line:84% take any small interval about it, 00:03:42.850 --> 00:03:51.970 align:middle line:84% I can find some S in there other than possibly x. 00:03:51.970 --> 00:03:54.730 align:middle line:84% And this should be able to be done for every interval. 00:03:54.730 --> 00:04:03.380 align:middle line:84% So let's take a look at some examples. 00:04:03.380 --> 00:04:12.250 align:middle line:84% Let's say we take S to be 1/n in a natural number, OK? 00:04:12.250 --> 00:04:15.000 align:middle line:90% 00:04:15.000 --> 00:04:25.300 align:middle line:84% And so since now I actually have people in front of me, 00:04:25.300 --> 00:04:27.100 align:middle line:90% I can actually ask questions. 00:04:27.100 --> 00:04:37.940 align:middle line:90% 00:04:37.940 --> 00:04:42.050 align:middle line:84% So whoever first says it, what would be 00:04:42.050 --> 00:04:44.103 align:middle line:90% a cluster point of this set S? 00:04:44.103 --> 00:04:46.881 align:middle line:90% 00:04:46.881 --> 00:04:49.280 align:middle line:84% So what would be a point in the real numbers 00:04:49.280 --> 00:04:51.340 align:middle line:84% So that there's a lot of S near it? 00:04:51.340 --> 00:04:55.330 align:middle line:90% 00:04:55.330 --> 00:04:57.580 align:middle line:84% Feel free to blurt it out if it comes to you. 00:04:57.580 --> 00:05:06.340 align:middle line:90% 00:05:06.340 --> 00:05:08.770 align:middle line:90% Why don't you just take a guess? 00:05:08.770 --> 00:05:09.270 align:middle line:90% 1? 00:05:09.270 --> 00:05:13.710 align:middle line:90% So that's one possibility. 00:05:13.710 --> 00:05:20.320 align:middle line:84% So there has to be a lot of S near this proposed cluster 00:05:20.320 --> 00:05:21.160 align:middle line:90% point. 00:05:21.160 --> 00:05:23.090 align:middle line:84% So let me mark out the points that we have. 00:05:23.090 --> 00:05:27.040 align:middle line:84% There's a-- I'll put 0, and then 1/2 00:05:27.040 --> 00:05:34.670 align:middle line:84% is there, and then 1/3, 1/4, 1/5, 1/6, so on, and so on. 00:05:34.670 --> 00:05:37.470 align:middle line:90% And so I see a guess of 0. 00:05:37.470 --> 00:05:40.860 align:middle line:90% So that's a good guess as well. 00:05:40.860 --> 00:05:43.710 align:middle line:84% So why would 1 maybe not be a cluster point? 00:05:43.710 --> 00:05:49.190 align:middle line:84% Well, it's because if I draw a little interval around 1, 00:05:49.190 --> 00:05:52.460 align:middle line:84% then there's only one point in S in here, 00:05:52.460 --> 00:05:55.820 align:middle line:84% and it's the proposed cluster point, 1, OK? 00:05:55.820 --> 00:05:57.770 align:middle line:84% So remember, what we should be able to do 00:05:57.770 --> 00:06:02.320 align:middle line:84% is find in every small interval [? sum ?] of S 00:06:02.320 --> 00:06:05.390 align:middle line:84% in the interval that's not equal to the point, OK? 00:06:05.390 --> 00:06:08.168 align:middle line:84% And we can't do that for the point 1. 00:06:08.168 --> 00:06:13.430 align:middle line:84% But for 0, if we draw any interval around 0, 00:06:13.430 --> 00:06:19.590 align:middle line:84% there's plenty of S in the interval not equal to 0, OK? 00:06:19.590 --> 00:06:31.220 align:middle line:84% So here, 0 is a cluster point of S. And let's 00:06:31.220 --> 00:06:33.170 align:middle line:90% give a quick proof of this. 00:06:33.170 --> 00:06:36.050 align:middle line:90% 00:06:36.050 --> 00:06:41.350 align:middle line:84% So we have to verify for every delta positive this interval, 0 00:06:41.350 --> 00:06:45.040 align:middle line:84% minus delta, 0 plus delta intersect S take away 0 00:06:45.040 --> 00:06:46.030 align:middle line:90% is non-empty. 00:06:46.030 --> 00:06:56.420 align:middle line:84% So the delta would be positive, and now here's the picture. 00:06:56.420 --> 00:06:58.040 align:middle line:90% Here's 0. 00:06:58.040 --> 00:07:02.540 align:middle line:90% Here's delta, minus delta. 00:07:02.540 --> 00:07:04.640 align:middle line:84% We have to show that there's some S in here. 00:07:04.640 --> 00:07:07.690 align:middle line:84% So we simply choose natural number n 00:07:07.690 --> 00:07:11.200 align:middle line:84% large enough so that 1/n is there, 00:07:11.200 --> 00:07:12.940 align:middle line:84% and we can always do this because 00:07:12.940 --> 00:07:16.602 align:middle line:84% of the Archimedean property of the real numbers, right? 00:07:16.602 --> 00:07:19.770 align:middle line:90% 00:07:19.770 --> 00:07:23.860 align:middle line:90% So let delta be positive. 00:07:23.860 --> 00:07:30.930 align:middle line:84% Choose natural number n so that 1/n is less than delta, 00:07:30.930 --> 00:07:32.700 align:middle line:84% and it's certainly bigger than 0. 00:07:32.700 --> 00:07:35.130 align:middle line:90% So then n is a natural number. 00:07:35.130 --> 00:07:37.910 align:middle line:90% 00:07:37.910 --> 00:07:46.330 align:middle line:84% Then 1/n is in x minus delta, x plus delta intersect 00:07:46.330 --> 00:07:48.640 align:middle line:90% S take away 0. 00:07:48.640 --> 00:07:57.770 align:middle line:84% So this should have been 0, which 00:07:57.770 --> 00:08:07.220 align:middle line:84% implies this set is non-empty because it has something in it, 00:08:07.220 --> 00:08:09.806 align:middle line:90% OK? 00:08:09.806 --> 00:08:12.220 align:middle line:90% All right, was that example OK? 00:08:12.220 --> 00:08:14.230 align:middle line:90% Did you all get that all right? 00:08:14.230 --> 00:08:17.670 align:middle line:90% 00:08:17.670 --> 00:08:21.825 align:middle line:84% OK, so feel free to ask questions as they come up. 00:08:21.825 --> 00:08:28.060 align:middle line:90% 00:08:28.060 --> 00:08:29.500 align:middle line:90% So let's do another example. 00:08:29.500 --> 00:08:34.320 align:middle line:84% This one, I will not give a full proof of, 00:08:34.320 --> 00:08:38.409 align:middle line:84% but we'll just kind of talk our way through it. 00:08:38.409 --> 00:08:40.975 align:middle line:84% Let's say S is now the open interval 0, 1. 00:08:40.975 --> 00:08:52.010 align:middle line:90% 00:08:52.010 --> 00:08:57.615 align:middle line:84% What is the set of cluster points of 0, 1 equal to? 00:08:57.615 --> 00:09:01.730 align:middle line:90% 00:09:01.730 --> 00:09:06.060 align:middle line:84% So what would be a reasonable guess or any guess? 00:09:06.060 --> 00:09:14.210 align:middle line:90% 00:09:14.210 --> 00:09:20.730 align:middle line:84% Yeah, so, certainly, this will contain 0, 1, right, since-- 00:09:20.730 --> 00:09:23.400 align:middle line:84% so 0, 1, everything between 0 and 1 00:09:23.400 --> 00:09:25.740 align:middle line:84% should be a cluster point of 0 and 1 00:09:25.740 --> 00:09:28.168 align:middle line:84% because there's always a lot of numbers. 00:09:28.168 --> 00:09:30.210 align:middle line:84% I mean, there's a lot of numbers between 0 and 1. 00:09:30.210 --> 00:09:44.240 align:middle line:84% So if I draw this 0, 1, and [? have ?] everything 00:09:44.240 --> 00:09:47.930 align:middle line:84% in between 0 and 1 is certainly a cluster point. 00:09:47.930 --> 00:09:49.310 align:middle line:90% What about the point 1? 00:09:49.310 --> 00:09:54.530 align:middle line:90% 00:09:54.530 --> 00:09:56.120 align:middle line:84% Is that a cluster point of this set? 00:09:56.120 --> 00:10:11.810 align:middle line:90% 00:10:11.810 --> 00:10:16.100 align:middle line:84% That's a good observation that, yeah, it's a sup of the set. 00:10:16.100 --> 00:10:38.280 align:middle line:84% So for every-- right, so for every delta positive, 00:10:38.280 --> 00:10:40.740 align:middle line:84% if I draw a little interval around 1, 00:10:40.740 --> 00:10:43.590 align:middle line:84% there's going to be a number less than 1 00:10:43.590 --> 00:10:47.670 align:middle line:84% that's bigger than 1 minus delta, right? 00:10:47.670 --> 00:10:51.720 align:middle line:84% The set of cluster points of the open interval 0, 1 00:10:51.720 --> 00:10:54.092 align:middle line:90% is the closed interval 0, 1. 00:10:54.092 --> 00:10:54.592 align:middle line:90% OK? 00:10:54.592 --> 00:10:57.540 align:middle line:90% 00:10:57.540 --> 00:11:03.790 align:middle line:90% Now, let's do another example. 00:11:03.790 --> 00:11:09.255 align:middle line:84% Let's say we look at now, instead of rational numbers-- 00:11:09.255 --> 00:11:12.160 align:middle line:90% 00:11:12.160 --> 00:11:14.340 align:middle line:84% so you should, again, think of cluster points 00:11:14.340 --> 00:11:22.360 align:middle line:84% as being the set of points where there is a lot of this set 00:11:22.360 --> 00:11:26.080 align:middle line:90% nearby, OK? 00:11:26.080 --> 00:11:28.705 align:middle line:84% Now, what is the set of cluster points of the rational numbers? 00:11:28.705 --> 00:11:37.880 align:middle line:90% 00:11:37.880 --> 00:11:42.750 align:middle line:84% OK, so perhaps it will contain the irrational numbers. 00:11:42.750 --> 00:11:45.060 align:middle line:84% So let's think about this for a little bit. 00:11:45.060 --> 00:11:48.020 align:middle line:84% So, for example, let's look at square root of 2, OK? 00:11:48.020 --> 00:11:49.700 align:middle line:90% Perfectly good rational number-- 00:11:49.700 --> 00:11:51.380 align:middle line:90% I mean, irrational number. 00:11:51.380 --> 00:11:57.010 align:middle line:84% If I draw a little interval around square root of 2, 00:11:57.010 --> 00:12:00.310 align:middle line:84% can I find a rational number in there 00:12:00.310 --> 00:12:02.350 align:middle line:84% that's not equal to a square root of 2? 00:12:02.350 --> 00:12:08.110 align:middle line:90% 00:12:08.110 --> 00:12:09.640 align:middle line:84% Right, because we have this theorem 00:12:09.640 --> 00:12:12.880 align:middle line:84% about the density of the rational numbers, right, 00:12:12.880 --> 00:12:13.760 align:middle line:90% that we have-- 00:12:13.760 --> 00:12:19.060 align:middle line:84% so remember that this was a theorem we proved at one point. 00:12:19.060 --> 00:12:24.460 align:middle line:84% For all x, y real numbers with x less than y, 00:12:24.460 --> 00:12:27.820 align:middle line:84% there exists a rational number r such 00:12:27.820 --> 00:12:33.946 align:middle line:84% that x is less than r is less than y, right? 00:12:33.946 --> 00:12:36.990 align:middle line:84% So we can always find a rational number here, 00:12:36.990 --> 00:12:42.180 align:middle line:84% or we could have taken another rational number here, yeah? 00:12:42.180 --> 00:12:45.000 align:middle line:84% So for every interval we draw around square root of 2, 00:12:45.000 --> 00:12:49.350 align:middle line:84% we can find a rational in there not equal to square root of 2, 00:12:49.350 --> 00:12:49.860 align:middle line:90% obviously. 00:12:49.860 --> 00:12:53.720 align:middle line:84% But we can find a rational in that interval, yeah? 00:12:53.720 --> 00:12:56.600 align:middle line:84% So this suggests that the set of cluster points 00:12:56.600 --> 00:12:59.033 align:middle line:90% contains the set of irrationals. 00:12:59.033 --> 00:13:00.200 align:middle line:90% What about rational numbers? 00:13:00.200 --> 00:13:01.940 align:middle line:90% Are those also cluster points? 00:13:01.940 --> 00:13:06.290 align:middle line:90% 00:13:06.290 --> 00:13:08.120 align:middle line:90% Let's say I take 0. 00:13:08.120 --> 00:13:11.390 align:middle line:84% Now I look at a small interval around 0. 00:13:11.390 --> 00:13:13.985 align:middle line:84% Can I find a rational number in there other than 0? 00:13:13.985 --> 00:13:19.560 align:middle line:90% 00:13:19.560 --> 00:13:21.780 align:middle line:84% I'm not asking like, actually give me one. 00:13:21.780 --> 00:13:25.860 align:middle line:84% I'm just saying, does there exist a rational number 00:13:25.860 --> 00:13:29.302 align:middle line:90% in this interval other than 0? 00:13:29.302 --> 00:13:30.080 align:middle line:90% AUDIENCE: Yeah. 00:13:30.080 --> 00:13:31.747 align:middle line:84% CASEY RODRIGUEZ: By this theorem, right? 00:13:31.747 --> 00:13:34.490 align:middle line:90% 00:13:34.490 --> 00:13:38.358 align:middle line:84% You can also use the theorem to find 00:13:38.358 --> 00:13:40.970 align:middle line:84% irrational in this interval as well. 00:13:40.970 --> 00:13:45.700 align:middle line:84% So no matter if it's square root of 2 or any irrational number, 00:13:45.700 --> 00:13:48.310 align:middle line:84% no matter if it's a rational number, 00:13:48.310 --> 00:13:51.200 align:middle line:90% everything is a cluster point. 00:13:51.200 --> 00:13:56.590 align:middle line:84% So the set of cluster points of rational numbers is equal to R, 00:13:56.590 --> 00:13:59.047 align:middle line:90% OK? 00:13:59.047 --> 00:13:59.630 align:middle line:90% Is that clear? 00:13:59.630 --> 00:14:05.550 align:middle line:90% 00:14:05.550 --> 00:14:08.290 align:middle line:84% OK, feel free to stop me if you have any questions. 00:14:08.290 --> 00:14:17.072 align:middle line:84% So now let's look at, for example, 00:14:17.072 --> 00:14:25.520 align:middle line:84% S equal to just a single guy, say, 0, OK? 00:14:25.520 --> 00:14:28.440 align:middle line:84% Remember, cluster points of a set 00:14:28.440 --> 00:14:30.240 align:middle line:84% means there's a lot of that set nearby. 00:14:30.240 --> 00:14:35.240 align:middle line:90% 00:14:35.240 --> 00:14:38.690 align:middle line:84% Now, I claim that this guy has no cluster 00:14:38.690 --> 00:14:45.800 align:middle line:84% points because if you think so intuitively why is this, 00:14:45.800 --> 00:14:48.160 align:middle line:84% this is because, for something to be a cluster point, 00:14:48.160 --> 00:14:52.400 align:middle line:84% it has to have a lot of the set near it, a lot. 00:14:52.400 --> 00:14:54.710 align:middle line:84% And there's not a lot of set to begin with. 00:14:54.710 --> 00:14:56.285 align:middle line:90% There's just one element. 00:14:56.285 --> 00:15:00.710 align:middle line:84% So let's see why this is the case, OK? 00:15:00.710 --> 00:15:02.450 align:middle line:84% And this gives us a chance to negate 00:15:02.450 --> 00:15:06.920 align:middle line:84% the definition of cluster point, which is always a good idea. 00:15:06.920 --> 00:15:14.030 align:middle line:84% OK, so S equals just the singleton set, 0. 00:15:14.030 --> 00:15:16.220 align:middle line:84% This has no cluster points, and so the negation 00:15:16.220 --> 00:15:17.850 align:middle line:84% of the definition of a cluster point-- 00:15:17.850 --> 00:15:32.590 align:middle line:84% so x is not a cluster point of S means if this is some bad 00:15:32.590 --> 00:15:33.100 align:middle line:90% delta-- 00:15:33.100 --> 00:15:36.100 align:middle line:84% remember, so the definition of cluster point is for all delta. 00:15:36.100 --> 00:15:40.060 align:middle line:84% So negation means there exists one bad delta 00:15:40.060 --> 00:15:45.430 align:middle line:84% so that x minus delta 0, x plus delta 0 00:15:45.430 --> 00:15:55.380 align:middle line:84% intersect S take away x equals the empty set, OK? 00:15:55.380 --> 00:15:57.570 align:middle line:84% So the picture that goes along with this 00:15:57.570 --> 00:16:04.850 align:middle line:84% is that x does not cluster point if there's 00:16:04.850 --> 00:16:08.540 align:middle line:84% some interval and possibly the only S there 00:16:08.540 --> 00:16:12.640 align:middle line:84% is the point x, but nothing else, OK? 00:16:12.640 --> 00:16:17.140 align:middle line:90% 00:16:17.140 --> 00:16:21.090 align:middle line:84% All right, so let's use the negation of this definition 00:16:21.090 --> 00:16:25.195 align:middle line:84% to show that there are no cluster points, OK? 00:16:25.195 --> 00:16:28.050 align:middle line:90% 00:16:28.050 --> 00:16:31.110 align:middle line:84% And we have to deal with two cases. 00:16:31.110 --> 00:16:50.980 align:middle line:84% I want to show x is in R, and x is not a cluster point of S. 00:16:50.980 --> 00:16:54.250 align:middle line:84% And I'll just do the case x not equal to 0. 00:16:54.250 --> 00:16:59.500 align:middle line:84% x equal to 0 is much easier, so we'll just do x not equal to 0. 00:16:59.500 --> 00:17:05.500 align:middle line:90% 00:17:05.500 --> 00:17:07.329 align:middle line:90% Let's prove this. 00:17:07.329 --> 00:17:09.849 align:middle line:90% And what's the idea? 00:17:09.849 --> 00:17:11.300 align:middle line:90% We have to be able to find-- 00:17:11.300 --> 00:17:13.030 align:middle line:90% so here's 0. 00:17:13.030 --> 00:17:16.540 align:middle line:84% This is all that S is, is just this point, 0. 00:17:16.540 --> 00:17:19.750 align:middle line:90% Now we have a point x. 00:17:19.750 --> 00:17:23.579 align:middle line:84% What would be an interval containing 00:17:23.579 --> 00:17:25.365 align:middle line:90% x that contains none of S? 00:17:25.365 --> 00:17:35.350 align:middle line:90% 00:17:35.350 --> 00:17:39.500 align:middle line:84% So x here is just some nonzero number. 00:17:39.500 --> 00:17:43.330 align:middle line:90% S is what's highlighted. 00:17:43.330 --> 00:17:56.810 align:middle line:84% The interval around x that doesn't contain any S. So 00:17:56.810 --> 00:17:57.370 align:middle line:90% like a x/2? 00:17:57.370 --> 00:18:00.440 align:middle line:90% 00:18:00.440 --> 00:18:07.810 align:middle line:84% 3x over 2, meaning delta equals x/2, at least for this picture, 00:18:07.810 --> 00:18:09.970 align:middle line:90% yeah? 00:18:09.970 --> 00:18:11.990 align:middle line:84% OK, and that's essentially the proof. 00:18:11.990 --> 00:18:20.410 align:middle line:84% So choose delta 0, and we'll simplify this further 00:18:20.410 --> 00:18:22.000 align:middle line:90% and suppose x is positive. 00:18:22.000 --> 00:18:24.010 align:middle line:84% For x less than 0, you choose delta 0 00:18:24.010 --> 00:18:26.290 align:middle line:90% to be the absolute value of x. 00:18:26.290 --> 00:18:28.420 align:middle line:84% But for this one, this is simple enough-- 00:18:28.420 --> 00:18:39.530 align:middle line:84% choose delta 0 to be x/2 and x minus delta 0, x plus delta 0 00:18:39.530 --> 00:18:43.260 align:middle line:90% is equal to x/2, 3x over 2. 00:18:43.260 --> 00:18:49.460 align:middle line:90% 00:18:49.460 --> 00:18:57.585 align:middle line:84% And since 0 is not in this set, in this set, 00:18:57.585 --> 00:19:14.540 align:middle line:84% [? implies ?] intersect S take away x equals empty, OK? 00:19:14.540 --> 00:19:18.620 align:middle line:84% So we had to find a delta so that this interval around x 00:19:18.620 --> 00:19:21.230 align:middle line:84% doesn't contain anything from S, and we've done that 00:19:21.230 --> 00:19:22.820 align:middle line:90% by choosing delta 0 to be x/2. 00:19:22.820 --> 00:19:31.630 align:middle line:90% 00:19:31.630 --> 00:19:32.130 align:middle line:90% OK. 00:19:32.130 --> 00:19:36.500 align:middle line:90% 00:19:36.500 --> 00:19:39.830 align:middle line:84% So I could have done it for just-- 00:19:39.830 --> 00:19:45.450 align:middle line:84% so I did it for a set containing just one element, 00:19:45.450 --> 00:19:48.360 align:middle line:84% showed that this set has no cluster points. 00:19:48.360 --> 00:19:52.050 align:middle line:84% You can also show that if it has finitely many points, 00:19:52.050 --> 00:19:55.560 align:middle line:90% it has no cluster points. 00:19:55.560 --> 00:20:00.200 align:middle line:84% And so therefore, any finite set has no cluster points, 00:20:00.200 --> 00:20:03.220 align:middle line:84% but being a cluster point has nothing 00:20:03.220 --> 00:20:04.420 align:middle line:90% to do with the cardinality. 00:20:04.420 --> 00:20:07.120 align:middle line:84% So if you see the example in the notes, which 00:20:07.120 --> 00:20:09.710 align:middle line:84% you can think about, and I'm not going to go over right here, 00:20:09.710 --> 00:20:14.650 align:middle line:84% but, for example, x equals z, the set of integers-- 00:20:14.650 --> 00:20:23.990 align:middle line:90% 00:20:23.990 --> 00:20:30.120 align:middle line:90% this has no cluster points, OK? 00:20:30.120 --> 00:20:35.890 align:middle line:84% So being a cluster point means, where is this set clustering? 00:20:35.890 --> 00:20:38.490 align:middle line:84% So it has nothing necessarily to do with the size of the set 00:20:38.490 --> 00:20:45.980 align:middle line:84% but where the set is taking up space in the real number line. 00:20:45.980 --> 00:20:55.990 align:middle line:84% All right, so with the definition of cluster points, 00:20:55.990 --> 00:20:59.020 align:middle line:84% let me first state the theorem that, in fact, you 00:20:59.020 --> 00:21:08.355 align:middle line:84% proved in the assignment, which is that for cluster points-- 00:21:08.355 --> 00:21:11.100 align:middle line:90% 00:21:11.100 --> 00:21:22.810 align:middle line:84% let S be a subset of R, and x is a cluster point of S 00:21:22.810 --> 00:21:30.880 align:middle line:84% if and only if there exist sequence 00:21:30.880 --> 00:21:43.600 align:middle line:84% xn of the elements of and S take away x such that xn converges 00:21:43.600 --> 00:21:46.288 align:middle line:90% to x, OK? 00:21:46.288 --> 00:21:47.580 align:middle line:90% And that was in the assignment. 00:21:47.580 --> 00:21:50.490 align:middle line:84% You proved that basically using the squeeze theorem 00:21:50.490 --> 00:21:53.720 align:middle line:90% in the definition. 00:21:53.720 --> 00:21:56.660 align:middle line:84% All right, so with the notion of a cluster point, 00:21:56.660 --> 00:21:59.360 align:middle line:84% these are the points in a set where 00:21:59.360 --> 00:22:02.420 align:middle line:84% we're going to start talking about how a function behaves 00:22:02.420 --> 00:22:09.700 align:middle line:84% near a point, which is the notion of a limit of a function 00:22:09.700 --> 00:22:11.550 align:middle line:90% is our next definition. 00:22:11.550 --> 00:22:14.232 align:middle line:90% 00:22:14.232 --> 00:22:19.400 align:middle line:84% Let S be a subset of R. And even though I was referring 00:22:19.400 --> 00:22:21.320 align:middle line:84% to cluster points earlier as x, we're 00:22:21.320 --> 00:22:24.830 align:middle line:90% now going to switch over to c. 00:22:24.830 --> 00:22:38.050 align:middle line:84% Let c be cluster point of S. And let f be a function from S 00:22:38.050 --> 00:22:41.900 align:middle line:90% to R, all right? 00:22:41.900 --> 00:22:55.880 align:middle line:84% So you say f of x converges to L as x goes to c. 00:22:55.880 --> 00:23:00.890 align:middle line:84% So this is the new terminology here. 00:23:00.890 --> 00:23:05.700 align:middle line:90% 00:23:05.700 --> 00:23:10.100 align:middle line:84% So here, L is a real number if, for every epsilon positive, 00:23:10.100 --> 00:23:22.485 align:middle line:84% there exists delta positive such that if x is in S, 00:23:22.485 --> 00:23:27.360 align:middle line:84% the absolute value of x minus c is bigger than 0 less 00:23:27.360 --> 00:23:34.170 align:middle line:84% than delta, then f of x minus L is less than epsilon. 00:23:34.170 --> 00:23:41.680 align:middle line:84% So let's try and compare this definition a little bit 00:23:41.680 --> 00:23:43.910 align:middle line:90% to what happens for a sequence. 00:23:43.910 --> 00:23:49.270 align:middle line:84% So for a sequence, the sequence is getting close to L 00:23:49.270 --> 00:23:52.390 align:middle line:84% as long as we go far enough out in the sequence, 00:23:52.390 --> 00:23:54.820 align:middle line:84% as long as we look at terms far enough out. 00:23:54.820 --> 00:23:58.540 align:middle line:84% Now, for limits of functions, that going far enough 00:23:58.540 --> 00:24:01.030 align:middle line:84% out along the sequence is replaced by, 00:24:01.030 --> 00:24:03.580 align:middle line:84% as long as we look close enough to the point c 00:24:03.580 --> 00:24:05.170 align:middle line:90% but not at the point c. 00:24:05.170 --> 00:24:16.630 align:middle line:84% So if you'd like, this says that if x is near c-- 00:24:16.630 --> 00:24:19.510 align:middle line:84% and so this is just an intuitive interpretation 00:24:19.510 --> 00:24:21.760 align:middle line:90% of what this definition says. 00:24:21.760 --> 00:24:32.360 align:middle line:84% If x is near c, then f of x is near L. And let me also throw 00:24:32.360 --> 00:24:40.495 align:middle line:84% in here that we're saying that x is near c but not equal to c. 00:24:40.495 --> 00:24:45.230 align:middle line:84% But for a limit, you don't look at what happens at f of c. 00:24:45.230 --> 00:24:49.160 align:middle line:84% The function doesn't even need to be defined there to be 00:24:49.160 --> 00:24:50.750 align:middle line:90% able to define what a limit is. 00:24:50.750 --> 00:24:54.260 align:middle line:84% We just care about what happens nearby, OK? 00:24:54.260 --> 00:24:58.640 align:middle line:90% So you could have-- 00:24:58.640 --> 00:25:02.000 align:middle line:84% the picture you should have in mind is that here's c, 00:25:02.000 --> 00:25:07.930 align:middle line:84% and maybe f is not even defined up to there, 00:25:07.930 --> 00:25:11.455 align:middle line:84% but there is f, the graph of the function f there. 00:25:11.455 --> 00:25:17.340 align:middle line:90% 00:25:17.340 --> 00:25:25.470 align:middle line:84% There's L, and as long as I get very close to c, 00:25:25.470 --> 00:25:30.820 align:middle line:84% I'll be very close to L, "very close" 00:25:30.820 --> 00:25:40.243 align:middle line:84% being measured within some strip here, OK? 00:25:40.243 --> 00:25:45.720 align:middle line:90% 00:25:45.720 --> 00:25:52.440 align:middle line:84% OK, so this is the notion of the limit of a function 00:25:52.440 --> 00:25:56.310 align:middle line:84% as we approach a cluster point of the set S where the function 00:25:56.310 --> 00:25:58.740 align:middle line:90% f is defined. 00:25:58.740 --> 00:26:00.330 align:middle line:90% Just a little bit of notation-- 00:26:00.330 --> 00:26:04.150 align:middle line:90% 00:26:04.150 --> 00:26:07.510 align:middle line:84% typically write like we did with sequences 00:26:07.510 --> 00:26:12.670 align:middle line:84% f of x converges to L as f of x arrow L. And then I'll write, 00:26:12.670 --> 00:26:14.380 align:middle line:90% as x goes to c. 00:26:14.380 --> 00:26:17.480 align:middle line:84% Maybe I won't write as x goes to c. 00:26:17.480 --> 00:26:22.970 align:middle line:84% We note this by limit of f of x equals L as well. 00:26:22.970 --> 00:26:27.180 align:middle line:90% 00:26:27.180 --> 00:26:32.910 align:middle line:84% OK, so first question about limits is, are they unique? 00:26:32.910 --> 00:26:38.610 align:middle line:84% And this is, in fact, why we require 00:26:38.610 --> 00:26:43.050 align:middle line:84% that c be a cluster point of S because if we didn't require 00:26:43.050 --> 00:26:45.240 align:middle line:84% c to be a cluster point of S in this definition, 00:26:45.240 --> 00:26:46.587 align:middle line:90% limits would not be unique. 00:26:46.587 --> 00:26:48.420 align:middle line:84% In fact, you could have functions converging 00:26:48.420 --> 00:26:52.170 align:middle line:84% to whatever you want because the definition would then 00:26:52.170 --> 00:26:52.920 align:middle line:90% be vacuous. 00:26:52.920 --> 00:26:57.400 align:middle line:84% But if you don't want to think about that, that's fine. 00:26:57.400 --> 00:26:58.300 align:middle line:90% It's not a big issue. 00:26:58.300 --> 00:27:00.670 align:middle line:84% But I'm just going to point out at one point, 00:27:00.670 --> 00:27:03.060 align:middle line:84% we're going to use the fact that c is a cluster point 00:27:03.060 --> 00:27:05.770 align:middle line:90% to prove the following theorem. 00:27:05.770 --> 00:27:26.240 align:middle line:84% So let c be a cluster point of S, which is a subset of R. 00:27:26.240 --> 00:27:30.652 align:middle line:84% And let f be a function from S to R. 00:27:30.652 --> 00:27:38.230 align:middle line:84% If f of x converges to L1 as x goes to c, 00:27:38.230 --> 00:27:48.630 align:middle line:84% and f of x converges to L2 as x goes to c, then L1 equals L2, 00:27:48.630 --> 00:27:49.130 align:middle line:90% OK? 00:27:49.130 --> 00:27:52.580 align:middle line:84% So this is what I mean by uniqueness, 00:27:52.580 --> 00:27:56.210 align:middle line:84% that a function can only have one limit as it approaches 00:27:56.210 --> 00:27:57.050 align:middle line:90% a point c. 00:27:57.050 --> 00:27:59.940 align:middle line:90% 00:27:59.940 --> 00:28:02.816 align:middle line:90% Let me give the proof. 00:28:02.816 --> 00:28:07.720 align:middle line:84% So we're going to play this game again, 00:28:07.720 --> 00:28:10.485 align:middle line:84% where instead of showing something as 0 or something is 00:28:10.485 --> 00:28:11.860 align:middle line:84% equal to something else directly, 00:28:11.860 --> 00:28:15.300 align:middle line:84% we'll give ourselves a little room and prove the following. 00:28:15.300 --> 00:28:20.850 align:middle line:84% We'll instead show that while epsilon positive, 00:28:20.850 --> 00:28:24.310 align:middle line:84% L1 minus L2 is less than epsilon. 00:28:24.310 --> 00:28:29.040 align:middle line:84% So L1 and L2, these are just two fixed real numbers. 00:28:29.040 --> 00:28:30.990 align:middle line:84% If their absolute value is less than epsilon 00:28:30.990 --> 00:28:34.020 align:middle line:84% for arbitrary epsilon, then this is 00:28:34.020 --> 00:28:38.040 align:middle line:84% a number which is smaller than any positive real number. 00:28:38.040 --> 00:28:41.730 align:middle line:84% That implies that that number is 0, so L1 equals L2, OK? 00:28:41.730 --> 00:28:44.880 align:middle line:90% 00:28:44.880 --> 00:28:46.650 align:middle line:84% Why we give ourselves a little room 00:28:46.650 --> 00:28:49.920 align:middle line:84% is because, in the definition, we have a small parameter there 00:28:49.920 --> 00:28:51.426 align:middle line:90% which we hope to use. 00:28:51.426 --> 00:28:56.730 align:middle line:90% So let's prove this. 00:28:56.730 --> 00:28:58.540 align:middle line:90% Let epsilon be positive. 00:28:58.540 --> 00:29:01.720 align:middle line:90% 00:29:01.720 --> 00:29:10.890 align:middle line:84% So since f of x converges to L1, there exists-- 00:29:10.890 --> 00:29:14.830 align:middle line:84% so just from the definition, given some small tolerance, 00:29:14.830 --> 00:29:23.410 align:middle line:84% I can find a little number delta so 00:29:23.410 --> 00:29:25.240 align:middle line:84% that if I look inside this interval, 00:29:25.240 --> 00:29:30.260 align:middle line:84% I get mapped to the interval L plus epsilon, L minus epsilon. 00:29:30.260 --> 00:29:36.450 align:middle line:84% So there exists a delta 1 positive 00:29:36.450 --> 00:29:43.253 align:middle line:84% such that if x is in S-- and I'm going to probably keep 00:29:43.253 --> 00:29:44.670 align:middle line:84% forgetting to write that, but that 00:29:44.670 --> 00:29:47.640 align:middle line:84% should be understood that x is in the domain of the function 00:29:47.640 --> 00:29:48.600 align:middle line:90% f. 00:29:48.600 --> 00:29:54.660 align:middle line:84% If x minus c is less than delta 1 and bigger than 0, 00:29:54.660 --> 00:30:07.950 align:middle line:84% then f of x minus L1 is less than epsilon/2, OK? 00:30:07.950 --> 00:30:10.500 align:middle line:90% And the same thing for L2. 00:30:10.500 --> 00:30:13.560 align:middle line:84% I mean, if f of x is converging to L2, 00:30:13.560 --> 00:30:17.560 align:middle line:84% then there exists a delta 2 so that I have the same statement. 00:30:17.560 --> 00:30:30.530 align:middle line:84% So instead, I'm just going to change this 1 to i, i, i, i. 00:30:30.530 --> 00:30:32.940 align:middle line:90% And here, i equals 1 too. 00:30:32.940 --> 00:30:37.340 align:middle line:84% So since f of x converges to each of these two numbers, L1 00:30:37.340 --> 00:30:40.430 align:middle line:84% and L2, there exists a delta 1 and a delta 00:30:40.430 --> 00:30:44.480 align:middle line:84% 2 so that if I'm within this tolerance delta of 1 00:30:44.480 --> 00:30:47.840 align:middle line:84% or delta 2, then f of x minus delta 1-- 00:30:47.840 --> 00:30:50.270 align:middle line:90% L1 or L2 is less than epsilon/2. 00:30:50.270 --> 00:30:55.030 align:middle line:90% 00:30:55.030 --> 00:30:57.985 align:middle line:84% Let delta be the minimum of these two numbers. 00:30:57.985 --> 00:31:03.010 align:middle line:90% 00:31:03.010 --> 00:31:13.390 align:middle line:84% Now, since c is a cluster point of S, in fact, 00:31:13.390 --> 00:31:21.130 align:middle line:84% there does exist an x in S such that 0 is bigger 00:31:21.130 --> 00:31:26.330 align:middle line:84% than x minus c is less than delta, all right? 00:31:26.330 --> 00:31:29.015 align:middle line:84% And I'm going to use this x to compare L1 and L2. 00:31:29.015 --> 00:31:31.880 align:middle line:90% 00:31:31.880 --> 00:31:37.810 align:middle line:84% Then since x is in this interval, or I should say, 00:31:37.810 --> 00:31:38.560 align:middle line:90% since it's-- 00:31:38.560 --> 00:31:41.969 align:middle line:90% 00:31:41.969 --> 00:31:45.090 align:middle line:84% so remember I had this, which means 00:31:45.090 --> 00:31:51.910 align:middle line:84% I have this inequality or i equals 1, 2. 00:31:51.910 --> 00:31:58.540 align:middle line:84% I get f of x, or I should say, L1 minus L2. 00:31:58.540 --> 00:32:10.510 align:middle line:84% This is equal to if I add and subtract f of x 00:32:10.510 --> 00:32:19.370 align:middle line:84% and now use the triangle inequality 00:32:19.370 --> 00:32:23.870 align:middle line:84% and use the inequality that's in green, the f of x minus L1 00:32:23.870 --> 00:32:26.330 align:middle line:90% is less than epsilon/2. 00:32:26.330 --> 00:32:28.220 align:middle line:90% And same for L2, I get-- 00:32:28.220 --> 00:32:33.110 align:middle line:90% 00:32:33.110 --> 00:32:34.770 align:middle line:90% equals epsilon, OK? 00:32:34.770 --> 00:32:36.680 align:middle line:84% And that's the end because remember 00:32:36.680 --> 00:32:42.995 align:middle line:84% we started off with the absolute value of L1 minus L2. 00:32:42.995 --> 00:32:44.555 align:middle line:84% Are there any questions about that? 00:32:44.555 --> 00:32:58.642 align:middle line:90% 00:32:58.642 --> 00:33:04.420 align:middle line:84% OK, so not the most exciting theorem 00:33:04.420 --> 00:33:08.530 align:middle line:84% in the world that this thing that we've defined. 00:33:08.530 --> 00:33:16.010 align:middle line:84% A limit is a unique thing for function. 00:33:16.010 --> 00:33:18.055 align:middle line:84% But let's start looking at some examples. 00:33:18.055 --> 00:33:25.160 align:middle line:90% 00:33:25.160 --> 00:33:27.350 align:middle line:84% And at one point, we'll look at the negation 00:33:27.350 --> 00:33:29.250 align:middle line:90% of this definition. 00:33:29.250 --> 00:33:33.890 align:middle line:84% So let's suppose-- so let's look at the limit 00:33:33.890 --> 00:33:40.060 align:middle line:84% as x goes to c of the function ax plus b. 00:33:40.060 --> 00:33:45.460 align:middle line:84% And I claim this is equal to ac plus b, OK? 00:33:45.460 --> 00:33:48.820 align:middle line:84% So here, S is, although I didn't write it out, 00:33:48.820 --> 00:33:51.430 align:middle line:84% this function ax plus b, this is-- 00:33:51.430 --> 00:33:57.370 align:middle line:84% I'm looking at it defined on all of R, 00:33:57.370 --> 00:34:02.570 align:middle line:90% and c is a real number, OK? 00:34:02.570 --> 00:34:05.170 align:middle line:84% So we have to verify this definition. 00:34:05.170 --> 00:34:11.670 align:middle line:90% 00:34:11.670 --> 00:34:14.110 align:middle line:84% And since it's for every epsilon, 00:34:14.110 --> 00:34:15.203 align:middle line:90% there exists something. 00:34:15.203 --> 00:34:16.620 align:middle line:84% The beginning of every proof, I'll 00:34:16.620 --> 00:34:18.370 align:middle line:84% give you a few points if you can just say, 00:34:18.370 --> 00:34:21.389 align:middle line:90% let epsilon be positive. 00:34:21.389 --> 00:34:22.830 align:middle line:90% How will we choose delta? 00:34:22.830 --> 00:34:27.600 align:middle line:84% Well, you do similar computations 00:34:27.600 --> 00:34:30.540 align:middle line:84% as you would if you were doing epsilon m 00:34:30.540 --> 00:34:34.000 align:middle line:90% arguments for sequences. 00:34:34.000 --> 00:34:36.840 align:middle line:90% So what are after? 00:34:36.840 --> 00:34:45.659 align:middle line:84% We want to find delta so that x minus c less than delta 00:34:45.659 --> 00:34:51.389 align:middle line:84% and also bigger than 0 implies f of x, which is ax plus b, 00:34:51.389 --> 00:34:57.360 align:middle line:84% minus ac plus b is less than epsilon. 00:34:57.360 --> 00:35:03.690 align:middle line:84% So let's start off with this thing, play with it, 00:35:03.690 --> 00:35:07.120 align:middle line:84% and see if we can find how to choose delta. 00:35:07.120 --> 00:35:09.220 align:middle line:84% Remember, we don't solve this inequality. 00:35:09.220 --> 00:35:11.920 align:middle line:84% We want to find how to choose delta 00:35:11.920 --> 00:35:15.580 align:middle line:84% based on estimating this thing and trying 00:35:15.580 --> 00:35:16.880 align:middle line:90% to make it less than epsilon. 00:35:16.880 --> 00:35:26.030 align:middle line:84% So f of x minus ac plus b, absolute value-- 00:35:26.030 --> 00:35:27.640 align:middle line:84% this is equal to if you just plug 00:35:27.640 --> 00:35:33.960 align:middle line:84% in f of x equals ax plus b, this equals a times x 00:35:33.960 --> 00:35:41.600 align:middle line:84% minus c, which equals absolute value of a times x minus c. 00:35:41.600 --> 00:35:43.850 align:middle line:84% And now we're in business because the thing 00:35:43.850 --> 00:35:47.660 align:middle line:84% that we do have control over is the delta, right? 00:35:47.660 --> 00:35:50.060 align:middle line:90% It is how big x minus c is. 00:35:50.060 --> 00:35:53.840 align:middle line:84% So this is less than absolute value of a times delta 00:35:53.840 --> 00:35:57.720 align:middle line:90% if x minus c is less than delta. 00:35:57.720 --> 00:36:02.340 align:middle line:84% And remember, the thing that we want in the end 00:36:02.340 --> 00:36:05.700 align:middle line:84% is the thing in yellow, and the thing we get to control 00:36:05.700 --> 00:36:06.825 align:middle line:90% is the thing in green. 00:36:06.825 --> 00:36:11.880 align:middle line:90% 00:36:11.880 --> 00:36:14.810 align:middle line:84% So if I want the thing in yellow, 00:36:14.810 --> 00:36:18.905 align:middle line:84% and I have the thing in green, how do I choose delta? 00:36:18.905 --> 00:36:36.300 align:middle line:90% 00:36:36.300 --> 00:36:39.180 align:middle line:84% So in the end, I want f of x minus ac plus b 00:36:39.180 --> 00:36:40.470 align:middle line:90% to be less than epsilon. 00:36:40.470 --> 00:36:44.980 align:middle line:84% Right now, I'm at the absolute value of a times delta. 00:36:44.980 --> 00:36:49.120 align:middle line:84% Delta is the thing I get to play with. 00:36:49.120 --> 00:36:50.230 align:middle line:90% So how do I choose it? 00:36:50.230 --> 00:36:53.710 align:middle line:90% 00:36:53.710 --> 00:36:55.465 align:middle line:90% Epsilon to be absolute value-- 00:36:55.465 --> 00:37:01.190 align:middle line:90% 00:37:01.190 --> 00:37:01.690 align:middle line:90% right? 00:37:01.690 --> 00:37:04.780 align:middle line:84% Because if I choose delta this way, then x 00:37:04.780 --> 00:37:10.540 align:middle line:84% minus c less than delta implies that, just by this computation 00:37:10.540 --> 00:37:21.860 align:middle line:84% here, I would get epsilon in the end, yeah? 00:37:21.860 --> 00:37:25.520 align:middle line:84% So that's kind of the scratch work that goes into it. 00:37:25.520 --> 00:37:27.660 align:middle line:84% Again, delta's the thing you get to choose. 00:37:27.660 --> 00:37:32.210 align:middle line:84% You get to choose based on epsilon. 00:37:32.210 --> 00:37:36.950 align:middle line:84% And so you look f of x minus ac plus b. 00:37:36.950 --> 00:37:41.570 align:middle line:84% You estimate it to be no bigger than the absolute value 00:37:41.570 --> 00:37:44.660 align:middle line:84% of a times x minus c, which is less than the absolute value 00:37:44.660 --> 00:37:46.140 align:middle line:90% of a times delta. 00:37:46.140 --> 00:37:49.130 align:middle line:84% And so if you want that thing to be less than epsilon, 00:37:49.130 --> 00:37:51.350 align:middle line:84% you should choose, for example, delta 00:37:51.350 --> 00:37:53.360 align:middle line:84% to be epsilon over the absolute value of a. 00:37:53.360 --> 00:37:54.620 align:middle line:90% There's not a unique choice. 00:37:54.620 --> 00:38:00.580 align:middle line:84% I could have chosen delta to be epsilon/2 absolute value of a, 00:38:00.580 --> 00:38:04.110 align:middle line:90% but this is a choice, OK? 00:38:04.110 --> 00:38:08.740 align:middle line:90% So let epsilon be positive. 00:38:08.740 --> 00:38:15.023 align:middle line:84% Choose delta to be epsilon over absolute value of a. 00:38:15.023 --> 00:38:17.801 align:middle line:90% 00:38:17.801 --> 00:38:24.410 align:middle line:84% If x minus c is less than delta, then-- 00:38:24.410 --> 00:38:27.330 align:middle line:90% 00:38:27.330 --> 00:38:29.070 align:middle line:84% and when you write the proof, it's 00:38:29.070 --> 00:38:32.640 align:middle line:84% essentially a rehashing of the best parts of your computation. 00:38:32.640 --> 00:38:36.930 align:middle line:84% And f of x minus a times c plus b, 00:38:36.930 --> 00:38:44.110 align:middle line:84% this is equal to absolute value of a times x minus c. 00:38:44.110 --> 00:38:47.860 align:middle line:84% So if we take f of x and subtract our proposed limit 00:38:47.860 --> 00:38:50.470 align:middle line:84% ac plus b, we get the absolute value of a times 00:38:50.470 --> 00:38:52.600 align:middle line:84% absolute value of x minus c, which 00:38:52.600 --> 00:38:56.110 align:middle line:84% is less than the absolute value of a times delta, which 00:38:56.110 --> 00:39:09.850 align:middle line:84% equals epsilon since we chose delta to be epsilon/a, OK? 00:39:09.850 --> 00:39:12.420 align:middle line:90% That's the proof. 00:39:12.420 --> 00:39:13.620 align:middle line:90% Are there any questions? 00:39:13.620 --> 00:39:33.780 align:middle line:90% 00:39:33.780 --> 00:39:36.570 align:middle line:90% OK, so let's do another example. 00:39:36.570 --> 00:39:43.440 align:middle line:90% 00:39:43.440 --> 00:39:45.140 align:middle line:84% Let's look at the limit as x goes 00:39:45.140 --> 00:39:49.040 align:middle line:84% to c, a square of the function square root of x. 00:39:49.040 --> 00:39:53.180 align:middle line:84% And I'm going to show this is equal to square root of c. 00:39:53.180 --> 00:39:55.460 align:middle line:84% And here, although I'm not writing it, 00:39:55.460 --> 00:39:59.870 align:middle line:84% S will be, for us, the domain of the square root, which 00:39:59.870 --> 00:40:03.200 align:middle line:84% is the closed interval 0, infinity, not including 00:40:03.200 --> 00:40:06.460 align:middle line:90% infinity, of course. 00:40:06.460 --> 00:40:09.070 align:middle line:84% And for what I'm going to do, c is positive, OK? 00:40:09.070 --> 00:40:12.610 align:middle line:84% You can also do c equal to 0, but for this proof, 00:40:12.610 --> 00:40:13.930 align:middle line:90% I'm going to do c positive. 00:40:13.930 --> 00:40:18.520 align:middle line:90% 00:40:18.520 --> 00:40:25.000 align:middle line:84% All right, so I have to verify this definition, which 00:40:25.000 --> 00:40:28.210 align:middle line:84% means for every epsilon, I can find a delta, 00:40:28.210 --> 00:40:29.720 align:middle line:90% so let epsilon be positive. 00:40:29.720 --> 00:40:37.990 align:middle line:84% Let's go over to our box of scribbling and scratching. 00:40:37.990 --> 00:40:41.290 align:middle line:84% So remember, our goal is to find a delta which 00:40:41.290 --> 00:40:45.460 align:middle line:84% I can choose so that f of x minus square root of c, 00:40:45.460 --> 00:40:47.450 align:middle line:84% in this case, is less than epsilon. 00:40:47.450 --> 00:40:52.620 align:middle line:84% So if I look at f of x minus square root of c, 00:40:52.620 --> 00:40:58.220 align:middle line:84% this is square root of x minus square root of c. 00:40:58.220 --> 00:41:01.610 align:middle line:84% And perhaps you remember from your days in calculus 00:41:01.610 --> 00:41:04.280 align:middle line:84% that whenever you see the difference of two square roots, 00:41:04.280 --> 00:41:10.700 align:middle line:84% it's a good idea maybe to multiply the top and bottom, 00:41:10.700 --> 00:41:13.920 align:middle line:84% or, I guess, multiply by 1 in a very special way, 00:41:13.920 --> 00:41:15.840 align:middle line:84% so that I get the difference of squares. 00:41:15.840 --> 00:41:17.630 align:middle line:84% So this is equal to the square root 00:41:17.630 --> 00:41:22.790 align:middle line:84% of x minus square root of c times square root of x 00:41:22.790 --> 00:41:27.850 align:middle line:84% plus square root of c all over square root of x 00:41:27.850 --> 00:41:33.920 align:middle line:90% plus square root of c, OK? 00:41:33.920 --> 00:41:36.530 align:middle line:90% And so now I have-- 00:41:36.530 --> 00:41:41.660 align:middle line:84% on the top, I have product a minus b times a plus b. 00:41:41.660 --> 00:41:44.210 align:middle line:84% So that's the difference of squares, which gives me 00:41:44.210 --> 00:41:48.230 align:middle line:84% x minus c on top over square root of x 00:41:48.230 --> 00:41:50.546 align:middle line:90% plus square root of c, OK? 00:41:50.546 --> 00:41:53.270 align:middle line:90% 00:41:53.270 --> 00:41:57.560 align:middle line:84% And remember, this is less than the thing 00:41:57.560 --> 00:41:59.360 align:middle line:84% that we're going to choose in the end. 00:41:59.360 --> 00:42:02.640 align:middle line:84% The absolute value of x minus c is less than delta, 00:42:02.640 --> 00:42:06.800 align:middle line:84% so this is less than square root of x plus square root of c. 00:42:06.800 --> 00:42:08.900 align:middle line:84% Now, in the end, you have to only-- 00:42:08.900 --> 00:42:11.510 align:middle line:84% you can choose delta depending only on epsilon 00:42:11.510 --> 00:42:14.150 align:middle line:90% and maybe the point c, OK? 00:42:14.150 --> 00:42:18.630 align:middle line:84% You cannot choose it to be dependent on the point x, 00:42:18.630 --> 00:42:20.720 align:middle line:90% which is changing, right? 00:42:20.720 --> 00:42:22.850 align:middle line:84% What maybe you're tempted to do here 00:42:22.850 --> 00:42:27.710 align:middle line:84% is to then choose delta to be square root of x 00:42:27.710 --> 00:42:32.870 align:middle line:84% plus square root of c times epsilon. 00:42:32.870 --> 00:42:37.070 align:middle line:84% Don't do that because delta, again, only 00:42:37.070 --> 00:42:39.680 align:middle line:84% depends on the point c and the point epsilon. 00:42:39.680 --> 00:42:43.290 align:middle line:84% It cannot depend on the thing that's changing, which is x, 00:42:43.290 --> 00:42:43.790 align:middle line:90% right? 00:42:43.790 --> 00:42:46.530 align:middle line:90% This is changing. 00:42:46.530 --> 00:42:49.995 align:middle line:90% So you don't make this choice. 00:42:49.995 --> 00:42:52.850 align:middle line:90% 00:42:52.850 --> 00:42:54.565 align:middle line:84% Now, if we just do one more thing, 00:42:54.565 --> 00:42:55.940 align:middle line:84% then we'll get to something where 00:42:55.940 --> 00:42:58.640 align:middle line:84% we can choose an independent of the thing that's changing, x. 00:42:58.640 --> 00:43:01.260 align:middle line:90% 00:43:01.260 --> 00:43:03.030 align:middle line:84% The square root of x on the bottom 00:43:03.030 --> 00:43:06.940 align:middle line:84% is only making things smaller because that's non-negative. 00:43:06.940 --> 00:43:13.098 align:middle line:84% So this is less than or equal to delta over square root of c, 00:43:13.098 --> 00:43:15.500 align:middle line:90% OK? 00:43:15.500 --> 00:43:18.470 align:middle line:84% So now, in the end, I have something which is delta 00:43:18.470 --> 00:43:21.260 align:middle line:84% and depends on the point c, and I 00:43:21.260 --> 00:43:24.630 align:middle line:84% want f of x minus square root of c to be less than epsilon. 00:43:24.630 --> 00:43:25.865 align:middle line:90% So how should I choose delta? 00:43:25.865 --> 00:43:35.430 align:middle line:90% 00:43:35.430 --> 00:43:36.510 align:middle line:90% Shout it if you know it. 00:43:36.510 --> 00:43:54.080 align:middle line:90% 00:43:54.080 --> 00:43:57.350 align:middle line:84% Right, epsilon times the square root of c. 00:43:57.350 --> 00:44:00.260 align:middle line:90% So now this will be our choice. 00:44:00.260 --> 00:44:07.880 align:middle line:84% So choose delta to be epsilon times square root of c. 00:44:07.880 --> 00:44:13.670 align:middle line:90% 00:44:13.670 --> 00:44:16.430 align:middle line:84% Now we have to show this delta works, which is essentially, 00:44:16.430 --> 00:44:19.070 align:middle line:84% again, just rehashing our computation 00:44:19.070 --> 00:44:22.280 align:middle line:84% over on the right, which gave us how to choose delta 00:44:22.280 --> 00:44:30.990 align:middle line:84% than if the absolute value of x minus c is less than delta, 00:44:30.990 --> 00:44:36.483 align:middle line:84% f of x minus square root of c-- and I'm not 00:44:36.483 --> 00:44:37.400 align:middle line:90% going to write f of x. 00:44:37.400 --> 00:44:57.610 align:middle line:84% I'm going to write the square root of x is equal to basically 00:44:57.610 --> 00:44:59.350 align:middle line:84% what we did over here in the box, which 00:44:59.350 --> 00:45:03.610 align:middle line:84% is absolute value of x minus c over square root of x 00:45:03.610 --> 00:45:09.970 align:middle line:84% plus square root of c, which is less than or equal to, 00:45:09.970 --> 00:45:12.730 align:middle line:84% if I take away the square root of x on the bottom like we did 00:45:12.730 --> 00:45:16.960 align:middle line:84% before, which is less than delta or square root of c, which 00:45:16.960 --> 00:45:20.500 align:middle line:84% equals epsilon square root of c over square root of c, which 00:45:20.500 --> 00:45:22.490 align:middle line:90% equals epsilon. 00:45:22.490 --> 00:45:23.730 align:middle line:90% So this delta works. 00:45:23.730 --> 00:45:32.198 align:middle line:90% 00:45:32.198 --> 00:45:34.350 align:middle line:84% OK, Is there any questions about that example? 00:45:34.350 --> 00:45:47.030 align:middle line:90% 00:45:47.030 --> 00:45:49.610 align:middle line:90% OK. 00:45:49.610 --> 00:45:55.147 align:middle line:84% So let's do one more example, which really illustrates what 00:45:55.147 --> 00:45:57.480 align:middle line:84% I was getting at when I was talking about the limit only 00:45:57.480 --> 00:46:01.470 align:middle line:84% cares about what the function is doing near a point 00:46:01.470 --> 00:46:02.590 align:middle line:90% but not at the point. 00:46:02.590 --> 00:46:08.340 align:middle line:84% So let's say we look at the function f of x given by 1, 00:46:08.340 --> 00:46:13.740 align:middle line:84% if x equals 0, and 2, if x does not equals 0. 00:46:13.740 --> 00:46:21.940 align:middle line:90% So this is-- here's the point 0. 00:46:21.940 --> 00:46:29.830 align:middle line:84% And where it's not equal to 0, it's equal to 2, 00:46:29.830 --> 00:46:36.700 align:middle line:84% and when x equals 0, it's equal to 1, all right? 00:46:36.700 --> 00:46:40.966 align:middle line:90% 00:46:40.966 --> 00:46:44.900 align:middle line:84% And they claim that the limit as x goes to 0 of f of x-- 00:46:44.900 --> 00:46:47.960 align:middle line:90% this equals 2. 00:46:47.960 --> 00:46:53.750 align:middle line:84% And the thing to note is that this 00:46:53.750 --> 00:46:59.830 align:middle line:84% limit it does not equal f of 0, which 00:46:59.830 --> 00:47:01.780 align:middle line:84% it did for the previous two cases, right? 00:47:01.780 --> 00:47:04.120 align:middle line:84% If we look back at these two examples 00:47:04.120 --> 00:47:07.090 align:middle line:84% that we did a minute ago, the limit 00:47:07.090 --> 00:47:09.250 align:middle line:84% as x goes to c of both of these examples 00:47:09.250 --> 00:47:13.910 align:middle line:84% was I just plug in c to the function, right? 00:47:13.910 --> 00:47:17.040 align:middle line:84% But for this example, that's not the case. 00:47:17.040 --> 00:47:21.260 align:middle line:84% The limit is not equal to the function evaluated at the point 00:47:21.260 --> 00:47:24.050 align:middle line:84% in highlighting that limits don't 00:47:24.050 --> 00:47:26.540 align:middle line:84% care about the function evaluated at the point. 00:47:26.540 --> 00:47:29.510 align:middle line:84% They care about what's happening near the point, OK? 00:47:29.510 --> 00:47:36.420 align:middle line:84% And near the point, f of x is just equal to 2 identically. 00:47:36.420 --> 00:47:41.030 align:middle line:84% So I'll give you the quick proof, 00:47:41.030 --> 00:47:42.980 align:middle line:84% or you can completely ignore it and just 00:47:42.980 --> 00:47:49.670 align:middle line:84% believe that the limit as x goes to 0 of this function equals 2. 00:47:49.670 --> 00:47:54.170 align:middle line:84% So the point is that for x not equal to 0, f of x 00:47:54.170 --> 00:47:55.400 align:middle line:90% is just a constant 2. 00:47:55.400 --> 00:47:59.245 align:middle line:90% So let epsilon be positive. 00:47:59.245 --> 00:48:00.870 align:middle line:84% And you can choose delta to be whatever 00:48:00.870 --> 00:48:04.530 align:middle line:84% you like because f of x when x is not equal to 0 00:48:04.530 --> 00:48:08.850 align:middle line:84% is going to be equal to 2, the proposed limit. 00:48:08.850 --> 00:48:14.360 align:middle line:84% Then if 0 is bigger than the absolute value of x minus 0, 00:48:14.360 --> 00:48:17.130 align:middle line:84% which is just x, is less than delta, 00:48:17.130 --> 00:48:20.690 align:middle line:84% then this implies x is not equal to 0. 00:48:20.690 --> 00:48:25.010 align:middle line:84% And therefore, if I look at f of x minus 2, 00:48:25.010 --> 00:48:36.520 align:middle line:84% this is just 2 minus 2 equals 0, which is less than epsilon, OK? 00:48:36.520 --> 00:48:37.930 align:middle line:84% Any questions about this example? 00:48:37.930 --> 00:48:56.550 align:middle line:90% 00:48:56.550 --> 00:48:57.480 align:middle line:90% No questions? 00:48:57.480 --> 00:49:11.140 align:middle line:90% 00:49:11.140 --> 00:49:16.120 align:middle line:84% All right, so what we'll talk about now is-- 00:49:16.120 --> 00:49:22.090 align:middle line:84% so before this section on continuous functions, 00:49:22.090 --> 00:49:24.490 align:middle line:84% or at least this section on limits of functions, 00:49:24.490 --> 00:49:29.130 align:middle line:84% we had the notion of limits of sequences. 00:49:29.130 --> 00:49:31.110 align:middle line:84% And so a natural question is, how 00:49:31.110 --> 00:49:37.320 align:middle line:84% do limits of functions and limits of sequences 00:49:37.320 --> 00:49:41.340 align:middle line:84% belong together, interact, relate to each other, I guess, 00:49:41.340 --> 00:49:43.890 align:middle line:84% is the best way to phrase that question. 00:49:43.890 --> 00:49:46.560 align:middle line:90% 00:49:46.560 --> 00:49:51.480 align:middle line:84% And the point is that to decide if a function has 00:49:51.480 --> 00:49:56.310 align:middle line:84% a limit it, suffices to stick in sequences that converge to that 00:49:56.310 --> 00:49:58.450 align:middle line:90% point that you're looking at. 00:49:58.450 --> 00:50:00.840 align:middle line:84% And that's the content of the next theorem. 00:50:00.840 --> 00:50:20.950 align:middle line:84% So let S be a subset of R, c cluster point of S. 00:50:20.950 --> 00:50:24.990 align:middle line:90% Let f be a function from S to R. 00:50:24.990 --> 00:50:33.180 align:middle line:90% Then the limit exists. 00:50:33.180 --> 00:50:39.270 align:middle line:84% Limit as x goes to c of f of x equals a number L. This is 00:50:39.270 --> 00:50:46.110 align:middle line:84% equivalent to the statement that for every sequence 00:50:46.110 --> 00:50:51.440 align:middle line:84% x sub n of elements of S take away 00:50:51.440 --> 00:51:08.160 align:middle line:84% c such that xn converges to c, we have f of xn converges to L, 00:51:08.160 --> 00:51:08.730 align:middle line:90% OK? 00:51:08.730 --> 00:51:10.355 align:middle line:84% So again, the statement of this theorem 00:51:10.355 --> 00:51:14.850 align:middle line:84% is that a function converges to L if and only 00:51:14.850 --> 00:51:18.000 align:middle line:84% if for every sequence I stick into the function which 00:51:18.000 --> 00:51:23.540 align:middle line:84% converges to c, f of x then converges to L. So 00:51:23.540 --> 00:51:27.980 align:middle line:84% to choose or to decide if a function converges, 00:51:27.980 --> 00:51:31.220 align:middle line:84% it suffices to look at convergence 00:51:31.220 --> 00:51:32.810 align:middle line:90% of certain sequences, OK? 00:51:32.810 --> 00:51:37.890 align:middle line:90% 00:51:37.890 --> 00:51:38.390 align:middle line:90% OK. 00:51:38.390 --> 00:51:41.130 align:middle line:90% 00:51:41.130 --> 00:51:44.260 align:middle line:84% And why is this theorem important, 00:51:44.260 --> 00:51:45.520 align:middle line:90% or at least very useful? 00:51:45.520 --> 00:51:51.540 align:middle line:84% It's because with this theorem, we now get basically 00:51:51.540 --> 00:51:55.620 align:middle line:84% every analog of theorems we proved for sequences for free, 00:51:55.620 --> 00:52:01.020 align:middle line:84% for example, limits of sums of functions, a squeeze 00:52:01.020 --> 00:52:08.600 align:middle line:84% theorem that can be stated for limits of functions, and so on. 00:52:08.600 --> 00:52:10.300 align:middle line:84% And I'll say a few comments about this 00:52:10.300 --> 00:52:11.500 align:middle line:90% after we proved the theorem. 00:52:11.500 --> 00:52:16.050 align:middle line:90% 00:52:16.050 --> 00:52:17.420 align:middle line:90% All right, so let's-- 00:52:17.420 --> 00:52:18.610 align:middle line:90% this is a two-way street. 00:52:18.610 --> 00:52:22.060 align:middle line:90% 00:52:22.060 --> 00:52:28.310 align:middle line:84% And I'm going to mark the second statement in green. 00:52:28.310 --> 00:52:30.810 align:middle line:84% That way, I don't have to rewrite it again. 00:52:30.810 --> 00:52:38.460 align:middle line:84% So first, assume that limit at x goes to c of f of x equals L, 00:52:38.460 --> 00:52:44.356 align:middle line:84% and now we want to prove the statement in green, OK? 00:52:44.356 --> 00:52:48.310 align:middle line:90% So here's the idea. 00:52:48.310 --> 00:52:49.620 align:middle line:90% So now we want to show-- 00:52:49.620 --> 00:53:02.180 align:middle line:84% so let xn be a sequence of elements in S 00:53:02.180 --> 00:53:09.550 align:middle line:84% take away c such that x sub n converges to c. 00:53:09.550 --> 00:53:14.490 align:middle line:84% And what we want to show is that f 00:53:14.490 --> 00:53:20.070 align:middle line:84% of xn, which is now a new sequence 00:53:20.070 --> 00:53:23.890 align:middle line:84% converges to L as n goes to infinity. 00:53:23.890 --> 00:53:27.520 align:middle line:90% 00:53:27.520 --> 00:53:31.050 align:middle line:84% OK, so here, you should write, as n 00:53:31.050 --> 00:53:40.205 align:middle line:90% goes to infinity, in here, OK? 00:53:40.205 --> 00:53:45.440 align:middle line:84% So let me move that, and let me draw the picture of why 00:53:45.440 --> 00:53:46.460 align:middle line:90% you should believe this. 00:53:46.460 --> 00:53:49.380 align:middle line:90% 00:53:49.380 --> 00:53:51.620 align:middle line:84% So remember, what it means for a sequence 00:53:51.620 --> 00:53:54.750 align:middle line:84% to converge to a number means, for every epsilon, 00:53:54.750 --> 00:53:57.470 align:middle line:84% there exists a capital M so that for all n bigger 00:53:57.470 --> 00:54:01.130 align:middle line:84% than or equal to capital M, the sequence is close to the number 00:54:01.130 --> 00:54:03.170 align:middle line:90% L within epsilon distance to it. 00:54:03.170 --> 00:54:08.760 align:middle line:84% So let's think about this a little bit. 00:54:08.760 --> 00:54:10.520 align:middle line:90% I'm not going to draw the graph. 00:54:10.520 --> 00:54:15.990 align:middle line:84% I'm just going to draw two copies of the number line. 00:54:15.990 --> 00:54:22.100 align:middle line:84% Here, we have the number L. Here's 00:54:22.100 --> 00:54:29.350 align:middle line:84% the number c and L plus epsilon, L minus epsilon. 00:54:29.350 --> 00:54:32.730 align:middle line:90% So let epsilon be positive. 00:54:32.730 --> 00:54:35.770 align:middle line:90% 00:54:35.770 --> 00:54:39.100 align:middle line:84% And the idea is we want to be able to find or say 00:54:39.100 --> 00:54:41.380 align:middle line:84% there exists a natural number capital M so 00:54:41.380 --> 00:54:45.070 align:middle line:84% that n bigger than or equal to capital M implies f of xn 00:54:45.070 --> 00:54:47.920 align:middle line:90% is inside this interval here. 00:54:47.920 --> 00:54:57.030 align:middle line:84% Now, since f of x converges to L as x converges to c, 00:54:57.030 --> 00:55:02.550 align:middle line:90% there exists a delta so that-- 00:55:02.550 --> 00:55:05.890 align:middle line:84% so this is the picture that goes along with it-- 00:55:05.890 --> 00:55:14.030 align:middle line:84% so that if I'm inside this interval here, 00:55:14.030 --> 00:55:19.708 align:middle line:84% then I get mapped into this interval here. 00:55:19.708 --> 00:55:22.500 align:middle line:84% So this is what I get from the assumption. 00:55:22.500 --> 00:55:27.170 align:middle line:90% 00:55:27.170 --> 00:55:30.350 align:middle line:84% So then how I should choose the natural number capital 00:55:30.350 --> 00:55:35.631 align:middle line:84% M is so that all of the x sub n's 00:55:35.631 --> 00:55:40.220 align:middle line:84% lie in this interval, c minus delta, c plus delta. 00:55:40.220 --> 00:55:42.230 align:middle line:84% And if I can arrange that, which I can 00:55:42.230 --> 00:55:46.910 align:middle line:84% because the x sub n's converge to c, 00:55:46.910 --> 00:55:51.160 align:middle line:84% then I will get that these guys get mapped 00:55:51.160 --> 00:55:55.700 align:middle line:84% into this interval, L plus epsilon, L minus epsilon, OK? 00:55:55.700 --> 00:55:57.400 align:middle line:84% And that's the whole idea of the proof. 00:55:57.400 --> 00:55:58.870 align:middle line:84% Now we just need to write it down. 00:55:58.870 --> 00:56:06.990 align:middle line:90% 00:56:06.990 --> 00:56:14.270 align:middle line:84% Since f of x converges to L, exists a delta positive such 00:56:14.270 --> 00:56:27.260 align:middle line:84% that if less than delta, then if x minus L is less than epsilon. 00:56:27.260 --> 00:56:29.280 align:middle line:84% That's what I drew in yellow there. 00:56:29.280 --> 00:56:39.590 align:middle line:84% Now, since the sequence x sub n equals c, 00:56:39.590 --> 00:56:42.350 align:middle line:84% this implies that there exists some natural number M sub 00:56:42.350 --> 00:56:50.030 align:middle line:84% 0 such that little n bigger than or equal to M sub 0-- 00:56:50.030 --> 00:56:53.660 align:middle line:84% since all of these x sub n's are not equal to c, by assumption, 00:56:53.660 --> 00:56:57.380 align:middle line:84% they're in the set S take away c. 00:56:57.380 --> 00:57:00.140 align:middle line:84% X sub n minus c, and absolute value is bigger than 0 00:57:00.140 --> 00:57:04.760 align:middle line:90% and less than delta, OK? 00:57:04.760 --> 00:57:08.810 align:middle line:84% This follows from the definition of convergence of sequences 00:57:08.810 --> 00:57:09.380 align:middle line:90% to a point. 00:57:09.380 --> 00:57:11.240 align:middle line:84% Here, if you like, I'm choosing epsilon 00:57:11.240 --> 00:57:13.460 align:middle line:90% in that definition to be delta. 00:57:13.460 --> 00:57:16.820 align:middle line:84% And this delta is coming from the definition 00:57:16.820 --> 00:57:18.650 align:middle line:90% for convergence of functions. 00:57:18.650 --> 00:57:23.420 align:middle line:90% 00:57:23.420 --> 00:57:29.365 align:middle line:84% And so we'll choose capital M to be this integer M sub 0. 00:57:29.365 --> 00:57:32.290 align:middle line:90% 00:57:32.290 --> 00:57:35.880 align:middle line:84% And if n is bigger than or equal to M, 00:57:35.880 --> 00:57:39.780 align:middle line:84% I get that, by our choice of it being M sub 0, 00:57:39.780 --> 00:57:43.320 align:middle line:84% you get that x sub n minus c is less than delta. 00:57:43.320 --> 00:57:46.350 align:middle line:84% That means that it falls in this little yellow highlighted 00:57:46.350 --> 00:57:53.170 align:middle line:84% part in the first number line, which implies-- 00:57:53.170 --> 00:57:58.390 align:middle line:90% 00:57:58.390 --> 00:58:05.070 align:middle line:84% so here, what I'm using is what's highlighted in blue. 00:58:05.070 --> 00:58:10.850 align:middle line:84% So this is content of this implication-- 00:58:10.850 --> 00:58:19.050 align:middle line:84% implies that f of xn minus L is less than epsilon, all right? 00:58:19.050 --> 00:58:20.860 align:middle line:84% And that's what we wanted to do, right? 00:58:20.860 --> 00:58:23.940 align:middle line:84% We wanted to show that there exists some natural number 00:58:23.940 --> 00:58:28.260 align:middle line:84% M so that n bigger than or equal to M implies f of x minus L 00:58:28.260 --> 00:58:31.070 align:middle line:84% is less than epsilon in absolute value. 00:58:31.070 --> 00:58:37.630 align:middle line:84% And therefore, we've proved that the limit as n 00:58:37.630 --> 00:58:48.365 align:middle line:84% goes to infinity of the sequence f of xn equals L. 00:58:48.365 --> 00:58:51.610 align:middle line:84% OK, so that's one direction of this if-and-only-if statement. 00:58:51.610 --> 00:58:54.710 align:middle line:84% Let's prove the opposite direction. 00:58:54.710 --> 00:59:00.155 align:middle line:84% So suppose what's highlighted in green holds. 00:59:00.155 --> 00:59:06.991 align:middle line:90% 00:59:06.991 --> 00:59:07.990 align:middle line:90% OK? 00:59:07.990 --> 00:59:10.700 align:middle line:84% And this proof is going to be by contradiction. 00:59:10.700 --> 00:59:15.430 align:middle line:84% So let's suppose that our assumption, what's in green, 00:59:15.430 --> 00:59:16.090 align:middle line:90% holds. 00:59:16.090 --> 00:59:20.761 align:middle line:84% But the limit as x goes to c of f of x does not equal L. 00:59:20.761 --> 00:59:27.740 align:middle line:84% And now we're going to do the proof by contradiction, 00:59:27.740 --> 00:59:29.825 align:middle line:84% meaning suppose the outcome is false. 00:59:29.825 --> 00:59:37.490 align:middle line:90% 00:59:37.490 --> 00:59:38.950 align:middle line:90% OK? 00:59:38.950 --> 00:59:40.780 align:middle line:84% We're still assuming what's in green holds. 00:59:40.780 --> 00:59:41.930 align:middle line:90% That's our assumption. 00:59:41.930 --> 00:59:44.590 align:middle line:84% And for proof by contradiction, we 00:59:44.590 --> 00:59:48.520 align:middle line:84% assume that the outcome or the conclusion 00:59:48.520 --> 00:59:51.790 align:middle line:84% is false, which is that the limit as x goes to c of f of x 00:59:51.790 --> 00:59:55.120 align:middle line:90% does not equal L. 00:59:55.120 --> 00:59:57.260 align:middle line:90% Now, what does this mean? 00:59:57.260 --> 00:59:59.080 align:middle line:84% So now we get to negate the definition 00:59:59.080 --> 01:00:04.870 align:middle line:84% and get better acquainted with it. 01:00:04.870 --> 01:00:09.030 align:middle line:84% This means that-- so the definition 01:00:09.030 --> 01:00:13.750 align:middle line:84% of limit, if you go back to the notes is, for all epsilon, 01:00:13.750 --> 01:00:17.160 align:middle line:90% there exists a delta statement. 01:00:17.160 --> 01:00:20.060 align:middle line:84% So the negation is that there exists 01:00:20.060 --> 01:00:24.900 align:middle line:84% some bad epsilon, epsilon 0, such 01:00:24.900 --> 01:00:33.610 align:middle line:84% that for all delta positive, there exists an x such 01:00:33.610 --> 01:00:50.300 align:middle line:84% that x is close to c, but f of x is far away from L. OK? 01:00:50.300 --> 01:00:51.680 align:middle line:84% So the negation of the definition 01:00:51.680 --> 01:00:53.600 align:middle line:84% of limit of something being the limit 01:00:53.600 --> 01:00:56.220 align:middle line:84% is that there exists a bad epsilon 01:00:56.220 --> 01:01:00.110 align:middle line:84% so that no matter how close I get to c, 01:01:00.110 --> 01:01:03.800 align:middle line:84% I can find something very close to c that's far away from L 01:01:03.800 --> 01:01:07.630 align:middle line:90% if I stick it into f, OK? 01:01:07.630 --> 01:01:12.660 align:middle line:84% That's the intuitive way of viewing the negation 01:01:12.660 --> 01:01:13.680 align:middle line:90% of this definition. 01:01:13.680 --> 01:01:17.070 align:middle line:90% 01:01:17.070 --> 01:01:19.860 align:middle line:84% OK, so now I'm going to apply this 01:01:19.860 --> 01:01:25.830 align:middle line:84% with delta chosen to be 1/n for in a natural number, OK? 01:01:25.830 --> 01:01:29.670 align:middle line:90% 01:01:29.670 --> 01:01:35.800 align:middle line:84% And this says-- so this statement here 01:01:35.800 --> 01:01:37.700 align:middle line:90% is a for-all-delta statement. 01:01:37.700 --> 01:01:42.640 align:middle line:84% So that holds for every delta I pick. 01:01:42.640 --> 01:01:52.020 align:middle line:84% So for all n, a natural number, there exists an x sub n such 01:01:52.020 --> 01:01:56.962 align:middle line:84% that 0 is bigger than x sub n minus c is less than delta-- 01:01:56.962 --> 01:01:58.170 align:middle line:90% I'm going to pick to be 1/n-- 01:01:58.170 --> 01:02:08.100 align:middle line:90% 01:02:08.100 --> 01:02:11.310 align:middle line:84% and f of x sub n minus L is bigger than or equal 01:02:11.310 --> 01:02:12.300 align:middle line:90% to epsilon 0. 01:02:12.300 --> 01:02:15.630 align:middle line:84% So first off, I made a for-all-n statement. 01:02:15.630 --> 01:02:17.710 align:middle line:84% Let's think about this for just a second. 01:02:17.710 --> 01:02:19.410 align:middle line:84% So certainly, for delta equals 1, 01:02:19.410 --> 01:02:24.270 align:middle line:84% I can find a number x sub 1 satisfying 0 is less than x 01:02:24.270 --> 01:02:28.740 align:middle line:84% minus c is less than 1, and f of x minus f of x sub 1 minus 01:02:28.740 --> 01:02:32.010 align:middle line:84% L is bigger than or equal to epsilon 0. 01:02:32.010 --> 01:02:36.300 align:middle line:84% But delta can be anything I choose, so for delta equal 1/2, 01:02:36.300 --> 01:02:38.940 align:middle line:84% I can find an x sub 2 such that x 01:02:38.940 --> 01:02:43.890 align:middle line:84% sub 2 minus c is less than 1/2, and f of x minus L 01:02:43.890 --> 01:02:45.840 align:middle line:84% is bigger than or equal to epsilon 0. 01:02:45.840 --> 01:02:51.047 align:middle line:84% So I'm just choosing delta to be 1/n for each n, 01:02:51.047 --> 01:02:51.755 align:middle line:90% a natural number. 01:02:51.755 --> 01:02:57.340 align:middle line:90% 01:02:57.340 --> 01:03:04.002 align:middle line:90% Choosing delta to be 1/n, OK? 01:03:04.002 --> 01:03:04.835 align:middle line:90% Is this point clear? 01:03:04.835 --> 01:03:11.530 align:middle line:90% 01:03:11.530 --> 01:03:15.430 align:middle line:84% OK, please stop me if you have questions and something 01:03:15.430 --> 01:03:17.620 align:middle line:90% is not clear. 01:03:17.620 --> 01:03:22.950 align:middle line:84% OK, so I have this sequence of elements of S 01:03:22.950 --> 01:03:24.750 align:middle line:84% that are not equal to c because they 01:03:24.750 --> 01:03:28.170 align:middle line:84% are an absolute value bigger than 0, 01:03:28.170 --> 01:03:32.610 align:middle line:84% or the absolute value of x sub n minus c is bigger than 0. 01:03:32.610 --> 01:03:36.690 align:middle line:84% And they satisfy these two inequalities, OK? 01:03:36.690 --> 01:03:41.500 align:middle line:90% 01:03:41.500 --> 01:03:44.035 align:middle line:84% Then let me just write this inequality again. 01:03:44.035 --> 01:03:50.430 align:middle line:90% 01:03:50.430 --> 01:03:54.390 align:middle line:84% So then the inequality that I have is, for all n, 01:03:54.390 --> 01:03:57.210 align:middle line:84% the natural number 0 is less than x sub n minus c 01:03:57.210 --> 01:03:59.460 align:middle line:90% is less than 1/n. 01:03:59.460 --> 01:04:02.760 align:middle line:84% I claim that the x sub n's must then converge to c. 01:04:02.760 --> 01:04:06.870 align:middle line:90% What tool can I use to say that? 01:04:06.870 --> 01:04:14.570 align:middle line:90% 01:04:14.570 --> 01:04:17.730 align:middle line:90% Squeeze theorem, OK? 01:04:17.730 --> 01:04:20.190 align:middle line:84% Because this thing on the left, 0-- 01:04:20.190 --> 01:04:22.770 align:middle line:84% this, if you like, is a sequence, just 01:04:22.770 --> 01:04:23.670 align:middle line:90% a constant sequence. 01:04:23.670 --> 01:04:25.470 align:middle line:90% This converges to 0. 01:04:25.470 --> 01:04:29.100 align:middle line:84% And 1/n, something on the right, converges to 0 01:04:29.100 --> 01:04:30.280 align:middle line:90% as n goes to infinity. 01:04:30.280 --> 01:04:32.280 align:middle line:84% So that implies the thing that's sandwiched 01:04:32.280 --> 01:04:34.470 align:middle line:90% in between gets squeezed to 0. 01:04:34.470 --> 01:04:40.640 align:middle line:84% So that implies limit as n goes to infinity of x sub n minus c 01:04:40.640 --> 01:04:45.860 align:middle line:84% equals 0, i.e., xn's converge to c. 01:04:45.860 --> 01:04:50.150 align:middle line:84% Now, I'm assuming what's in green holds, which 01:04:50.150 --> 01:04:54.170 align:middle line:84% is, if I take any sequence converging to c, f of xn 01:04:54.170 --> 01:04:58.680 align:middle line:90% has to converge to L, OK? 01:04:58.680 --> 01:05:05.080 align:middle line:84% So by what's in green has given me this implication-- 01:05:05.080 --> 01:05:09.860 align:middle line:84% 0 must be equal to limit as n goes to infinity 01:05:09.860 --> 01:05:14.730 align:middle line:90% of f of x sub n minus L. 01:05:14.730 --> 01:05:16.200 align:middle line:90% But What. 01:05:16.200 --> 01:05:21.450 align:middle line:84% I know is that, since I've assumed 01:05:21.450 --> 01:05:24.690 align:middle line:84% f of x does not converge to L, along this sequence, 01:05:24.690 --> 01:05:27.000 align:middle line:84% each of these guys is bigger than or equal to epsilon 01:05:27.000 --> 01:05:28.920 align:middle line:90% 0, which is positive. 01:05:28.920 --> 01:05:33.530 align:middle line:90% 01:05:33.530 --> 01:05:37.800 align:middle line:84% So now I've concluded that 0 is bigger than 0. 01:05:37.800 --> 01:05:44.760 align:middle line:90% That's a contradiction, OK? 01:05:44.760 --> 01:05:47.370 align:middle line:90% 01:05:47.370 --> 01:05:51.000 align:middle line:84% OK, so next time, we'll talk about the applications 01:05:51.000 --> 01:05:52.320 align:middle line:90% of this theorem. 01:05:52.320 --> 01:05:55.800 align:middle line:84% Do you all have any questions about this proof or anything 01:05:55.800 --> 01:05:57.975 align:middle line:90% that we've covered so far today? 01:05:57.975 --> 01:06:20.302 align:middle line:90% 01:06:20.302 --> 01:06:22.510 align:middle line:84% You mean how you get the choice of delta from doing-- 01:06:22.510 --> 01:06:23.772 align:middle line:90% AUDIENCE: Yeah. 01:06:23.772 --> 01:06:24.730 align:middle line:90% CASEY RODRIGUEZ: Right. 01:06:24.730 --> 01:06:27.570 align:middle line:90% 01:06:27.570 --> 01:06:29.030 align:middle line:90% So let me phrase this a-- 01:06:29.030 --> 01:06:46.960 align:middle line:90% 01:06:46.960 --> 01:06:48.860 align:middle line:90% Yeah, I mean, so-- 01:06:48.860 --> 01:06:51.640 align:middle line:84% and probably in most classes, you 01:06:51.640 --> 01:06:54.640 align:middle line:84% would just be given this proof that's over here on the left. 01:06:54.640 --> 01:06:56.350 align:middle line:84% And somehow it looks like magic that I 01:06:56.350 --> 01:07:01.270 align:middle line:84% chose this delta this way, and it ends up working, right? 01:07:01.270 --> 01:07:05.110 align:middle line:84% But the thinking that goes in behind it is-- 01:07:05.110 --> 01:07:06.610 align:middle line:90% so what's the thing you want? 01:07:06.610 --> 01:07:08.290 align:middle line:90% You want what's in yellow. 01:07:08.290 --> 01:07:12.010 align:middle line:84% So let me erase some of what's in yellow. 01:07:12.010 --> 01:07:15.880 align:middle line:84% You want to choose delta so that if you 01:07:15.880 --> 01:07:19.180 align:middle line:84% take f of x minus ac plus b, that thing in absolute value 01:07:19.180 --> 01:07:22.420 align:middle line:90% will be less than epsilon, OK? 01:07:22.420 --> 01:07:26.480 align:middle line:84% So start with that thing that you want to make small 01:07:26.480 --> 01:07:31.650 align:middle line:84% and that you want to ensure is small, f of x minus ac plus b, 01:07:31.650 --> 01:07:34.390 align:middle line:90% and start computing. 01:07:34.390 --> 01:07:37.690 align:middle line:84% And when you compute it, you get what we got here 01:07:37.690 --> 01:07:41.440 align:middle line:90% in the second bit of yellow. 01:07:41.440 --> 01:07:44.890 align:middle line:84% And we're assuming that the absolute value of x minus c 01:07:44.890 --> 01:07:46.450 align:middle line:90% is less than delta, right? 01:07:46.450 --> 01:07:48.310 align:middle line:84% We're trying to choose delta so that what's 01:07:48.310 --> 01:07:51.550 align:middle line:84% in green implies what's in yellow. 01:07:51.550 --> 01:07:53.720 align:middle line:90% So far, we haven't chosen delta. 01:07:53.720 --> 01:07:57.100 align:middle line:84% We just know that the absolute value of x minus c 01:07:57.100 --> 01:07:58.660 align:middle line:90% will be less than delta. 01:07:58.660 --> 01:08:00.460 align:middle line:84% And we want to choose delta so that that 01:08:00.460 --> 01:08:06.550 align:middle line:84% implies what's in yellow after the arrow there. 01:08:06.550 --> 01:08:09.660 align:middle line:84% So when we do this computation, and assuming 01:08:09.660 --> 01:08:12.820 align:middle line:84% that the absolute value of x minus c is less than delta, 01:08:12.820 --> 01:08:22.180 align:middle line:84% we get this thing right here, the absolute value 01:08:22.180 --> 01:08:23.260 align:middle line:90% of a time delta. 01:08:23.260 --> 01:08:26.800 align:middle line:84% This is just assuming that the absolute value of x minus c 01:08:26.800 --> 01:08:32.040 align:middle line:84% is less than delta, which is what will it be assuming. 01:08:32.040 --> 01:08:34.319 align:middle line:84% And we want what's in yellow, which is that thing 01:08:34.319 --> 01:08:35.560 align:middle line:90% to be less than epsilon. 01:08:35.560 --> 01:08:39.550 align:middle line:84% So what we've arrived at is this thing here, 01:08:39.550 --> 01:08:41.700 align:middle line:84% which we want less than epsilon, is less 01:08:41.700 --> 01:08:46.250 align:middle line:90% than this thing in red, OK? 01:08:46.250 --> 01:08:48.890 align:middle line:84% And therefore, we want to choose the delta 01:08:48.890 --> 01:08:51.740 align:middle line:84% so that the thing in red is less than epsilon 01:08:51.740 --> 01:08:53.960 align:middle line:84% or less than or equal to epsilon. 01:08:53.960 --> 01:08:56.960 align:middle line:84% That will ensure that f of x minus ac 01:08:56.960 --> 01:09:00.390 align:middle line:84% plus b in absolute value will be less than epsilon, OK? 01:09:00.390 --> 01:09:02.960 align:middle line:84% So I'm not solving an inequality. 01:09:02.960 --> 01:09:05.160 align:middle line:84% I'm not solving the inequality I want, 01:09:05.160 --> 01:09:07.490 align:middle line:90% which is f of x minus ac plus b. 01:09:07.490 --> 01:09:10.189 align:middle line:84% I'm starting with what's on the left, 01:09:10.189 --> 01:09:12.920 align:middle line:84% estimating it using my assumptions, and then, 01:09:12.920 --> 01:09:15.140 align:middle line:84% at the end, choosing delta so that I come up 01:09:15.140 --> 01:09:17.220 align:middle line:90% with epsilon in the end. 01:09:17.220 --> 01:09:20.540 align:middle line:84% So in the last step, I have absolute value 01:09:20.540 --> 01:09:22.439 align:middle line:90% of a times delta. 01:09:22.439 --> 01:09:25.380 align:middle line:84% And I want that to be less than epsilon. 01:09:25.380 --> 01:09:31.149 align:middle line:84% So I could have chosen delta to be less than 01:09:31.149 --> 01:09:32.875 align:middle line:90% or equal to epsilon/a. 01:09:32.875 --> 01:09:35.790 align:middle line:90% 01:09:35.790 --> 01:09:38.410 align:middle line:84% So anything, any delta like that would have worked. 01:09:38.410 --> 01:09:42.149 align:middle line:84% So I could have put delta to be epsilon over 2a. 01:09:42.149 --> 01:09:44.677 align:middle line:84% That still would have worked, or 3a, OK? 01:09:44.677 --> 01:09:57.870 align:middle line:90% 01:09:57.870 --> 01:09:59.370 align:middle line:84% Yes, [? what ?] [? they're ?] green. 01:09:59.370 --> 01:10:02.720 align:middle line:84% That's the thing we're doing, thing we 01:10:02.720 --> 01:10:05.840 align:middle line:90% get to use in this computation. 01:10:05.840 --> 01:10:08.940 align:middle line:90% 01:10:08.940 --> 01:10:10.290 align:middle line:90% Are there any other questions? 01:10:10.290 --> 01:10:43.770 align:middle line:90% 01:10:43.770 --> 01:10:47.400 align:middle line:84% This is a style thing, all right? 01:10:47.400 --> 01:10:53.130 align:middle line:84% So throughout this entire semester, for all epsilon, 01:10:53.130 --> 01:10:56.890 align:middle line:84% you have to be able to find the M so that something happens, 01:10:56.890 --> 01:10:57.390 align:middle line:90% right? 01:10:57.390 --> 01:10:59.880 align:middle line:84% So at some point, you always say, choose 01:10:59.880 --> 01:11:02.170 align:middle line:90% M to be something, right? 01:11:02.170 --> 01:11:05.970 align:middle line:84% Maybe it's the maximum of some M0, M1 that you had. 01:11:05.970 --> 01:11:11.010 align:middle line:84% Or choose M to be so that 1/M is less than epsilon or something 01:11:11.010 --> 01:11:11.800 align:middle line:90% like that. 01:11:11.800 --> 01:11:16.080 align:middle line:84% So I'm just sticking with that style of saying, 01:11:16.080 --> 01:11:17.400 align:middle line:90% choose M to be this. 01:11:17.400 --> 01:11:20.820 align:middle line:84% Otherwise, I could have said, there 01:11:20.820 --> 01:11:24.750 align:middle line:84% exists a capital number M so that for all n bigger 01:11:24.750 --> 01:11:28.200 align:middle line:84% than or equal to capital M, x sub n minus 01:11:28.200 --> 01:11:31.860 align:middle line:84% c is less than delta, and then move to the next part. 01:11:31.860 --> 01:11:35.190 align:middle line:84% But throughout the course, I've always, at least in the proofs 01:11:35.190 --> 01:11:40.830 align:middle line:84% I presented, you're making a choice of capital M, right? 01:11:40.830 --> 01:11:43.320 align:middle line:84% It's for all epsilon, there exists 01:11:43.320 --> 01:11:45.510 align:middle line:84% a capital M. So that means you have to tell me 01:11:45.510 --> 01:11:47.040 align:middle line:90% how to choose capital M. 01:11:47.040 --> 01:11:49.680 align:middle line:84% And it's the same thing with these epsilon 01:11:49.680 --> 01:11:52.350 align:middle line:84% delta proofs is that, for all epsilon, there exists a delta. 01:11:52.350 --> 01:11:54.510 align:middle line:84% So at some point, you have to tell me how 01:11:54.510 --> 01:11:57.820 align:middle line:90% to choose delta in the proof. 01:11:57.820 --> 01:12:00.090 align:middle line:84% So I've just kind of stuck with that style 01:12:00.090 --> 01:12:04.775 align:middle line:84% of saying, choose M to be something, although you're 01:12:04.775 --> 01:12:06.650 align:middle line:84% absolutely right that I could have just said, 01:12:06.650 --> 01:12:08.960 align:middle line:84% for all n bigger than or equal to capital M0, 01:12:08.960 --> 01:12:10.340 align:middle line:90% I have what I want. 01:12:10.340 --> 01:12:13.190 align:middle line:84% And therefore, implicitly, I'm saying 01:12:13.190 --> 01:12:16.490 align:middle line:84% that M equals M0 is the thing that works. 01:12:16.490 --> 01:12:33.210 align:middle line:90% 01:12:33.210 --> 01:12:35.753 align:middle line:90% Yeah, that's the same delta. 01:12:35.753 --> 01:12:36.920 align:middle line:90% Yeah, that's the same delta. 01:12:36.920 --> 01:12:43.490 align:middle line:90% 01:12:43.490 --> 01:12:45.040 align:middle line:90% Mhm. 01:12:45.040 --> 01:12:53.000 align:middle line:90%