WEBVTT 00:00:00.000 --> 00:00:00.994 align:middle line:90% [SQUEAKING] 00:00:00.994 --> 00:00:02.485 align:middle line:90% 00:00:02.485 --> 00:00:03.479 align:middle line:90% [RUSTLING] 00:00:03.479 --> 00:00:04.473 align:middle line:90% 00:00:04.473 --> 00:00:05.467 align:middle line:90% [CLICKING] 00:00:05.467 --> 00:00:13.420 align:middle line:90% 00:00:13.420 --> 00:00:14.710 align:middle line:90% CASEY RODRIGUEZ: OK. 00:00:14.710 --> 00:00:22.370 align:middle line:84% So we're going to continue our discussion of-- 00:00:22.370 --> 00:00:25.150 align:middle line:90% 00:00:25.150 --> 00:00:28.023 align:middle line:90% is it 1M or 2? 00:00:28.023 --> 00:00:28.523 align:middle line:90% It's 1. 00:00:28.523 --> 00:00:31.694 align:middle line:90% 00:00:31.694 --> 00:00:42.290 align:middle line:84% Of the Riemann integral, which, remember, from the discussion 00:00:42.290 --> 00:00:44.150 align:middle line:84% at the end of the last lecture, is 00:00:44.150 --> 00:00:49.345 align:middle line:84% a theory of area underneath the graph of a function. 00:00:49.345 --> 00:00:52.730 align:middle line:90% 00:00:52.730 --> 00:00:56.240 align:middle line:90% So what is that theory? 00:00:56.240 --> 00:00:58.800 align:middle line:84% The theory is built up as follows. 00:00:58.800 --> 00:01:01.790 align:middle line:90% Given a continuous function f-- 00:01:01.790 --> 00:01:04.550 align:middle line:84% so this is just a refresher on the definitions 00:01:04.550 --> 00:01:06.800 align:middle line:90% I introduced last time-- 00:01:06.800 --> 00:01:12.740 align:middle line:84% a partition is just points between-- 00:01:12.740 --> 00:01:23.450 align:middle line:84% just breaking up the interval from a to b into points, 00:01:23.450 --> 00:01:28.100 align:middle line:84% into little subintervals, with the norm of this partition 00:01:28.100 --> 00:01:33.650 align:middle line:84% being the length of the longest subinterval [INAUDIBLE] 00:01:33.650 --> 00:01:35.240 align:middle line:90% which we refer to as a tag. 00:01:35.240 --> 00:01:37.790 align:middle line:84% It's just a set of points with one 00:01:37.790 --> 00:01:41.200 align:middle line:84% lying in each of these subintervals-- 00:01:41.200 --> 00:01:46.180 align:middle line:84% so for me, let's take them all to be the right endpoint 00:01:46.180 --> 00:01:56.090 align:middle line:90% of these intervals-- 00:01:56.090 --> 00:01:59.810 align:middle line:84% and then associated to a tagged partition, which 00:01:59.810 --> 00:02:04.040 align:middle line:84% is a pair, x and [? xe. ?] We associated a Riemann sum, 00:02:04.040 --> 00:02:07.363 align:middle line:84% which is f evaluated at these points times the length 00:02:07.363 --> 00:02:08.030 align:middle line:90% of the interval. 00:02:08.030 --> 00:02:11.800 align:middle line:90% 00:02:11.800 --> 00:02:15.460 align:middle line:84% So if we draw the graph of a function f and-- 00:02:15.460 --> 00:02:16.390 align:middle line:90% say, in this picture-- 00:02:16.390 --> 00:02:24.260 align:middle line:90% 00:02:24.260 --> 00:02:30.680 align:middle line:84% this number this Riemann sum would then 00:02:30.680 --> 00:02:37.160 align:middle line:84% be equal to the area of these three boxes here. 00:02:37.160 --> 00:02:39.200 align:middle line:90% OK? 00:02:39.200 --> 00:02:41.720 align:middle line:84% And so our theory of Riemann integration, 00:02:41.720 --> 00:02:45.110 align:middle line:84% or our theory of the area underneath the curve, 00:02:45.110 --> 00:02:50.570 align:middle line:84% is built on the following goal, which is to show that, 00:02:50.570 --> 00:02:53.990 align:middle line:84% as these partitions get finer and finer, 00:02:53.990 --> 00:02:57.950 align:middle line:84% as the norm of these partitions goes to 0, 00:02:57.950 --> 00:03:02.440 align:middle line:84% these Riemann sums should converge to some number. 00:03:02.440 --> 00:03:05.830 align:middle line:84% And that number, which we call the Riemann integral of f, 00:03:05.830 --> 00:03:10.570 align:middle line:84% we interpret as the area underneath the graph of f. 00:03:10.570 --> 00:03:12.760 align:middle line:90% So the first goal-- 00:03:12.760 --> 00:03:15.490 align:middle line:84% the main goal, really, to start off with-- 00:03:15.490 --> 00:03:19.180 align:middle line:84% is to show that this is a reasonable-- 00:03:19.180 --> 00:03:25.400 align:middle line:84% that this is actually true that, these Riemann sums do, in fact, 00:03:25.400 --> 00:03:30.010 align:middle line:84% converge to some number as they become 00:03:30.010 --> 00:03:34.370 align:middle line:84% finer and finer for a given continuous function. 00:03:34.370 --> 00:03:38.770 align:middle line:90% 00:03:38.770 --> 00:03:40.980 align:middle line:84% So that is the main theorem, which 00:03:40.980 --> 00:03:46.500 align:middle line:84% I think is going to be the main thing that we prove today. 00:03:46.500 --> 00:03:55.560 align:middle line:84% So theorem of the Riemann integral, 00:03:55.560 --> 00:03:56.720 align:middle line:90% which is the following-- 00:03:56.720 --> 00:04:00.080 align:middle line:90% 00:04:00.080 --> 00:04:05.120 align:middle line:84% let f be a continuous function from a b to r-- 00:04:05.120 --> 00:04:07.010 align:middle line:84% which, remember, this is the notation 00:04:07.010 --> 00:04:09.750 align:middle line:90% we used from last time. 00:04:09.750 --> 00:04:13.200 align:middle line:84% Let me move my picture over here. 00:04:13.200 --> 00:04:18.980 align:middle line:90% 00:04:18.980 --> 00:04:28.360 align:middle line:84% Then there exists a unique real number, which 00:04:28.360 --> 00:04:38.540 align:middle line:84% we denote as the usual symbol, integral a b f 00:04:38.540 --> 00:04:45.730 align:middle line:84% of x dx with the following property-- 00:04:45.730 --> 00:04:53.020 align:middle line:90% 00:04:53.020 --> 00:05:13.400 align:middle line:84% for every sequence of partitions xr cr, 00:05:13.400 --> 00:05:20.750 align:middle line:84% such that the norm of these partitions is going to 0 as r 00:05:20.750 --> 00:05:24.320 align:middle line:84% goes to infinity-- so these partitions are 00:05:24.320 --> 00:05:28.760 align:middle line:84% getting finer and finer if the norm is going to 0. 00:05:28.760 --> 00:05:31.370 align:middle line:84% Remember, the norm is the length of the longest subinterval 00:05:31.370 --> 00:05:33.500 align:middle line:90% I have. 00:05:33.500 --> 00:05:36.080 align:middle line:84% So for all sequence of partitions, 00:05:36.080 --> 00:05:41.310 align:middle line:84% with norms converging to 0, we have 00:05:41.310 --> 00:05:44.880 align:middle line:84% that the limit of the Riemann sums 00:05:44.880 --> 00:06:00.830 align:middle line:84% exists and equals this number, which 00:06:00.830 --> 00:06:04.210 align:middle line:84% we refer to as the Riemann integral of f. 00:06:04.210 --> 00:06:05.866 align:middle line:90% OK? 00:06:05.866 --> 00:06:08.120 align:middle line:90% So there's a lot to unpack here. 00:06:08.120 --> 00:06:12.970 align:middle line:84% First off, what this number, integral from a b 00:06:12.970 --> 00:06:15.850 align:middle line:84% to f of x dx-- this property that it has is, 00:06:15.850 --> 00:06:17.680 align:middle line:84% no matter what sequence of partitions 00:06:17.680 --> 00:06:21.310 align:middle line:84% you take, as long as they're becoming finer and finer, 00:06:21.310 --> 00:06:24.940 align:middle line:84% this limit actually exist, and it equals the same number. 00:06:24.940 --> 00:06:28.360 align:middle line:84% It equals this number, integral a b f of x dx. 00:06:28.360 --> 00:06:32.260 align:middle line:84% So I could take two different sequences of partitions 00:06:32.260 --> 00:06:35.680 align:middle line:84% becoming finer and finer, look at the Riemann sums, 00:06:35.680 --> 00:06:40.360 align:middle line:84% and those two sequences of Riemann sums converge, 00:06:40.360 --> 00:06:43.630 align:middle line:84% and they converge to the same number-- 00:06:43.630 --> 00:06:45.765 align:middle line:84% again, which we denote by integral a b f 00:06:45.765 --> 00:06:49.480 align:middle line:84% of x dx, the Riemann integral of f. 00:06:49.480 --> 00:06:52.150 align:middle line:84% And so it's this number, which we 00:06:52.150 --> 00:06:57.340 align:middle line:84% interpret as the area underneath the graph of f. 00:06:57.340 --> 00:06:57.840 align:middle line:90% OK? 00:06:57.840 --> 00:07:02.270 align:middle line:90% 00:07:02.270 --> 00:07:05.915 align:middle line:84% All right, so our main goal for today is to prove this theorem. 00:07:05.915 --> 00:07:14.400 align:middle line:90% 00:07:14.400 --> 00:07:21.450 align:middle line:84% So first off, there's two parts of the statement here. 00:07:21.450 --> 00:07:23.040 align:middle line:84% There exists a unique real number, 00:07:23.040 --> 00:07:26.550 align:middle line:84% which we denoted by that with this property. 00:07:26.550 --> 00:07:28.680 align:middle line:90% Uniqueness is immediate. 00:07:28.680 --> 00:07:34.510 align:middle line:84% If I have two real numbers, such that, 00:07:34.510 --> 00:07:37.810 align:middle line:84% for all sequence of partitions with norm going to 0, 00:07:37.810 --> 00:07:39.640 align:middle line:84% the limit of the Riemann sums equals 00:07:39.640 --> 00:07:44.830 align:middle line:84% that limit, well, limits of sequences are unique, 00:07:44.830 --> 00:07:47.330 align:middle line:84% and therefore, those two numbers have to be the same. 00:07:47.330 --> 00:07:48.580 align:middle line:90% So uniqueness is clear. 00:07:48.580 --> 00:07:51.850 align:middle line:84% It's really the existence part that we have to prove, 00:07:51.850 --> 00:07:54.250 align:middle line:84% that there exists a real number so that, 00:07:54.250 --> 00:07:57.700 align:middle line:84% no matter what sequence of partitions we take, 00:07:57.700 --> 00:07:59.920 align:middle line:84% the corresponding Riemann sums converges to 0-- 00:07:59.920 --> 00:08:01.090 align:middle line:90% as long as they become-- 00:08:01.090 --> 00:08:03.760 align:middle line:84% the partitions become finer and finer. 00:08:03.760 --> 00:08:04.260 align:middle line:90% OK? 00:08:04.260 --> 00:08:08.110 align:middle line:90% 00:08:08.110 --> 00:08:08.610 align:middle line:90% OK. 00:08:08.610 --> 00:08:11.700 align:middle line:90% 00:08:11.700 --> 00:08:15.430 align:middle line:84% Now, we're not going to prove this theorem just yet. 00:08:15.430 --> 00:08:19.540 align:middle line:84% We need some facts that will be used in the proof, 00:08:19.540 --> 00:08:21.420 align:middle line:84% so we're going to put off proving 00:08:21.420 --> 00:08:25.590 align:middle line:84% this theorem for a few minutes and first prove 00:08:25.590 --> 00:08:27.120 align:middle line:90% some necessary facts. 00:08:27.120 --> 00:08:41.419 align:middle line:84% So let me first define a useful metric or number associated 00:08:41.419 --> 00:08:44.510 align:middle line:84% to-- or function associated to a continuous function called 00:08:44.510 --> 00:08:46.190 align:middle line:90% the modulus of continuity. 00:08:46.190 --> 00:09:00.520 align:middle line:84% So for f continuous function and eta positive, 00:09:00.520 --> 00:09:07.710 align:middle line:84% we define the modulus continuity omega f of eta-- 00:09:07.710 --> 00:09:15.060 align:middle line:84% this is equal to the supremum of f of x minus f of y [INAUDIBLE] 00:09:15.060 --> 00:09:18.870 align:middle line:84% absolute value, such that the absolute value of x minus y 00:09:18.870 --> 00:09:21.890 align:middle line:84% is less than or equal to [? eta. ?] OK? 00:09:21.890 --> 00:09:24.970 align:middle line:90% 00:09:24.970 --> 00:09:28.720 align:middle line:84% So what does this modulus of continuity measure? 00:09:28.720 --> 00:09:35.700 align:middle line:84% It measures, given-- if I take any two points less than 00:09:35.700 --> 00:09:40.800 align:middle line:84% or equal to eta in distance, and I look f of x minus f of y 00:09:40.800 --> 00:09:43.710 align:middle line:84% in absolute value, and take the sup over all that, 00:09:43.710 --> 00:09:48.480 align:middle line:84% that gives me the modulus of continuity. 00:09:48.480 --> 00:09:53.220 align:middle line:84% Now, for example, let's do the simplest example 00:09:53.220 --> 00:09:58.860 align:middle line:84% possible, f of x equals ax plus b. 00:09:58.860 --> 00:10:03.260 align:middle line:90% 00:10:03.260 --> 00:10:06.310 align:middle line:90% Then what do we get? 00:10:06.310 --> 00:10:09.460 align:middle line:90% f of x minus f of y-- 00:10:09.460 --> 00:10:14.515 align:middle line:84% this is equal to a times x minus y. 00:10:14.515 --> 00:10:20.550 align:middle line:90% 00:10:20.550 --> 00:10:27.540 align:middle line:84% So if x minus y is less than or equal to eta, 00:10:27.540 --> 00:10:35.200 align:middle line:84% we get that f of x minus f of y in absolute value 00:10:35.200 --> 00:10:36.910 align:middle line:90% is less than or equal to-- 00:10:36.910 --> 00:10:41.770 align:middle line:84% well, it's equal to a times x minus y 00:10:41.770 --> 00:10:46.610 align:middle line:84% is less than or equal to a times eta. 00:10:46.610 --> 00:10:49.000 align:middle line:90% Right? 00:10:49.000 --> 00:10:52.950 align:middle line:84% And therefore, absolute value of a times eta 00:10:52.950 --> 00:10:57.810 align:middle line:84% is an upper bound for this set for this function. 00:10:57.810 --> 00:10:59.310 align:middle line:90% And it's also achieved. 00:10:59.310 --> 00:11:02.490 align:middle line:84% If I take absolute value of x minus y equal to eta, 00:11:02.490 --> 00:11:05.190 align:middle line:90% then this will be equality. 00:11:05.190 --> 00:11:10.410 align:middle line:84% So what I'm saying is that, for this example, 00:11:10.410 --> 00:11:15.630 align:middle line:84% for f equals ax plus b, the modulus of continuity 00:11:15.630 --> 00:11:19.470 align:middle line:84% is equal to a absolute value of a times eta. 00:11:19.470 --> 00:11:22.490 align:middle line:90% 00:11:22.490 --> 00:11:22.990 align:middle line:90% OK. 00:11:22.990 --> 00:11:27.510 align:middle line:90% 00:11:27.510 --> 00:11:31.620 align:middle line:84% Now, something to note here is that, for this guy right here, 00:11:31.620 --> 00:11:35.955 align:middle line:84% as eta goes to 0, the modulus of continuity goes to 0 as well. 00:11:35.955 --> 00:11:38.720 align:middle line:90% 00:11:38.720 --> 00:11:43.710 align:middle line:84% And this is not special to this function. 00:11:43.710 --> 00:11:46.290 align:middle line:84% In fact, it's true for all continuous functions. 00:11:46.290 --> 00:11:49.160 align:middle line:90% 00:11:49.160 --> 00:11:51.050 align:middle line:84% So let me state the theorem and then give you 00:11:51.050 --> 00:11:52.770 align:middle line:90% an interpretation of it. 00:11:52.770 --> 00:11:54.380 align:middle line:90% So let me call this theorem 1. 00:11:54.380 --> 00:11:58.450 align:middle line:90% 00:11:58.450 --> 00:12:04.150 align:middle line:84% So for all f continuous on the closed and bounded interval 00:12:04.150 --> 00:12:15.940 align:middle line:84% in a, b, limit as eta goes to 0 of omega f of eta equals 0. 00:12:15.940 --> 00:12:16.440 align:middle line:90% OK. 00:12:16.440 --> 00:12:24.270 align:middle line:90% 00:12:24.270 --> 00:12:27.690 align:middle line:84% So let me make one more comment here. 00:12:27.690 --> 00:12:29.712 align:middle line:90% Maybe this will also-- 00:12:29.712 --> 00:12:31.170 align:middle line:84% just to connect to something we did 00:12:31.170 --> 00:12:33.630 align:middle line:84% earlier, which follows immediately 00:12:33.630 --> 00:12:43.880 align:middle line:84% from the definition of this guy, that for all x and y, 00:12:43.880 --> 00:12:49.270 align:middle line:90% this is true. 00:12:49.270 --> 00:12:51.800 align:middle line:90% OK? 00:12:51.800 --> 00:12:54.170 align:middle line:84% And therefore, you should think of, 00:12:54.170 --> 00:12:58.100 align:middle line:84% I'm trying to make this small, then the modulus of continuity 00:12:58.100 --> 00:13:01.940 align:middle line:84% is something that controls how close f of x and f of y 00:13:01.940 --> 00:13:02.640 align:middle line:90% are together. 00:13:02.640 --> 00:13:06.260 align:middle line:84% So somehow continuity is controlled by this function, 00:13:06.260 --> 00:13:08.600 align:middle line:90% the modulus of continuity. 00:13:08.600 --> 00:13:10.595 align:middle line:90% So if this goes to 0, then-- 00:13:10.595 --> 00:13:13.860 align:middle line:90% 00:13:13.860 --> 00:13:16.110 align:middle line:84% as this goes to 0, this goes to 0, 00:13:16.110 --> 00:13:19.295 align:middle line:84% then somehow I'm saying f of x and f of y 00:13:19.295 --> 00:13:20.295 align:middle line:90% are very close together. 00:13:20.295 --> 00:13:23.040 align:middle line:90% 00:13:23.040 --> 00:13:25.050 align:middle line:84% So again, the modulus of continuity 00:13:25.050 --> 00:13:30.160 align:middle line:84% is a way of measuring continuity of a function. 00:13:30.160 --> 00:13:31.410 align:middle line:90% That's one way to think of it. 00:13:31.410 --> 00:13:37.530 align:middle line:90% 00:13:37.530 --> 00:13:43.476 align:middle line:90% All right, so let's prove this. 00:13:43.476 --> 00:13:47.000 align:middle line:84% This is a limited statement, and what we're going to do 00:13:47.000 --> 00:13:50.090 align:middle line:84% is we'll just prove it the old-fashioned way. 00:13:50.090 --> 00:13:51.650 align:middle line:90% Let epsilon be positive. 00:13:51.650 --> 00:13:56.680 align:middle line:90% 00:13:56.680 --> 00:13:58.390 align:middle line:84% What do we know about functions which 00:13:58.390 --> 00:14:00.610 align:middle line:84% are continuous on a closed and bounded interval? 00:14:00.610 --> 00:14:03.790 align:middle line:84% While, they're also uniformly continuous. 00:14:03.790 --> 00:14:13.010 align:middle line:84% So since f is continuous on this closed and bounded interval, 00:14:13.010 --> 00:14:18.310 align:middle line:90% f is uniformly continuous. 00:14:18.310 --> 00:14:26.740 align:middle line:84% This means that there exists delta 0 positive, 00:14:26.740 --> 00:14:35.580 align:middle line:84% such that, for all x and y, x and y less than delta 0 00:14:35.580 --> 00:14:43.557 align:middle line:84% implies that f of x minus f of y is less than-- 00:14:43.557 --> 00:14:44.640 align:middle line:90% let me put epsilon over 2. 00:14:44.640 --> 00:14:46.780 align:middle line:84% I'm going to give myself a little space here. 00:14:46.780 --> 00:14:47.280 align:middle line:90% OK? 00:14:47.280 --> 00:14:51.210 align:middle line:90% 00:14:51.210 --> 00:14:54.570 align:middle line:84% So just to recall, what we're trying 00:14:54.570 --> 00:14:57.450 align:middle line:84% to do here in terms of the definition of what this means-- 00:14:57.450 --> 00:15:00.930 align:middle line:84% we're trying to show that, for all epsilon positive, 00:15:00.930 --> 00:15:09.965 align:middle line:84% there exists delta positive, such that, for all eta-- 00:15:09.965 --> 00:15:12.660 align:middle line:90% 00:15:12.660 --> 00:15:14.940 align:middle line:84% eta's actually a positive number here, so-- 00:15:14.940 --> 00:15:17.930 align:middle line:90% 00:15:17.930 --> 00:15:22.480 align:middle line:84% implies omega of eta is less than epsilon. 00:15:22.480 --> 00:15:22.980 align:middle line:90% OK? 00:15:22.980 --> 00:15:31.580 align:middle line:90% 00:15:31.580 --> 00:15:32.080 align:middle line:90% OK. 00:15:32.080 --> 00:15:35.794 align:middle line:90% 00:15:35.794 --> 00:15:39.130 align:middle line:84% All right, so this is what we're trying to prove. 00:15:39.130 --> 00:15:41.030 align:middle line:84% We know that, for a continuous function, 00:15:41.030 --> 00:15:43.330 align:middle line:84% it's uniformly continuous, and therefore, there exists 00:15:43.330 --> 00:15:47.410 align:middle line:84% is delta 0 so that, when x minus y is less than delta 0, 00:15:47.410 --> 00:15:50.380 align:middle line:84% f of x minus f of y is less than epsilon over 2 for all 00:15:50.380 --> 00:15:52.010 align:middle line:90% x, y in the interval. 00:15:52.010 --> 00:15:52.510 align:middle line:90% OK? 00:15:52.510 --> 00:15:58.710 align:middle line:90% 00:15:58.710 --> 00:16:01.170 align:middle line:84% So basically, I'm going to choose delta 00:16:01.170 --> 00:16:08.568 align:middle line:84% to be this delta 0, the delta I need for this. 00:16:08.568 --> 00:16:09.985 align:middle line:84% And now let me show that it works. 00:16:09.985 --> 00:16:13.050 align:middle line:90% 00:16:13.050 --> 00:16:17.480 align:middle line:84% Suppose eta is less than delta, which, remember, is delta 0. 00:16:17.480 --> 00:16:23.990 align:middle line:84% Then, if x minus y is less than eta-- 00:16:23.990 --> 00:16:26.640 align:middle line:90% which, again, is less than-- 00:16:26.640 --> 00:16:29.420 align:middle line:84% less than or equal to eta, which is less than delta 0-- 00:16:29.420 --> 00:16:37.970 align:middle line:84% I get that f of x minus f of y is less than epsilon over 2. 00:16:37.970 --> 00:16:41.390 align:middle line:84% Basically, I just rewrote what I had right there. 00:16:41.390 --> 00:16:50.860 align:middle line:84% And therefore, this implies that epsilon over 2 is an upper 00:16:50.860 --> 00:17:03.440 align:middle line:84% bound for the set f of x minus f of y, x minus y, 00:17:03.440 --> 00:17:09.380 align:middle line:84% which is less than eta, which implies that the supremum 00:17:09.380 --> 00:17:11.510 align:middle line:84% of this set-- which is, by definition-- 00:17:11.510 --> 00:17:18.760 align:middle line:90% 00:17:18.760 --> 00:17:21.569 align:middle line:84% so I'm just going to put brackets here, 00:17:21.569 --> 00:17:24.240 align:middle line:84% and I hope that's clear that I'm referring to this set 00:17:24.240 --> 00:17:26.220 align:middle line:90% here when I put that here-- 00:17:26.220 --> 00:17:27.780 align:middle line:84% that the supremum of this set has 00:17:27.780 --> 00:17:31.440 align:middle line:84% to be less than or equal to this upper bound, which 00:17:31.440 --> 00:17:32.860 align:middle line:90% is less than epsilon. 00:17:32.860 --> 00:17:33.360 align:middle line:90% OK? 00:17:33.360 --> 00:17:38.530 align:middle line:84% So we've proven that, if eta is less than delta, 00:17:38.530 --> 00:17:44.440 align:middle line:84% then the modulus of continuity of f-- 00:17:44.440 --> 00:17:46.760 align:middle line:90% of eta is less than epsilon. 00:17:46.760 --> 00:17:49.225 align:middle line:84% And therefore, this is what we wanted to prove. 00:17:49.225 --> 00:18:07.440 align:middle line:90% 00:18:07.440 --> 00:18:07.940 align:middle line:90% OK. 00:18:07.940 --> 00:18:22.010 align:middle line:90% 00:18:22.010 --> 00:18:28.810 align:middle line:84% All right, so that's one fact that I need, 00:18:28.810 --> 00:18:31.420 align:middle line:84% that this modulus of continuity-- which, remember, 00:18:31.420 --> 00:18:33.670 align:middle line:84% is the supremum of the difference of f of x and f 00:18:33.670 --> 00:18:36.970 align:middle line:84% of y, as long as x and y are bounded by eta-- 00:18:36.970 --> 00:18:39.250 align:middle line:84% that this converges to 0 as eta converges 00:18:39.250 --> 00:18:42.110 align:middle line:84% to 0 for all continuous functions. 00:18:42.110 --> 00:18:42.610 align:middle line:90% OK? 00:18:42.610 --> 00:18:49.980 align:middle line:84% And this is key in showing that the Riemann integral exists 00:18:49.980 --> 00:18:51.420 align:middle line:90% for continuous functions. 00:18:51.420 --> 00:18:54.570 align:middle line:90% 00:18:54.570 --> 00:18:56.880 align:middle line:90% So theorem two is the following. 00:18:56.880 --> 00:19:02.540 align:middle line:84% So this is going to tell us how two Riemann 00:19:02.540 --> 00:19:06.860 align:middle line:84% sums are comparable if one partition is 00:19:06.860 --> 00:19:08.640 align:middle line:90% finer than the other. 00:19:08.640 --> 00:19:28.480 align:middle line:84% So if I have two tagged partitions of a, b, 00:19:28.480 --> 00:19:34.850 align:middle line:84% such that x prime is a subset of x-- 00:19:34.850 --> 00:19:37.800 align:middle line:90% so this means that x prime-- 00:19:37.800 --> 00:19:40.370 align:middle line:84% so this is a partition-- a breakup of a, b-- 00:19:40.370 --> 00:19:44.160 align:middle line:84% that contains all the points of x and more. 00:19:44.160 --> 00:19:44.660 align:middle line:90% OK? 00:19:44.660 --> 00:19:46.220 align:middle line:90% So it's a finer partition. 00:19:46.220 --> 00:19:49.170 align:middle line:90% 00:19:49.170 --> 00:19:54.700 align:middle line:84% And we refer to x prime as a refinement of the partition x. 00:19:54.700 --> 00:20:03.050 align:middle line:90% 00:20:03.050 --> 00:20:05.480 align:middle line:84% All right, so we have two partitions. 00:20:05.480 --> 00:20:10.340 align:middle line:84% One is finer than the other, and a continuous function f. 00:20:10.340 --> 00:20:13.760 align:middle line:84% Then we can estimate the difference in the Riemann sums. 00:20:13.760 --> 00:20:21.660 align:middle line:90% 00:20:21.660 --> 00:20:26.300 align:middle line:84% This is less than or equal to the modulus of continuity 00:20:26.300 --> 00:20:33.770 align:middle line:84% of f evaluated at the norm of the courser 00:20:33.770 --> 00:20:36.305 align:middle line:84% partition times the length of the interval. 00:20:36.305 --> 00:20:38.810 align:middle line:90% 00:20:38.810 --> 00:20:39.310 align:middle line:90% OK? 00:20:39.310 --> 00:20:50.860 align:middle line:90% 00:20:50.860 --> 00:20:53.950 align:middle line:90% OK, so this says, as long as-- 00:20:53.950 --> 00:21:00.820 align:middle line:84% one way to think about it is, if the partition x is very fine, 00:21:00.820 --> 00:21:03.220 align:middle line:84% and then I take a finer partition, that's not 00:21:03.220 --> 00:21:05.470 align:middle line:84% going to change the Riemann sum too much, 00:21:05.470 --> 00:21:06.970 align:middle line:84% because remember, this is converging 00:21:06.970 --> 00:21:10.710 align:middle line:84% to 0 as the norm of the partition-- how fine it is, 00:21:10.710 --> 00:21:13.990 align:middle line:84% or you can think how coarse it is-- is going to 0. 00:21:13.990 --> 00:21:20.270 align:middle line:90% 00:21:20.270 --> 00:21:22.350 align:middle line:90% OK, so the idea is very simple. 00:21:22.350 --> 00:21:25.220 align:middle line:84% It's just going to take me a little bit just to write down. 00:21:25.220 --> 00:21:34.730 align:middle line:90% 00:21:34.730 --> 00:21:39.020 align:middle line:84% Like I said, this partition being contained in the other 00:21:39.020 --> 00:21:45.170 align:middle line:84% means this partition has all the points in x and then some more. 00:21:45.170 --> 00:21:48.770 align:middle line:84% We're going to break these partitions up into-- 00:21:48.770 --> 00:21:52.400 align:middle line:84% or at least this one up into parts 00:21:52.400 --> 00:21:57.500 align:middle line:84% where it's points in x and then plus the extra one. 00:21:57.500 --> 00:22:01.205 align:middle line:90% So for k equals 1 to n-- 00:22:01.205 --> 00:22:03.080 align:middle line:84% so let me draw a picture first, and then I'll 00:22:03.080 --> 00:22:06.020 align:middle line:90% define the concepts that I have. 00:22:06.020 --> 00:22:15.060 align:middle line:84% So here we have the partition points of x. 00:22:15.060 --> 00:22:17.470 align:middle line:90% OK? 00:22:17.470 --> 00:22:21.325 align:middle line:84% And above I'm going to write now-- 00:22:21.325 --> 00:22:22.700 align:middle line:84% I used to write [? xe, ?] but I'm 00:22:22.700 --> 00:22:25.190 align:middle line:84% going to write the partition points 00:22:25.190 --> 00:22:26.690 align:middle line:90% corresponding to x prime. 00:22:26.690 --> 00:22:31.820 align:middle line:84% So this should be x prime sub l for some l, because remember, 00:22:31.820 --> 00:22:39.315 align:middle line:84% [INAUDIBLE] this one is contained in the other. 00:22:39.315 --> 00:22:43.610 align:middle line:90% 00:22:43.610 --> 00:22:48.610 align:middle line:84% So this partition point will be x prime sub l for some l. 00:22:48.610 --> 00:22:53.960 align:middle line:84% And then I'll have some other ones, 00:22:53.960 --> 00:22:57.560 align:middle line:84% and so on, until I reach the last partition 00:22:57.560 --> 00:23:01.220 align:middle line:84% point from x prime, which is contained 00:23:01.220 --> 00:23:05.856 align:middle line:84% in this subinterval from the partition x. 00:23:05.856 --> 00:23:06.356 align:middle line:90% OK? 00:23:06.356 --> 00:23:09.480 align:middle line:90% 00:23:09.480 --> 00:23:15.245 align:middle line:90% So I let y with a k upper-- 00:23:15.245 --> 00:23:19.360 align:middle line:90% 00:23:19.360 --> 00:23:28.840 align:middle line:84% this is equal to the partition x sub k minus 1 equals x prime l 00:23:28.840 --> 00:23:33.910 align:middle line:84% plus 1, until I get to some partition point x sub 00:23:33.910 --> 00:23:37.250 align:middle line:90% m, which equals x sub k. 00:23:37.250 --> 00:23:37.750 align:middle line:90% OK? 00:23:37.750 --> 00:23:41.740 align:middle line:84% So this is just a part of the partition x prime, contained 00:23:41.740 --> 00:23:47.010 align:middle line:84% in this subinterval, where these are partition points in x. 00:23:47.010 --> 00:23:51.220 align:middle line:84% And then I'm going to write eta k. 00:23:51.220 --> 00:23:55.940 align:middle line:84% This will be the tags that come with these points. 00:23:55.940 --> 00:24:06.580 align:middle line:90% 00:24:06.580 --> 00:24:07.080 align:middle line:90% OK. 00:24:07.080 --> 00:24:11.430 align:middle line:90% 00:24:11.430 --> 00:24:13.950 align:middle line:84% So what I want to draw attention to is 00:24:13.950 --> 00:24:21.660 align:middle line:84% that the tagged partition x, prime c prime-- 00:24:21.660 --> 00:24:24.070 align:middle line:84% this is equal to let me write it this way. 00:24:24.070 --> 00:24:27.150 align:middle line:84% Since these are sets what I mean by-- 00:24:27.150 --> 00:24:32.410 align:middle line:84% let me write it as the union of the tagged partitions. 00:24:32.410 --> 00:24:42.690 align:middle line:90% 00:24:42.690 --> 00:24:45.790 align:middle line:84% And also, this eta has nothing to do with the eta from before. 00:24:45.790 --> 00:24:46.290 align:middle line:90% All right? 00:24:46.290 --> 00:24:49.080 align:middle line:90% I'm just using this as notation. 00:24:49.080 --> 00:24:55.100 align:middle line:84% So the full partition of x prime and [INAUDIBLE] 00:24:55.100 --> 00:24:58.990 align:middle line:84% prime tagged partition is equal to the union of these smaller 00:24:58.990 --> 00:25:04.280 align:middle line:84% partitions of xk minus 1 and xk as k runs from 1 to k. 00:25:04.280 --> 00:25:04.780 align:middle line:90% OK? 00:25:04.780 --> 00:25:06.500 align:middle line:90% I hope that's clear. 00:25:06.500 --> 00:25:07.000 align:middle line:90% All right. 00:25:07.000 --> 00:25:09.920 align:middle line:90% 00:25:09.920 --> 00:25:14.150 align:middle line:84% All right, so now let's compare the part of the agreement 00:25:14.150 --> 00:25:16.617 align:middle line:84% Riemann for x [? xe ?] on this interval. 00:25:16.617 --> 00:25:18.200 align:middle line:84% So remember, somewhere sitting in here 00:25:18.200 --> 00:25:26.060 align:middle line:84% is the tag corresponding to the k-th tag for this guy. 00:25:26.060 --> 00:25:29.600 align:middle line:84% Let's compare it to the part of the Riemann sum 00:25:29.600 --> 00:25:32.820 align:middle line:84% for the x prime [? xe ?] prime also coming from this interval. 00:25:32.820 --> 00:25:33.320 align:middle line:90% OK. 00:25:33.320 --> 00:25:39.320 align:middle line:90% 00:25:39.320 --> 00:25:45.020 align:middle line:84% In fact, let me make one more remark 00:25:45.020 --> 00:25:46.790 align:middle line:84% that, since this tagged partition is 00:25:46.790 --> 00:25:50.540 align:middle line:84% equal to the union of these tagged partitions, 00:25:50.540 --> 00:25:56.650 align:middle line:84% this implies that the Riemann sum 00:25:56.650 --> 00:25:58.510 align:middle line:84% corresponding to this tagged partition 00:25:58.510 --> 00:26:09.340 align:middle line:84% is equal to the sum of the Riemann sums 00:26:09.340 --> 00:26:15.390 align:middle line:84% corresponding to these partitions of these intervals, 00:26:15.390 --> 00:26:16.840 align:middle line:90% xk minus 1 and xk. 00:26:16.840 --> 00:26:17.340 align:middle line:90% All right? 00:26:17.340 --> 00:26:19.230 align:middle line:84% So these are not partitions of a, b, 00:26:19.230 --> 00:26:23.220 align:middle line:84% but they are partitions of xk minus 1 and xk. 00:26:23.220 --> 00:26:25.920 align:middle line:90% All right. 00:26:25.920 --> 00:26:28.200 align:middle line:90% So I hope all this is clear. 00:26:28.200 --> 00:26:32.760 align:middle line:84% Again, I'm just breaking this Riemann sum up into parts where 00:26:32.760 --> 00:26:35.340 align:middle line:84% now I'm looking at what's happening between each 00:26:35.340 --> 00:26:37.500 align:middle line:84% of the partition points x sub k minus 1 00:26:37.500 --> 00:26:43.710 align:middle line:84% and x sub k coming from the original courser partition x. 00:26:43.710 --> 00:26:44.710 align:middle line:90% All right? 00:26:44.710 --> 00:26:49.320 align:middle line:90% 00:26:49.320 --> 00:26:49.945 align:middle line:90% Now we compute. 00:26:49.945 --> 00:27:02.470 align:middle line:90% 00:27:02.470 --> 00:27:06.790 align:middle line:84% So this is a part of the Riemann sum 00:27:06.790 --> 00:27:14.050 align:middle line:84% of x [? xe ?] from this interval minus the part of the agreement 00:27:14.050 --> 00:27:22.410 align:middle line:84% some for x prime [? xe ?] prime coming from this interval. 00:27:22.410 --> 00:27:25.100 align:middle line:90% 00:27:25.100 --> 00:27:30.280 align:middle line:84% Now, this is equal to f of [? xe ?] k. 00:27:30.280 --> 00:27:34.855 align:middle line:84% And so let me just rewrite this sum here. 00:27:34.855 --> 00:27:37.610 align:middle line:90% 00:27:37.610 --> 00:27:40.340 align:middle line:84% x sub k minus x sub k minus 1-- this 00:27:40.340 --> 00:27:54.340 align:middle line:84% is equal to sum from k equals, let's say, j equals l plus 1 00:27:54.340 --> 00:28:01.910 align:middle line:84% to m x prime sub j minus [? x of ?] [INAUDIBLE] minus 1. 00:28:01.910 --> 00:28:06.690 align:middle line:84% So of the smaller ones that are in there, 00:28:06.690 --> 00:28:08.070 align:middle line:90% this is just a telescoping sum. 00:28:08.070 --> 00:28:12.260 align:middle line:84% So all I pick up is x sub j prime sub m, which is x sub k, 00:28:12.260 --> 00:28:16.370 align:middle line:84% minus the first one, which is x sub-- x prime sub l, 00:28:16.370 --> 00:28:19.190 align:middle line:84% which is x sub k minus 1, minus-- 00:28:19.190 --> 00:28:23.720 align:middle line:84% and then this is, by definition, equal to j equals l plus 1 00:28:23.720 --> 00:28:35.000 align:middle line:84% [INAUDIBLE] f of prime j, x prime j minus x minus 1-- 00:28:35.000 --> 00:28:39.820 align:middle line:90% 00:28:39.820 --> 00:28:42.010 align:middle line:84% and then a absolute value sign on the outside. 00:28:42.010 --> 00:28:47.430 align:middle line:84% And then I combine these two, and-- 00:28:47.430 --> 00:28:55.930 align:middle line:84% this is equal to sum from j equals l plus 1 to k of k 00:28:55.930 --> 00:29:10.190 align:middle line:84% minus j times x prime j minus x prime j minus 1-- 00:29:10.190 --> 00:29:10.865 align:middle line:90% absolute value. 00:29:10.865 --> 00:29:15.220 align:middle line:90% 00:29:15.220 --> 00:29:17.968 align:middle line:84% Now, by the triangle inequality, the absolute value of the sum 00:29:17.968 --> 00:29:20.260 align:middle line:84% is less than or equal to the sum of the absolute value. 00:29:20.260 --> 00:29:22.143 align:middle line:90% So save myself writing-- 00:29:22.143 --> 00:29:24.060 align:middle line:84% I'm going to bring the absolute values inside. 00:29:24.060 --> 00:29:31.280 align:middle line:90% 00:29:31.280 --> 00:29:31.780 align:middle line:90% OK. 00:29:31.780 --> 00:29:34.450 align:middle line:90% 00:29:34.450 --> 00:29:37.150 align:middle line:84% Now, [? xe ?] sub k and xe sub j-- 00:29:37.150 --> 00:29:42.070 align:middle line:84% they're both in the interval x sub k, x sub k minus 1. 00:29:42.070 --> 00:29:45.820 align:middle line:84% And by what I wrote up there, another way of-- 00:29:45.820 --> 00:29:48.340 align:middle line:84% or what follows simply from the definition 00:29:48.340 --> 00:29:51.580 align:middle line:84% of the modulus of continuity-- this is less than or equal 00:29:51.580 --> 00:30:06.280 align:middle line:84% to omega f of x sub k [? x sub ?] 00:30:06.280 --> 00:30:14.275 align:middle line:84% k minus 1 times x times j minus x prime j minus 1. 00:30:14.275 --> 00:30:17.160 align:middle line:90% 00:30:17.160 --> 00:30:22.570 align:middle line:84% And this equals-- now, this doesn't depend on j, 00:30:22.570 --> 00:30:24.890 align:middle line:84% so this is, again just a telescoping sum. 00:30:24.890 --> 00:30:33.350 align:middle line:90% So I just pick up times x-- 00:30:33.350 --> 00:30:38.150 align:middle line:90% 00:30:38.150 --> 00:30:42.550 align:middle line:84% so that's for one piece, and now I just add them up. 00:30:42.550 --> 00:30:48.580 align:middle line:90% 00:30:48.580 --> 00:30:50.400 align:middle line:90% Then I look at-- 00:30:50.400 --> 00:30:59.950 align:middle line:90% 00:30:59.950 --> 00:31:04.410 align:middle line:84% and so like I said, this is equal to the sum of these terms 00:31:04.410 --> 00:31:08.490 align:middle line:84% that look like this from k equals 1 to n. 00:31:08.490 --> 00:31:11.940 align:middle line:84% This is equal to the sum of terms like this from k 00:31:11.940 --> 00:31:12.840 align:middle line:90% equals 1 to n. 00:31:12.840 --> 00:31:14.380 align:middle line:90% That's what I wrote up there. 00:31:14.380 --> 00:31:16.650 align:middle line:84% And so if I apply the triangle inequality, 00:31:16.650 --> 00:31:21.510 align:middle line:84% this is less than or equal to sum from k equals 1 to n of f 00:31:21.510 --> 00:31:37.990 align:middle line:84% of [? xe ?] k times xk minus xk minus 1 minus s [? sub ?] f 00:31:37.990 --> 00:31:42.805 align:middle line:90% eta, which-- 00:31:42.805 --> 00:31:46.240 align:middle line:90% 00:31:46.240 --> 00:31:49.420 align:middle line:90% I have this estimate right here. 00:31:49.420 --> 00:31:54.990 align:middle line:84% I've already estimated-- so started off with this. 00:31:54.990 --> 00:31:57.090 align:middle line:84% And I showed it was less than or equal to this. 00:31:57.090 --> 00:32:01.330 align:middle line:84% So now let me stick that estimate into here-- 00:32:01.330 --> 00:32:07.750 align:middle line:84% less than or equal to sum from k equals 1 to n omega 00:32:07.750 --> 00:32:17.635 align:middle line:84% f of xk minus xk minus 1 times xk minus xk minus 1. 00:32:17.635 --> 00:32:20.320 align:middle line:90% 00:32:20.320 --> 00:32:25.040 align:middle line:84% So another thing which you should 00:32:25.040 --> 00:32:26.810 align:middle line:84% notice about the modulus of continuity 00:32:26.810 --> 00:32:33.170 align:middle line:84% is, if I have something here and then something bigger, 00:32:33.170 --> 00:32:37.630 align:middle line:84% this set gets bigger, and therefore, this sup 00:32:37.630 --> 00:32:38.320 align:middle line:90% gets bigger. 00:32:38.320 --> 00:32:43.790 align:middle line:84% So another property of the modulus of continuity 00:32:43.790 --> 00:32:45.110 align:middle line:90% is that it's increasing. 00:32:45.110 --> 00:32:48.110 align:middle line:84% This just follows from looking at the definition. 00:32:48.110 --> 00:32:50.270 align:middle line:84% There's more points to take a sup over 00:32:50.270 --> 00:32:51.890 align:middle line:90% f this thing gets bigger. 00:32:51.890 --> 00:33:06.390 align:middle line:90% So implies-- OK? 00:33:06.390 --> 00:33:11.790 align:middle line:84% So all of these lengths of these subintervals, the distance 00:33:11.790 --> 00:33:13.920 align:middle line:84% between partition points, those are all 00:33:13.920 --> 00:33:18.630 align:middle line:84% bounded by the norm of the partition, the biggest one. 00:33:18.630 --> 00:33:24.240 align:middle line:84% So this is less than or equal to sum from k equals 1 to n omega 00:33:24.240 --> 00:33:33.330 align:middle line:84% f of norm of x times xk minus xk minus 1. 00:33:33.330 --> 00:33:35.580 align:middle line:84% And now, this doesn't depend on k, 00:33:35.580 --> 00:33:38.040 align:middle line:84% so I just pick up the sum from k equals 1 to n 00:33:38.040 --> 00:33:40.590 align:middle line:84% of this, which is a telescoping sum. 00:33:40.590 --> 00:33:54.310 align:middle line:84% All I pick up is x sub n minus x sub 0, which is just b minus a, 00:33:54.310 --> 00:33:56.305 align:middle line:84% which is the estimate we wanted to prove. 00:33:56.305 --> 00:33:59.740 align:middle line:90% 00:33:59.740 --> 00:34:02.505 align:middle line:90% OK? 00:34:02.505 --> 00:34:04.880 align:middle line:84% OK, so now I'm going to do something that's sacrilegious, 00:34:04.880 --> 00:34:06.710 align:middle line:90% as far as board work goes. 00:34:06.710 --> 00:34:11.000 align:middle line:84% And I'm going to erase this board 00:34:11.000 --> 00:34:12.980 align:middle line:84% and prove the next theorem on it, because I 00:34:12.980 --> 00:34:15.199 align:middle line:90% want to keep what I prove-- 00:34:15.199 --> 00:34:17.719 align:middle line:84% or at least a statement of what I prove around 00:34:17.719 --> 00:34:24.590 align:middle line:84% for when we finally tackled the proof of the existence 00:34:24.590 --> 00:34:25.590 align:middle line:90% of the Riemann integral. 00:34:25.590 --> 00:34:31.070 align:middle line:84% So this is in permanent form on-- 00:34:31.070 --> 00:34:36.350 align:middle line:84% in digital form, and also in the notes. 00:34:36.350 --> 00:34:40.790 align:middle line:84% Also, I should mention, we're not following the textbook now. 00:34:40.790 --> 00:34:47.630 align:middle line:84% So you have to read the lecture notes or watch the lecture. 00:34:47.630 --> 00:34:51.760 align:middle line:90% 00:34:51.760 --> 00:34:58.250 align:middle line:84% All right, so this theorem here, theorem two, 00:34:58.250 --> 00:35:03.230 align:middle line:84% allowed us to compare two Riemann sums, 00:35:03.230 --> 00:35:08.060 align:middle line:84% as long as one partition is finer than the other. 00:35:08.060 --> 00:35:10.730 align:middle line:84% But there's a simple trick which we can now 00:35:10.730 --> 00:35:18.080 align:middle line:84% use to prove how to compare any two Riemann sums for any two 00:35:18.080 --> 00:35:18.930 align:middle line:90% partitions. 00:35:18.930 --> 00:35:24.755 align:middle line:84% So we have the following, theorem three. 00:35:24.755 --> 00:35:30.820 align:middle line:90% 00:35:30.820 --> 00:35:42.550 align:middle line:84% If xc and-- now they're any tagged partitions-- 00:35:42.550 --> 00:35:48.230 align:middle line:90% 00:35:48.230 --> 00:35:57.520 align:middle line:84% and f is a continuous function, then 00:35:57.520 --> 00:36:01.810 align:middle line:84% I look at the Riemann sum of this guy 00:36:01.810 --> 00:36:08.800 align:middle line:84% and then compare it to the Riemann sum of the other guy. 00:36:08.800 --> 00:36:13.840 align:middle line:84% This is less than or equal to the modulus of continuity 00:36:13.840 --> 00:36:21.790 align:middle line:84% evaluated at the first partition plus modulus of continuity 00:36:21.790 --> 00:36:26.890 align:middle line:84% applied to the second partition times the length 00:36:26.890 --> 00:36:29.530 align:middle line:90% of the interval. 00:36:29.530 --> 00:36:30.820 align:middle line:90% OK? 00:36:30.820 --> 00:36:32.120 align:middle line:90% What does this say? 00:36:32.120 --> 00:36:36.890 align:middle line:84% This says that, if I take two very fine partitions, 00:36:36.890 --> 00:36:39.920 align:middle line:84% so that the norms are very small, 00:36:39.920 --> 00:36:42.190 align:middle line:84% and therefore, the modulus of continuity evaluated 00:36:42.190 --> 00:36:43.840 align:middle line:84% at these norms a small-- remember, 00:36:43.840 --> 00:36:48.100 align:middle line:84% the modulus of continuity converges to 0 as the argument 00:36:48.100 --> 00:36:48.820 align:middle line:90% goes to 0. 00:36:48.820 --> 00:36:51.370 align:middle line:84% So if I take two very fine partitions, 00:36:51.370 --> 00:36:56.110 align:middle line:84% then the Riemann sums are very close together. 00:36:56.110 --> 00:36:58.550 align:middle line:90% The Riemann sums barely change. 00:36:58.550 --> 00:37:03.460 align:middle line:84% And so now we have some hope to see that sequences of Riemann 00:37:03.460 --> 00:37:05.100 align:middle line:90% sum do, in fact, converge. 00:37:05.100 --> 00:37:09.790 align:middle line:90% 00:37:09.790 --> 00:37:10.990 align:middle line:90% OK, so how do we prove this? 00:37:10.990 --> 00:37:19.670 align:middle line:84% It's a simple trick using the previous theorem 00:37:19.670 --> 00:37:26.130 align:middle line:84% to find a third partition to be the common refinement 00:37:26.130 --> 00:37:26.700 align:middle line:90% of these two. 00:37:26.700 --> 00:37:34.140 align:middle line:84% So take this partition and union it with the second one, 00:37:34.140 --> 00:37:44.020 align:middle line:84% and take a new set of tags to be the union of the two tags. 00:37:44.020 --> 00:37:48.680 align:middle line:90% 00:37:48.680 --> 00:37:51.590 align:middle line:84% So all the partition points of x and x prime-- 00:37:51.590 --> 00:37:56.390 align:middle line:84% I throw them together to get a new partition, x double prime-- 00:37:56.390 --> 00:37:58.970 align:middle line:90% and then the tags as well. 00:37:58.970 --> 00:38:03.230 align:middle line:90% 00:38:03.230 --> 00:38:10.770 align:middle line:90% Then x-- backwards-- 00:38:10.770 --> 00:38:15.120 align:middle line:84% this new partition is finer than x. 00:38:15.120 --> 00:38:17.220 align:middle line:90% It's finer than x prime. 00:38:17.220 --> 00:38:25.380 align:middle line:84% And therefore, by theorem two, we 00:38:25.380 --> 00:38:32.390 align:middle line:84% have this estimate which we can use. 00:38:32.390 --> 00:38:44.390 align:middle line:84% So if I look at the difference of the Riemann sums, 00:38:44.390 --> 00:38:53.220 align:middle line:84% and I add and subtract the Riemann sum corresponding to x 00:38:53.220 --> 00:38:55.020 align:middle line:84% double prime and [? xe ?] double prime, 00:38:55.020 --> 00:38:56.890 align:middle line:84% and then I use the triangle inequality-- 00:38:56.890 --> 00:38:58.500 align:middle line:90% so I'm doing two steps here-- 00:38:58.500 --> 00:39:02.400 align:middle line:84% so add and subtract the Riemann sum corresponding to-- 00:39:02.400 --> 00:39:12.790 align:middle line:90% 00:39:12.790 --> 00:39:15.040 align:middle line:84% and then apply the triangle inequality. 00:39:15.040 --> 00:39:28.760 align:middle line:90% 00:39:28.760 --> 00:39:30.620 align:middle line:84% For both of these, I can now insert 00:39:30.620 --> 00:39:33.890 align:middle line:84% this estimate here from theorem two, where-- 00:39:33.890 --> 00:39:39.750 align:middle line:84% what's the modulus of continuity being evaluated at? 00:39:39.750 --> 00:39:46.350 align:middle line:84% It's being evaluated at the courser partition. 00:39:46.350 --> 00:39:49.530 align:middle line:90% 00:39:49.530 --> 00:39:52.680 align:middle line:84% So x double prime is contained in x, so this is less than 00:39:52.680 --> 00:39:55.140 align:middle line:84% or equal to the modulus of continuity evaluated 00:39:55.140 --> 00:39:58.290 align:middle line:90% at x prime times b minus a. 00:39:58.290 --> 00:40:04.240 align:middle line:84% And then, for this one, x prime is coarser than x double prime. 00:40:04.240 --> 00:40:06.210 align:middle line:84% This one is contained in x prime-- so 00:40:06.210 --> 00:40:13.130 align:middle line:84% plus the modulus of continuity evaluated at x prime times b 00:40:13.130 --> 00:40:15.032 align:middle line:90% minus a. 00:40:15.032 --> 00:40:16.490 align:middle line:84% And that's what we wanted to prove. 00:40:16.490 --> 00:40:21.570 align:middle line:90% 00:40:21.570 --> 00:40:22.878 align:middle line:90% OK. 00:40:22.878 --> 00:40:24.420 align:middle line:84% All right, so now we're in a position 00:40:24.420 --> 00:40:29.145 align:middle line:84% to prove this theorem of the Riemann integral. 00:40:29.145 --> 00:40:44.230 align:middle line:90% 00:40:44.230 --> 00:40:46.830 align:middle line:84% And so what we're going to do is we're first 00:40:46.830 --> 00:40:48.580 align:middle line:84% going to come up with a candidate for what 00:40:48.580 --> 00:40:50.900 align:middle line:90% the integral could be. 00:40:50.900 --> 00:40:52.900 align:middle line:84% And then we're going to show that's the actual-- 00:40:52.900 --> 00:40:57.790 align:middle line:84% that that candidate satisfies the conclusion-- 00:40:57.790 --> 00:40:59.590 align:middle line:84% or this property that, no matter what 00:40:59.590 --> 00:41:02.950 align:middle line:84% partition I take, with norms converging to 0, 00:41:02.950 --> 00:41:10.510 align:middle line:84% the Riemann sums converge to this number. 00:41:10.510 --> 00:41:11.010 align:middle line:90% OK? 00:41:11.010 --> 00:41:14.530 align:middle line:84% So first we have to come up with a candidate number. 00:41:14.530 --> 00:41:15.100 align:middle line:90% All right. 00:41:15.100 --> 00:41:17.800 align:middle line:90% So first, take any-- 00:41:17.800 --> 00:41:20.785 align:middle line:90% so let's fix some partition of-- 00:41:20.785 --> 00:41:32.290 align:middle line:90% 00:41:32.290 --> 00:41:35.770 align:middle line:84% for the Greek letter aficionados out there, this is zeta. 00:41:35.770 --> 00:41:41.932 align:middle line:90% 00:41:41.932 --> 00:41:53.780 align:middle line:84% Let this be a partition of a, b with norm converting 00:41:53.780 --> 00:41:55.790 align:middle line:90% to 0 as r goes to infinity. 00:41:55.790 --> 00:41:59.280 align:middle line:90% 00:41:59.280 --> 00:42:03.330 align:middle line:84% You can always come up with some partition. 00:42:03.330 --> 00:42:06.370 align:middle line:90% So this [INAUDIBLE] tagged. 00:42:06.370 --> 00:42:12.430 align:middle line:90% 00:42:12.430 --> 00:42:17.260 align:middle line:84% You can always come up with one, at least one partition-- 00:42:17.260 --> 00:42:20.050 align:middle line:90% or sequence of partitions. 00:42:20.050 --> 00:42:28.590 align:middle line:84% So this is [? be a ?] sequence of tagged partitions. 00:42:28.590 --> 00:42:30.540 align:middle line:90% OK, now this is finally right. 00:42:30.540 --> 00:42:32.250 align:middle line:84% So you can always come up with a sequence 00:42:32.250 --> 00:42:36.840 align:middle line:84% of tagged partitions of a, b with the norms converging to 0. 00:42:36.840 --> 00:42:39.460 align:middle line:84% Take the first one to be just the whole interval-- 00:42:39.460 --> 00:42:42.640 align:middle line:84% so just left endpoint, right endpoint. 00:42:42.640 --> 00:42:44.560 align:middle line:90% Next one add-- the midpoint. 00:42:44.560 --> 00:42:46.410 align:middle line:84% That's now three partition points. 00:42:46.410 --> 00:42:51.990 align:middle line:84% Now add the midpoint of the previous two intervals you had. 00:42:51.990 --> 00:42:54.150 align:middle line:84% Now add the midpoint of the intervals 00:42:54.150 --> 00:42:56.070 align:middle line:84% you had before, and so on, and that'll 00:42:56.070 --> 00:42:58.080 align:middle line:84% build up a sequence of partitions with norm 00:42:58.080 --> 00:42:58.920 align:middle line:90% converging to 0. 00:42:58.920 --> 00:43:06.940 align:middle line:90% 00:43:06.940 --> 00:43:11.440 align:middle line:84% So I have this fixed sequence of tag partitions, 00:43:11.440 --> 00:43:15.500 align:middle line:84% and I claim that the Riemann sums converge for this guy. 00:43:15.500 --> 00:43:16.000 align:middle line:90% All right? 00:43:16.000 --> 00:43:24.960 align:middle line:84% So claim one-- the sequence of sums-- 00:43:24.960 --> 00:43:27.840 align:middle line:84% this is a sequence of numbers now-- 00:43:27.840 --> 00:43:41.530 align:middle line:90% 00:43:41.530 --> 00:43:42.850 align:middle line:90% OK. 00:43:42.850 --> 00:43:46.205 align:middle line:90% So it converges to some number. 00:43:46.205 --> 00:43:48.205 align:middle line:84% We're going to prove this by showing its Cauchy. 00:43:48.205 --> 00:43:52.050 align:middle line:90% 00:43:52.050 --> 00:43:58.890 align:middle line:84% Remember, Cauchy sequences of real numbers always converge. 00:43:58.890 --> 00:44:06.030 align:middle line:84% That's this completeness property of the real numbers. 00:44:06.030 --> 00:44:13.030 align:middle line:90% 00:44:13.030 --> 00:44:14.140 align:middle line:90% OK. 00:44:14.140 --> 00:44:15.700 align:middle line:84% So there's no other way to do this. 00:44:15.700 --> 00:44:20.720 align:middle line:84% Then, using the definition-- the epsilon [? m ?] definition-- 00:44:20.720 --> 00:44:22.210 align:middle line:90% so let epsilon be positive. 00:44:22.210 --> 00:44:27.730 align:middle line:90% 00:44:27.730 --> 00:44:39.700 align:middle line:84% So by theorem one, there exists a delta positive, 00:44:39.700 --> 00:44:44.740 align:middle line:84% such that if eta is less than delta-- 00:44:44.740 --> 00:44:47.300 align:middle line:90% 00:44:47.300 --> 00:44:52.780 align:middle line:84% I should say-- so we know that the modulus of continuity for f 00:44:52.780 --> 00:44:54.910 align:middle line:90% converges to 0 as eta goes to 0. 00:44:54.910 --> 00:45:00.700 align:middle line:84% So for all eta less than delta, omega f of eta 00:45:00.700 --> 00:45:09.440 align:middle line:84% is less than epsilon over 2 times B minus a. 00:45:09.440 --> 00:45:10.130 align:middle line:90% OK? 00:45:10.130 --> 00:45:12.080 align:middle line:90% Let me put a star by this guy. 00:45:12.080 --> 00:45:29.960 align:middle line:90% 00:45:29.960 --> 00:45:37.400 align:middle line:84% Now, since the norms of these partitions is-- 00:45:37.400 --> 00:45:44.300 align:middle line:84% are converging to 0, there exists natural number m0 such 00:45:44.300 --> 00:45:49.190 align:middle line:84% that, for all r bigger than or equal m0, 00:45:49.190 --> 00:45:58.015 align:middle line:90% the norm is less than delta. 00:45:58.015 --> 00:46:01.620 align:middle line:90% 00:46:01.620 --> 00:46:03.640 align:middle line:90% OK. 00:46:03.640 --> 00:46:13.390 align:middle line:84% And thus, for all r bigger than or equal to m0, 00:46:13.390 --> 00:46:17.800 align:middle line:84% if I look at the modulus of continuity evaluated 00:46:17.800 --> 00:46:26.790 align:middle line:84% at this norm of this partition-- so the norm of this partition 00:46:26.790 --> 00:46:28.650 align:middle line:90% is less than delta. 00:46:28.650 --> 00:46:31.950 align:middle line:84% And if I stick that into the modulus of continuity, 00:46:31.950 --> 00:46:35.340 align:middle line:84% anything I stick into the modulus of continuity which is 00:46:35.340 --> 00:46:37.500 align:middle line:84% less than delta-- which this is-- 00:46:37.500 --> 00:46:47.910 align:middle line:84% I should be less than epsilon over 2 times b minus a. 00:46:47.910 --> 00:46:49.740 align:middle line:84% So really, this is the one I want 00:46:49.740 --> 00:46:52.020 align:middle line:90% to star, maybe not this one. 00:46:52.020 --> 00:46:56.040 align:middle line:84% I am using this one to get it, but this is the key point here. 00:46:56.040 --> 00:46:59.960 align:middle line:90% 00:46:59.960 --> 00:47:02.560 align:middle line:90% OK. 00:47:02.560 --> 00:47:05.500 align:middle line:84% So here we're just using the fact that modulus of continuity 00:47:05.500 --> 00:47:08.740 align:middle line:84% converges to 0 as the argument converges to 0, 00:47:08.740 --> 00:47:11.870 align:middle line:84% and that the norms are converging to 0. 00:47:11.870 --> 00:47:12.370 align:middle line:90% OK? 00:47:12.370 --> 00:47:28.270 align:middle line:90% 00:47:28.270 --> 00:47:29.600 align:middle line:90% OK. 00:47:29.600 --> 00:47:33.890 align:middle line:84% So [? to show something's ?] Cauchy, we 00:47:33.890 --> 00:47:34.848 align:middle line:90% have to now choose m. 00:47:34.848 --> 00:47:36.140 align:middle line:90% I'm going to choose m to be m0. 00:47:36.140 --> 00:47:44.900 align:middle line:90% 00:47:44.900 --> 00:47:48.080 align:middle line:84% [INAUDIBLE] r, r prime bigger than or equal to m-- which, 00:47:48.080 --> 00:47:51.680 align:middle line:90% remember, we've chosen to m0. 00:47:51.680 --> 00:47:59.832 align:middle line:84% We have that the absolute value of s of f xr-- 00:47:59.832 --> 00:48:05.745 align:middle line:84% r, minus the Riemann sum, now with r prime-- 00:48:05.745 --> 00:48:13.080 align:middle line:90% 00:48:13.080 --> 00:48:14.490 align:middle line:90% absolute value. 00:48:14.490 --> 00:48:21.800 align:middle line:84% This is by theorem three, which we proved over there-- 00:48:21.800 --> 00:48:25.520 align:middle line:84% is bounded by the sum of the modulus of continuities 00:48:25.520 --> 00:48:29.150 align:middle line:84% evaluated at the norms times b minus a. 00:48:29.150 --> 00:48:47.391 align:middle line:90% 00:48:47.391 --> 00:48:52.280 align:middle line:84% And now r and r prime are bigger than or equal to m0, and so 00:48:52.280 --> 00:48:53.210 align:middle line:90% by star-- 00:48:53.210 --> 00:48:54.920 align:middle line:90% so this line here. 00:48:54.920 --> 00:48:59.750 align:middle line:90% This is by theorem three. 00:48:59.750 --> 00:49:02.690 align:middle line:84% But now, by star, by our choice of m0, 00:49:02.690 --> 00:49:07.880 align:middle line:84% this is less than epsilon over 2 times 00:49:07.880 --> 00:49:16.610 align:middle line:84% b minus a plus epsilon over 2 b minus a times b minus a. 00:49:16.610 --> 00:49:22.395 align:middle line:84% This is now by star, which equals epsilon. 00:49:22.395 --> 00:49:25.380 align:middle line:90% 00:49:25.380 --> 00:49:28.770 align:middle line:84% And therefore, this sequence of Riemann 00:49:28.770 --> 00:49:33.410 align:middle line:84% sums with respect to this one sequence converges. 00:49:33.410 --> 00:49:33.910 align:middle line:90% OK? 00:49:33.910 --> 00:49:37.030 align:middle line:90% 00:49:37.030 --> 00:49:38.280 align:middle line:90% And so we've proven the claim. 00:49:38.280 --> 00:49:43.210 align:middle line:90% 00:49:43.210 --> 00:49:45.370 align:middle line:84% Let me call the limit of this sequence something. 00:49:45.370 --> 00:49:50.920 align:middle line:84% I'm going to call it L. In fact, let's call it I for integral. 00:49:50.920 --> 00:49:55.360 align:middle line:84% Let I be limit as r goes to infinity of-- 00:49:55.360 --> 00:50:00.190 align:middle line:90% 00:50:00.190 --> 00:50:04.790 align:middle line:84% oh, I messed up my notation earlier. 00:50:04.790 --> 00:50:05.600 align:middle line:90% Sorry about that. 00:50:05.600 --> 00:50:12.780 align:middle line:84% The y's, and, x's and [? xe's ?] should have been y's and zetas. 00:50:12.780 --> 00:50:20.310 align:middle line:84% So that should have been y, zeta, y, zeta, and then 00:50:20.310 --> 00:50:23.340 align:middle line:90% y here, y here. 00:50:23.340 --> 00:50:23.840 align:middle line:90% OK? 00:50:23.840 --> 00:50:27.660 align:middle line:90% 00:50:27.660 --> 00:50:29.710 align:middle line:90% Now that's right. 00:50:29.710 --> 00:50:34.555 align:middle line:84% So I is defined to be the limit of these guys. 00:50:34.555 --> 00:50:40.400 align:middle line:90% 00:50:40.400 --> 00:50:42.930 align:middle line:84% So now we have to do one last thing in order 00:50:42.930 --> 00:50:43.805 align:middle line:90% to prove the theorem. 00:50:43.805 --> 00:50:48.250 align:middle line:90% 00:50:48.250 --> 00:50:54.340 align:middle line:84% I have this number I, and I claim now that I satisfies 00:50:54.340 --> 00:50:56.560 align:middle line:84% the properties of the theorem-- namely, 00:50:56.560 --> 00:51:00.370 align:middle line:84% that if I take any sequence of Riemann sums-- 00:51:00.370 --> 00:51:04.090 align:middle line:84% or any sequence of tagged partitions 00:51:04.090 --> 00:51:06.910 align:middle line:84% and take the Riemann sums, that converges to I. 00:51:06.910 --> 00:51:10.210 align:middle line:84% We've just shown that there is one that converges to. 00:51:10.210 --> 00:51:11.470 align:middle line:90% That's how I is defined. 00:51:11.470 --> 00:51:13.930 align:middle line:90% We took one partition-- 00:51:13.930 --> 00:51:16.630 align:middle line:84% sequence of tagged partitions and showed that the Riemann 00:51:16.630 --> 00:51:19.840 align:middle line:84% sums converge to some number I. So this number 00:51:19.840 --> 00:51:24.970 align:middle line:84% I depends on the partition, which I chose in the beginning. 00:51:24.970 --> 00:51:30.165 align:middle line:84% Now we want to show that this I, in fact, 00:51:30.165 --> 00:51:31.540 align:middle line:84% has that property, that no matter 00:51:31.540 --> 00:51:33.710 align:middle line:84% what partition-- sequence of partitions 00:51:33.710 --> 00:51:36.280 align:middle line:84% I take the Riemann sums converge to I. 00:51:36.280 --> 00:51:39.670 align:middle line:84% So that's the second and last thing I need 00:51:39.670 --> 00:51:41.410 align:middle line:90% to do to prove this theorem. 00:51:41.410 --> 00:51:44.590 align:middle line:90% 00:51:44.590 --> 00:52:08.630 align:middle line:84% So claim-- let x be now any sequence of tagged partitions, 00:52:08.630 --> 00:52:10.940 align:middle line:84% which are becoming finer and finer-- so 00:52:10.940 --> 00:52:12.560 align:middle line:90% with the norm converging to 0. 00:52:12.560 --> 00:52:21.780 align:middle line:90% 00:52:21.780 --> 00:52:26.190 align:middle line:84% Then I want to show that limit as r 00:52:26.190 --> 00:52:29.670 align:middle line:84% goes to infinity of the Riemann sums 00:52:29.670 --> 00:52:35.010 align:middle line:84% corresponding to this sequence of partial sums-- 00:52:35.010 --> 00:52:38.850 align:middle line:84% [INAUDIBLE] partial sums-- this sequence of partitions 00:52:38.850 --> 00:52:42.270 align:middle line:84% exists, and equals this number I, 00:52:42.270 --> 00:52:47.840 align:middle line:84% which I obtained from this one sequence of partitions. 00:52:47.840 --> 00:52:52.630 align:middle line:84% So once I've proven this [INAUDIBLE] 00:52:52.630 --> 00:52:55.600 align:middle line:84% once I've proven this, then I'm done. 00:52:55.600 --> 00:52:56.100 align:middle line:90% OK? 00:52:56.100 --> 00:52:58.170 align:middle line:90% This number I is therefore-- 00:52:58.170 --> 00:53:03.272 align:middle line:90% 00:53:03.272 --> 00:53:05.480 align:middle line:84% satisfies that property that, no matter what sequence 00:53:05.480 --> 00:53:10.290 align:middle line:84% of partitions I take, the sequence 00:53:10.290 --> 00:53:12.870 align:middle line:84% of Riemann sums with respect to these partitions 00:53:12.870 --> 00:53:14.705 align:middle line:90% converges to that number. 00:53:14.705 --> 00:53:16.080 align:middle line:84% Of course, here I'm calling it I, 00:53:16.080 --> 00:53:20.580 align:middle line:84% but we denoted by the integral from a, b of f of x dx. 00:53:20.580 --> 00:53:25.480 align:middle line:90% 00:53:25.480 --> 00:53:31.620 align:middle line:84% So to prove this is not too difficult using 00:53:31.620 --> 00:53:32.830 align:middle line:90% what we have on the board. 00:53:32.830 --> 00:53:37.800 align:middle line:84% So remember, I is the limit with respect to this one sequence 00:53:37.800 --> 00:53:40.200 align:middle line:84% of partitions, and now we have an arbitrary sequence 00:53:40.200 --> 00:53:45.120 align:middle line:84% of partitions, which we want to show the Riemann sums converge 00:53:45.120 --> 00:53:46.560 align:middle line:90% to this number as well. 00:53:46.560 --> 00:53:51.810 align:middle line:90% 00:53:51.810 --> 00:54:02.760 align:middle line:84% So with y sub r-- or yr and zeta r, the partitions from before, 00:54:02.760 --> 00:54:03.330 align:middle line:90% we have-- 00:54:03.330 --> 00:54:07.550 align:middle line:90% 00:54:07.550 --> 00:54:13.540 align:middle line:84% by the triangle inequality, if I look 00:54:13.540 --> 00:54:26.250 align:middle line:84% at the Riemann sum with respect to this arbitrary 00:54:26.250 --> 00:54:27.600 align:middle line:90% partition now-- 00:54:27.600 --> 00:54:30.720 align:middle line:84% subtract I-- I want to show this convergence to 0, 00:54:30.720 --> 00:54:33.240 align:middle line:84% so I'm just going to bound this by something 00:54:33.240 --> 00:54:34.290 align:middle line:90% that converges to 0. 00:54:34.290 --> 00:54:37.560 align:middle line:84% And then, by the usual argument with the squeeze theorem, 00:54:37.560 --> 00:54:40.710 align:middle line:84% that implies this thing converges to 0. 00:54:40.710 --> 00:54:47.270 align:middle line:84% So adding and subtracting the Riemann sum associated 00:54:47.270 --> 00:54:50.540 align:middle line:90% to the partition y and zetas-- 00:54:50.540 --> 00:54:59.150 align:middle line:84% this is less than or equal to in the triangle inequality, 00:54:59.150 --> 00:55:11.705 align:middle line:84% minus sf of [INAUDIBLE] plus now-- 00:55:11.705 --> 00:55:18.050 align:middle line:90% 00:55:18.050 --> 00:55:20.930 align:middle line:90% minus I. 00:55:20.930 --> 00:55:23.760 align:middle line:90% And so I'm almost there. 00:55:23.760 --> 00:55:27.050 align:middle line:84% This thing now-- I use theorem three. 00:55:27.050 --> 00:55:28.715 align:middle line:84% So this is less than or equal to-- 00:55:28.715 --> 00:55:44.720 align:middle line:90% 00:55:44.720 --> 00:55:47.600 align:middle line:90% times b minus a plus-- 00:55:47.600 --> 00:55:56.590 align:middle line:90% 00:55:56.590 --> 00:56:06.170 align:middle line:90% s sub f of i sub r minus I. OK? 00:56:06.170 --> 00:56:16.980 align:middle line:90% 00:56:16.980 --> 00:56:21.330 align:middle line:84% The norm of the x sub r-- this is converging to 0. 00:56:21.330 --> 00:56:23.910 align:middle line:90% So this converges to-- 00:56:23.910 --> 00:56:27.780 align:middle line:84% since the modulus of continuity goes to 0 as the argument goes 00:56:27.780 --> 00:56:30.750 align:middle line:90% to 0, this goes to 0, plus-- 00:56:30.750 --> 00:56:35.700 align:middle line:84% remember, the same thing for this guy-- 00:56:35.700 --> 00:56:42.480 align:middle line:90% this also converges to 0, plus-- 00:56:42.480 --> 00:56:43.650 align:middle line:90% how did we define I? 00:56:43.650 --> 00:56:47.190 align:middle line:84% It was defined as the limit of the Riemann sums corresponding 00:56:47.190 --> 00:56:49.710 align:middle line:90% to this first fixed partition. 00:56:49.710 --> 00:56:52.350 align:middle line:84% And therefore, this thing in absolute value 00:56:52.350 --> 00:56:59.130 align:middle line:84% converges to 0 as r convergence is to infinity. 00:56:59.130 --> 00:57:04.080 align:middle line:84% All right, so the absolute value of this thing, 00:57:04.080 --> 00:57:07.950 align:middle line:84% this Riemann sum, minus I is bounded by something converging 00:57:07.950 --> 00:57:11.160 align:middle line:90% to 0 as r goes to infinity. 00:57:11.160 --> 00:57:19.900 align:middle line:84% And therefore, by the squeeze theorem, 00:57:19.900 --> 00:57:23.595 align:middle line:84% we conclude that the limit as r goes to infinity of s-- 00:57:23.595 --> 00:57:33.220 align:middle line:90% 00:57:33.220 --> 00:57:37.510 align:middle line:84% so in other words, no matter what sequence of partitions 00:57:37.510 --> 00:57:41.000 align:middle line:84% we take, with norms converging to 0, 00:57:41.000 --> 00:57:45.987 align:middle line:84% the Riemann sums converge to this same number I. 00:57:45.987 --> 00:57:47.320 align:middle line:90% And that's the end of the proof. 00:57:47.320 --> 00:57:55.000 align:middle line:90% 00:57:55.000 --> 00:57:57.685 align:middle line:90% OK. 00:57:57.685 --> 00:57:59.580 align:middle line:84% It took a lot of work to do this. 00:57:59.580 --> 00:58:01.250 align:middle line:84% This is the work of Raymond, so-- 00:58:01.250 --> 00:58:03.800 align:middle line:90% 00:58:03.800 --> 00:58:11.165 align:middle line:84% who was one of the greatest mathematicians of all time. 00:58:11.165 --> 00:58:14.440 align:middle line:90% 00:58:14.440 --> 00:58:17.020 align:middle line:84% Weierstrass was definitely one of the greatest analysts 00:58:17.020 --> 00:58:17.590 align:middle line:90% of all time. 00:58:17.590 --> 00:58:20.380 align:middle line:84% Riemann, just as a mathematician in general, 00:58:20.380 --> 00:58:22.795 align:middle line:84% was one of the greatest of all time. 00:58:22.795 --> 00:58:25.636 align:middle line:90% 00:58:25.636 --> 00:58:27.390 align:middle line:90% He not only did analysis-- 00:58:27.390 --> 00:58:30.360 align:middle line:90% 00:58:30.360 --> 00:58:35.170 align:middle line:84% he applied analysis to number theory. 00:58:35.170 --> 00:58:38.060 align:middle line:84% This is the content of the famous Riemann hypothesis, 00:58:38.060 --> 00:58:40.750 align:middle line:84% which maybe you've heard about, which 00:58:40.750 --> 00:58:43.740 align:middle line:90% is worth a million dollars-- 00:58:43.740 --> 00:58:48.520 align:middle line:84% says something about the zeros of a certain function, which 00:58:48.520 --> 00:58:52.190 align:middle line:84% then gives you information about prime numbers. 00:58:52.190 --> 00:58:56.200 align:middle line:84% And he also had deep contributions 00:58:56.200 --> 00:58:59.040 align:middle line:90% to the foundations of geometry-- 00:58:59.040 --> 00:59:01.710 align:middle line:84% or I should say the foundations of differential geometry-- 00:59:01.710 --> 00:59:05.040 align:middle line:90% 00:59:05.040 --> 00:59:11.820 align:middle line:84% but truly some deep stuff going on. 00:59:11.820 --> 00:59:15.900 align:middle line:84% All right, so now we have this notion of a "remand integral. 00:59:15.900 --> 00:59:21.000 align:middle line:84% It's this no integral f of x, which has this property. 00:59:21.000 --> 00:59:22.960 align:middle line:84% No matter what sequence of partitions 00:59:22.960 --> 00:59:26.280 align:middle line:84% I take, with norms converging to 0, 00:59:26.280 --> 00:59:33.070 align:middle line:84% the partial sums converge to this number. 00:59:33.070 --> 00:59:34.800 align:middle line:90% So I'm often going to denote-- 00:59:34.800 --> 00:59:43.685 align:middle line:84% so if you like, this is some alternative notation I'll use. 00:59:43.685 --> 00:59:51.770 align:middle line:90% 00:59:51.770 --> 00:59:55.040 align:middle line:84% Instead of writing f of x dx, I may just 00:59:55.040 --> 01:00:00.030 align:middle line:90% write integral a, b, f. 01:00:00.030 --> 01:00:07.650 align:middle line:84% So this definition of the Riemann integral looks 01:00:07.650 --> 01:00:10.440 align:middle line:84% terrifying if you want to actually try and compute it. 01:00:10.440 --> 01:00:11.220 align:middle line:90% And it is. 01:00:11.220 --> 01:00:14.490 align:middle line:84% So that's the miracle of the fundamental theorem 01:00:14.490 --> 01:00:17.310 align:middle line:84% of calculus is that it gives us a way to compute it. 01:00:17.310 --> 01:00:20.370 align:middle line:84% And that's why I'm not doing any examples of computing it 01:00:20.370 --> 01:00:21.300 align:middle line:90% right now. 01:00:21.300 --> 01:00:27.790 align:middle line:84% But we can still learn about some properties of the Riemann 01:00:27.790 --> 01:00:32.280 align:middle line:90% integral without that to start. 01:00:32.280 --> 01:00:39.280 align:middle line:84% So now we're moving on to some properties of the integral. 01:00:39.280 --> 01:00:41.830 align:middle line:90% 01:00:41.830 --> 01:00:44.620 align:middle line:90% And the first is that it-- 01:00:44.620 --> 01:00:47.740 align:middle line:84% so this is a limiting process, but this limiting process 01:00:47.740 --> 01:00:49.450 align:middle line:90% is linear in f. 01:00:49.450 --> 01:00:54.370 align:middle line:84% If I look at Riemann sums of, let's say, f plus g, 01:00:54.370 --> 01:00:58.960 align:middle line:84% that's equal to the Riemann sum of f plus the Riemann sum of g. 01:00:58.960 --> 01:01:03.904 align:middle line:84% And therefore, since integration is a limit of this-- 01:01:03.904 --> 01:01:06.940 align:middle line:84% of Riemann sums, then integration 01:01:06.940 --> 01:01:08.650 align:middle line:90% should also be linear as well. 01:01:08.650 --> 01:01:17.530 align:middle line:84% And that's the first theorem is the linearity 01:01:17.530 --> 01:01:21.820 align:middle line:84% of the Riemann sum, Riemann integration-- 01:01:21.820 --> 01:01:25.240 align:middle line:90% f and g are a, b. 01:01:25.240 --> 01:01:32.080 align:middle line:90% 01:01:32.080 --> 01:01:41.290 align:middle line:90% And alpha is in R. 01:01:41.290 --> 01:01:46.270 align:middle line:84% So first off, alpha f plus g-- that's going to be-- 01:01:46.270 --> 01:01:48.310 align:middle line:90% so that should not be in a, b. 01:01:48.310 --> 01:01:50.980 align:middle line:90% This should be a continuous-- 01:01:50.980 --> 01:01:52.600 align:middle line:90% it's the end of a long day, OK? 01:01:52.600 --> 01:01:58.140 align:middle line:84% So sorry if I'm losing a little steam here towards the end. 01:01:58.140 --> 01:02:02.567 align:middle line:84% Anyways, alpha f plus g, if f and g are continuous-- 01:02:02.567 --> 01:02:04.150 align:middle line:84% this is a continuous function as well, 01:02:04.150 --> 01:02:07.110 align:middle line:90% so its integral is meaningful. 01:02:07.110 --> 01:02:11.910 align:middle line:84% And its integral is equal to alpha times the integral f 01:02:11.910 --> 01:02:17.030 align:middle line:90% plus the integral of g. 01:02:17.030 --> 01:02:17.530 align:middle line:90% OK? 01:02:17.530 --> 01:02:24.960 align:middle line:90% 01:02:24.960 --> 01:02:25.843 align:middle line:90% So what's the proof? 01:02:25.843 --> 01:02:27.510 align:middle line:84% The proof is basically what I said right 01:02:27.510 --> 01:02:32.760 align:middle line:84% before I stated the theorem, that Riemann sums are 01:02:32.760 --> 01:02:34.020 align:middle line:90% linear in the function. 01:02:34.020 --> 01:02:38.910 align:middle line:90% 01:02:38.910 --> 01:02:42.102 align:middle line:90% And then we just take a limit. 01:02:42.102 --> 01:02:58.130 align:middle line:84% Let's take a sequence of tagged partitions 01:02:58.130 --> 01:03:05.650 align:middle line:84% with norms converting to 0 as r goes to infinity. 01:03:05.650 --> 01:03:12.030 align:middle line:90% 01:03:12.030 --> 01:03:18.790 align:middle line:84% Then, if I look at the Riemann sum associated 01:03:18.790 --> 01:03:27.790 align:middle line:84% to alpha f plus g, it is easy to see from the definition, right? 01:03:27.790 --> 01:03:33.260 align:middle line:84% This is a sum from k equals 1 to [? in-- ?] 01:03:33.260 --> 01:03:35.480 align:middle line:84% so this is easy to see that this is 01:03:35.480 --> 01:03:38.930 align:middle line:84% equal to alpha times the Riemann some 01:03:38.930 --> 01:03:55.122 align:middle line:90% associated to f plus that guy. 01:03:55.122 --> 01:03:56.330 align:middle line:90% And now I just take a limit-- 01:03:56.330 --> 01:03:59.360 align:middle line:90% 01:03:59.360 --> 01:04:04.190 align:middle line:84% so take the limit as r goes to infinity. 01:04:04.190 --> 01:04:10.460 align:middle line:84% The left-hand side converges to the Riemann integral 01:04:10.460 --> 01:04:12.680 align:middle line:90% of alpha f plus g. 01:04:12.680 --> 01:04:16.010 align:middle line:84% The limit as r goes to infinity converges to-- 01:04:16.010 --> 01:04:22.250 align:middle line:84% because we know limits respect linear operations, 01:04:22.250 --> 01:04:23.870 align:middle line:84% the limit of the right-hand side is 01:04:23.870 --> 01:04:27.260 align:middle line:84% going to be alpha times the limit plus the limit. 01:04:27.260 --> 01:04:31.630 align:middle line:84% And therefore, this is equal to alpha times integral a, b, 01:04:31.630 --> 01:04:35.937 align:middle line:90% f plus integral a, b, g. 01:04:35.937 --> 01:04:39.600 align:middle line:90% 01:04:39.600 --> 01:04:41.570 align:middle line:90% OK. 01:04:41.570 --> 01:04:47.090 align:middle line:84% So like differentiation, it's also 01:04:47.090 --> 01:04:51.080 align:middle line:84% what one would call a linear operator. 01:04:51.080 --> 01:04:53.540 align:middle line:84% It takes a function and spits something out, 01:04:53.540 --> 01:04:55.565 align:middle line:90% but does it in a linear way. 01:04:55.565 --> 01:05:02.850 align:middle line:84% So for us, it takes a continuous function 01:05:02.850 --> 01:05:06.570 align:middle line:84% and spits out a real number in a linear way. 01:05:06.570 --> 01:05:08.490 align:middle line:84% The integral of alpha f plus g is 01:05:08.490 --> 01:05:10.410 align:middle line:90% equal to alpha times f plus g. 01:05:10.410 --> 01:05:13.870 align:middle line:84% Just like differentiation, the derivative of alpha f plus g 01:05:13.870 --> 01:05:15.990 align:middle line:84% if everything's differentiable, is 01:05:15.990 --> 01:05:18.570 align:middle line:84% equal to alpha times the derivative of f 01:05:18.570 --> 01:05:21.810 align:middle line:90% plus the derivative of g. 01:05:21.810 --> 01:05:24.600 align:middle line:84% But Riemann integration, the Riemann integral, 01:05:24.600 --> 01:05:32.075 align:middle line:84% or integration in general, is in some sense-- 01:05:32.075 --> 01:05:35.260 align:middle line:90% 01:05:35.260 --> 01:05:36.940 align:middle line:84% I don't want to say it's not miraculous. 01:05:36.940 --> 01:05:39.340 align:middle line:84% It still is a bit of a miracle that it 01:05:39.340 --> 01:05:42.500 align:middle line:84% exists for all continuous functions, 01:05:42.500 --> 01:05:45.610 align:middle line:84% but it's not as destructive an operator 01:05:45.610 --> 01:05:48.610 align:middle line:84% as taking the derivative, if you take 01:05:48.610 --> 01:05:52.660 align:middle line:84% as your baseline continuous functions. 01:05:52.660 --> 01:05:58.380 align:middle line:84% Last lecture we constructed a continuous function which 01:05:58.380 --> 01:06:01.950 align:middle line:90% is differentiable nowhere. 01:06:01.950 --> 01:06:06.660 align:middle line:84% So with differentiation, you can take a continuous function 01:06:06.660 --> 01:06:09.950 align:middle line:84% and not be able to take its derivative anywhere. 01:06:09.950 --> 01:06:11.930 align:middle line:84% Well, with integration, if you start off 01:06:11.930 --> 01:06:15.600 align:middle line:84% with a continuous function, you can always take it's integral. 01:06:15.600 --> 01:06:22.950 align:middle line:84% So somehow integration is a much more smoother process 01:06:22.950 --> 01:06:24.860 align:middle line:90% than differentiation. 01:06:24.860 --> 01:06:28.000 align:middle line:84% All right, so I think we'll stop there. 01:06:28.000 --> 01:06:34.000 align:middle line:90%