| [{"text": "[SQUEAKING]\n[RUSTLING] [CLICKING]", "start": 0.0, "duration": 13.05}, {"text": "CASEY RODRIGUEZ:\nOK, so last time, we", "start": 13.05, "duration": 3.93}, {"text": "were talking about sequences,\nand I introduced the notion", "start": 16.98, "duration": 5.4}, {"text": "of a limit of a sequence.", "start": 22.38, "duration": 3.16}, {"text": "So we say that x n converges to\nx if for all epsilon positive", "start": 25.54, "duration": 4.04}, {"text": "there exists an M,\na natural number,", "start": 29.58, "duration": 1.68}, {"text": "such that for all little n\nbigger than or equal to capital", "start": 31.26, "duration": 2.7}, {"text": "M we have x sub n minus\nx is less than epsilon.", "start": 33.96, "duration": 3.76}, {"text": "So what this definition\nsays is that given", "start": 37.72, "duration": 5.83}, {"text": "any little bit of tolerance,\nas long as I go far enough", "start": 43.55, "duration": 3.03}, {"text": "out in the sequence,\nthe entries are getting", "start": 46.58, "duration": 2.58}, {"text": "within that tolerance to x.", "start": 49.16, "duration": 2.46}, {"text": "And when x then\nconverges to x, we'll", "start": 51.62, "duration": 3.06}, {"text": "write x is equal to the limit as\nn goes to infinity of x sub n,", "start": 54.68, "duration": 3.63}, {"text": "or using this notation,\nx sub n arrow x.", "start": 58.31, "duration": 6.25}, {"text": "And so I've said\nbefore that if you ever", "start": 64.56, "duration": 3.97}, {"text": "have a reasonably complicated\nor interesting definition,", "start": 68.53, "duration": 5.1}, {"text": "you should try to come up\nwith examples and negate it.", "start": 73.63, "duration": 3.64}, {"text": "So the end of last time,\nwe saw a few examples of--", "start": 77.27, "duration": 6.98}, {"text": "at least one example of--\na convergent sequence.", "start": 84.25, "duration": 2.88}, {"text": "Let's negate this definition\nbecause then we'll", "start": 87.13, "duration": 2.7}, {"text": "also come up with an\nexample of a sequence which", "start": 89.83, "duration": 2.13}, {"text": "does not converge.", "start": 91.96, "duration": 1.36}, {"text": "And so the negation\nof this definition", "start": 93.32, "duration": 5.24}, {"text": "is that x n does\nnot converge to x.", "start": 98.56, "duration": 10.48}, {"text": "What does this mean?", "start": 109.04, "duration": 0.94}, {"text": "So this is the\nactual negation part.", "start": 109.98, "duration": 1.97}, {"text": "If there exists an\nepsilon positive-- so", "start": 111.95, "duration": 6.03}, {"text": "whenever we negate a \"for all,\"\nit becomes \"there exists,\"", "start": 117.98, "duration": 3.45}, {"text": "and whenever we negate\na \"there exists,\"", "start": 121.43, "duration": 1.92}, {"text": "it becomes a \"for all.\"", "start": 123.35, "duration": 1.53}, {"text": "So there exists some bad\nepsilon 0 positive such", "start": 124.88, "duration": 5.64}, {"text": "that for all natural numbers\nN, there exists an n bigger", "start": 130.52, "duration": 8.22}, {"text": "than or equal to M\nsuch that x n minus", "start": 138.74, "duration": 6.54}, {"text": "x is bigger than or equal to\nthis bad epsilon 0 in distance.", "start": 145.28, "duration": 7.14}, {"text": "So we have a picture that\ngoes along with convergence.", "start": 152.42, "duration": 2.61}, {"text": "We can have a picture that\ngoes along with a sequence not", "start": 155.03, "duration": 3.3}, {"text": "converging to x.", "start": 158.33, "duration": 0.81}, {"text": "So this means if I go out\nwithin this bad epsilon 0,", "start": 159.14, "duration": 7.38}, {"text": "and if I go arbitrarily\nout in the sequence,", "start": 166.52, "duration": 4.26}, {"text": "I can always find an\nelement x sub n which", "start": 170.78, "duration": 3.21}, {"text": "is not in that interval.", "start": 173.99, "duration": 4.605}, {"text": "And we use this negation\nto prove a certain sequence", "start": 181.33, "duration": 2.7}, {"text": "does not converge to x.", "start": 184.03, "duration": 3.09}, {"text": "But let's go back\ndo another example", "start": 187.12, "duration": 2.76}, {"text": "of a sequence which\ndoes converge, so--", "start": 189.88, "duration": 3.585}, {"text": "equals 0.", "start": 205.31, "duration": 0.6}, {"text": "So again, how does the proof go?", "start": 208.78, "duration": 2.13}, {"text": "You have to verify the\ndefinition of the limit.", "start": 210.91, "duration": 3.6}, {"text": "I mean, there's nothing else\nthat we have to use right now.", "start": 214.51, "duration": 4.26}, {"text": "So let epsilon be positive.", "start": 218.77, "duration": 3.492}, {"text": "And now, off to\nthe side over here,", "start": 222.262, "duration": 1.458}, {"text": "I'm going to do a\nlittle bit of work", "start": 223.72, "duration": 4.38}, {"text": "to show you how\nexactly in practice one", "start": 228.1, "duration": 3.54}, {"text": "would find such a capital M\nor choose a capital M to get", "start": 231.64, "duration": 3.54}, {"text": "that 1 over n squared plus\n30n plus 1 would be less", "start": 235.18, "duration": 4.29}, {"text": "than epsilon for n bigger\nthan or equal to capital M.", "start": 239.47, "duration": 4.91}, {"text": "So typically how this works\nis, unlike in calculus", "start": 244.38, "duration": 4.26}, {"text": "where you have some inequality\nthat you want to get", "start": 248.64, "duration": 3.39}, {"text": "and then you solve for\nn, in analysis what's", "start": 252.03, "duration": 4.619}, {"text": "great is that you can\nreplace that thing", "start": 256.649, "duration": 2.851}, {"text": "you want to make small\nby something simpler,", "start": 259.5, "duration": 2.55}, {"text": "and then solve that inequality.", "start": 262.05, "duration": 1.44}, {"text": "What do I mean?", "start": 263.49, "duration": 1.53}, {"text": "So what I want to do,\nI want to find M so", "start": 265.02, "duration": 8.43}, {"text": "that if n it is bigger\nthan or equal to M, then--", "start": 273.45, "duration": 4.8}, {"text": "so I'm saying the limit is\n0, so I want to show 1 over n", "start": 278.25, "duration": 5.5}, {"text": "squared plus 30n plus\n1 is less than epsilon.", "start": 283.75, "duration": 6.94}, {"text": "Now, I could try to solve\nthis inequality for n,", "start": 290.69, "duration": 3.3}, {"text": "but this is a quadratic function\nand imagine I put 20 there,", "start": 293.99, "duration": 6.03}, {"text": "so I can't exactly solve\nexactly for n to guarantee this.", "start": 300.02, "duration": 4.92}, {"text": "But what's great about\nanalysis is that I", "start": 304.94, "duration": 4.73}, {"text": "don't have to work that hard.", "start": 309.67, "duration": 3.96}, {"text": "I could start with\nthis thing which", "start": 313.63, "duration": 2.13}, {"text": "I want to bound by epsilon,\nreplace it by something bigger,", "start": 315.76, "duration": 4.62}, {"text": "and then find capital\nM so that that bigger", "start": 320.38, "duration": 2.88}, {"text": "thing is less than epsilon.", "start": 323.26, "duration": 1.3}, {"text": "So what do I mean?", "start": 324.56, "duration": 0.9}, {"text": "So let's start with this thing--", "start": 325.46, "duration": 1.67}, {"text": "1 over n squared\nplus 30n plus 1.", "start": 327.13, "duration": 6.47}, {"text": "So this one is only\nmaking things bigger.", "start": 333.6, "duration": 2.42}, {"text": "I'm going to do\nthis very slowly.", "start": 336.02, "duration": 1.375}, {"text": "So this is less than\nor equal to this", "start": 342.52, "duration": 4.08}, {"text": "because for this, I've just\nadded 1 to the denominator.", "start": 346.6, "duration": 5.28}, {"text": "And 1 over n squared plus 30n--", "start": 351.88, "duration": 2.582}, {"text": "this certainly looks\na little bit simpler", "start": 354.462, "duration": 1.708}, {"text": "to solve for n\nless than epsilon.", "start": 356.17, "duration": 2.11}, {"text": "But I can make it even simpler\nby dropping the n squared,", "start": 358.28, "duration": 2.78}, {"text": "because n squared is\npositive, so that's", "start": 361.06, "duration": 1.68}, {"text": "only making the bottom bigger.", "start": 362.74, "duration": 1.25}, {"text": "So I've 1 over 30n,\nand this is certainly", "start": 367.46, "duration": 3.24}, {"text": "less than or equal to 1 over n.", "start": 370.7, "duration": 3.54}, {"text": "So what does this\ncomputation show?", "start": 374.24, "duration": 4.95}, {"text": "This shows that if 1 over\nn is less than epsilon,", "start": 379.19, "duration": 5.49}, {"text": "then this implies that 1 over\nn squared plus 30n plus 1 is", "start": 384.68, "duration": 5.82}, {"text": "less than epsilon.", "start": 390.5, "duration": 0.75}, {"text": "If this thing is\nless than epsilon,", "start": 395.38, "duration": 1.56}, {"text": "then by this string\nof inequalities,", "start": 396.94, "duration": 1.89}, {"text": "this thing will be\nless than epsilon.", "start": 398.83, "duration": 1.6}, {"text": "So we choose capital\nM so that we get this.", "start": 400.43, "duration": 5.487}, {"text": "Now, when we actually\nwrite the proof,", "start": 405.917, "duration": 1.583}, {"text": "it'll look something like this.", "start": 407.5, "duration": 1.292}, {"text": "But the actual\nstyle of the proof", "start": 408.792, "duration": 2.668}, {"text": "is a little bit different.", "start": 411.46, "duration": 3.062}, {"text": "And if you didn't see\nthis, it would just", "start": 414.522, "duration": 1.708}, {"text": "seem like I pulled this\ncapital M out of nowhere,", "start": 416.23, "duration": 4.345}, {"text": "but this is the thinking\nthat goes behind it.", "start": 420.575, "duration": 1.875}, {"text": "So let epsilon be positive.", "start": 422.45, "duration": 1.89}, {"text": "Now I'll tell you how to\nchoose capital M. Choose", "start": 424.34, "duration": 5.39}, {"text": "M a natural number so\nthat-- so we want it", "start": 429.73, "duration": 4.68}, {"text": "for all n bigger than or\nequal to capital M, 1 over n", "start": 434.41, "duration": 2.55}, {"text": "is less than epsilon.", "start": 436.96, "duration": 1.45}, {"text": "So this will\ncertainly be true if n", "start": 438.41, "duration": 2.99}, {"text": "is bigger than or\nequal to capital M", "start": 441.4, "duration": 2.34}, {"text": "so that 1 over capital\nM is less than epsilon.", "start": 443.74, "duration": 2.07}, {"text": "And the reason we\ncan choose capital", "start": 449.985, "duration": 1.5}, {"text": "M like this-- again,\nwhere is this coming from?", "start": 451.485, "duration": 2.425}, {"text": "It's coming from the Archimedean\nproperty of the real numbers.", "start": 453.91, "duration": 3.605}, {"text": "So we've chosen\ncapital M. Now we", "start": 460.71, "duration": 2.55}, {"text": "need to show that\nthis capital M works.", "start": 463.26, "duration": 1.665}, {"text": "Then for all n bigger than\nor equal to capital M,", "start": 468.03, "duration": 5.16}, {"text": "if I look at 1 over n\nsquared plus 30n plus 1", "start": 473.19, "duration": 6.0}, {"text": "minus 0, my proposed\nlimit, this is equal to 1", "start": 479.19, "duration": 3.9}, {"text": "over n squared plus 1.", "start": 483.09, "duration": 4.8}, {"text": "And this is less\nthan or equal to,", "start": 487.89, "duration": 1.5}, {"text": "if I drop n squared\nand 1, 30n, which", "start": 489.39, "duration": 3.6}, {"text": "is less than or equal to 1\nover n, which is less than", "start": 492.99, "duration": 3.12}, {"text": "or equal to 1 over capital M,\nwhich is less than epsilon.", "start": 496.11, "duration": 2.73}, {"text": "So something of a non-example--", "start": 517.03, "duration": 5.94}, {"text": "let's show that the sequence\nminus 1 to the n DNC", "start": 522.97, "duration": 6.99}, {"text": "does not converge--", "start": 529.96, "duration": 1.35}, {"text": "not Democratic\nNational Committee--", "start": 531.31, "duration": 1.86}, {"text": "does not converge.", "start": 536.13, "duration": 1.44}, {"text": "So this is just a\nsequence minus 1,", "start": 537.57, "duration": 4.43}, {"text": "1, minus 1, 1,\nminus 1, and so on.", "start": 542.0, "duration": 4.87}, {"text": "So we have to show\nit does not converge,", "start": 546.87, "duration": 2.31}, {"text": "meaning for every\nx in R, we'll show", "start": 549.18, "duration": 2.4}, {"text": "that x n does not converge\nto x using the negation", "start": 551.58, "duration": 3.6}, {"text": "of the definition--", "start": 555.18, "duration": 2.13}, {"text": "so proof.", "start": 557.31, "duration": 3.285}, {"text": "Let x be in R. And what\ndo we want to show?", "start": 574.64, "duration": 1.905}, {"text": "Minus 1 to the n-th\ndoes not converge to x.", "start": 581.2, "duration": 11.82}, {"text": "And we'll use this definition.", "start": 595.7, "duration": 1.42}, {"text": "So this definition means we\nneed to find a bad epsilon 0.", "start": 597.12, "duration": 5.54}, {"text": "And here's the kind of thinking\nthat goes along with this--", "start": 606.78, "duration": 4.69}, {"text": "there's 1, there's minus 1.", "start": 611.47, "duration": 2.46}, {"text": "This sequence just keeps\nalternating, so intuitively", "start": 613.93, "duration": 3.72}, {"text": "there's no way that every\nelement of the sequence", "start": 617.65, "duration": 3.63}, {"text": "can be getting--", "start": 621.28, "duration": 0.93}, {"text": "or every entry in the\nsequence can be getting--", "start": 622.21, "duration": 1.98}, {"text": "close to some single x.", "start": 624.19, "duration": 1.83}, {"text": "So what's the idea?", "start": 626.02, "duration": 1.36}, {"text": "The idea is that the distance\nbetween these two guys", "start": 627.38, "duration": 4.4}, {"text": "is always 2.", "start": 631.78, "duration": 0.84}, {"text": "So somehow, the distance\nbetween every entry,", "start": 635.21, "duration": 9.01}, {"text": "or basically, some entry\nhas to be within distance--", "start": 644.22, "duration": 4.11}, {"text": "if this is within distance\n1, say, to a given x,", "start": 648.33, "duration": 4.5}, {"text": "then this will be greater than\nor equal to 1 in distance to x.", "start": 652.83, "duration": 5.37}, {"text": "So you can imagine\nx is over here.", "start": 658.2, "duration": 1.89}, {"text": "Minus 1 is within\ndistance 1 to x,", "start": 660.09, "duration": 2.97}, {"text": "but then 1 would be greater\nthan distance 1 to x.", "start": 663.06, "duration": 6.06}, {"text": "So our bad epsilon 0 was\ngoing to be 1, as we'll see.", "start": 669.12, "duration": 4.17}, {"text": "Let epsilon 0 equals 1.", "start": 677.01, "duration": 1.29}, {"text": "That's our bad epsilon.", "start": 678.3, "duration": 1.33}, {"text": "So now we have to\nshow for all capital", "start": 679.63, "duration": 1.67}, {"text": "M in the natural numbers,\nthere exists a little n", "start": 681.3, "duration": 1.89}, {"text": "bigger than or\nequal to capital M", "start": 683.19, "duration": 1.375}, {"text": "so that we have that inequality.", "start": 684.565, "duration": 3.645}, {"text": "So let M be a natural number.", "start": 688.21, "duration": 3.365}, {"text": "Then what do we see?", "start": 702.02, "duration": 1.425}, {"text": "So then it says, they're\nexists an n greater than", "start": 706.33, "duration": 2.97}, {"text": "or equal to capital\nM. And I will", "start": 709.3, "duration": 4.88}, {"text": "show that n is, in fact,\neither capital M or capital", "start": 714.18, "duration": 3.15}, {"text": "M plus 1, then 2, which is\nthe distance between minus 1", "start": 717.33, "duration": 11.4}, {"text": "to the M and minus\n1 to the M plus 1 is", "start": 728.73, "duration": 6.04}, {"text": "less than or equal to-- now\nusing the triangle inequality,", "start": 734.77, "duration": 2.67}, {"text": "I add and subtract x and then\nuse the triangle inequality.", "start": 737.44, "duration": 3.57}, {"text": "So I have the sum of two numbers\nis bigger than or equal to 2.", "start": 751.32, "duration": 5.9}, {"text": "And the only way that can happen\nis if one of these numbers", "start": 757.22, "duration": 2.58}, {"text": "is bigger than or equal to 1.", "start": 759.8, "duration": 1.47}, {"text": "If both of these\nnumbers are less than 1,", "start": 761.27, "duration": 2.13}, {"text": "then the sum is less than 2, and\nI would have 2 is less than 2.", "start": 763.4, "duration": 3.18}, {"text": "That's not possible.", "start": 766.58, "duration": 1.21}, {"text": "So one of these numbers has to\nbe bigger than or equal to 1.", "start": 767.79, "duration": 2.99}, {"text": "And therefore, I could\ntake n equal to capital M", "start": 789.14, "duration": 3.0}, {"text": "if this is bigger\nthan or equal to 1,", "start": 792.14, "duration": 1.8}, {"text": "or if this is bigger\nthan or equal to 1,", "start": 793.94, "duration": 2.34}, {"text": "I take little n to\nbe capital M plus 1.", "start": 796.28, "duration": 2.31}, {"text": "So that's the end of that.", "start": 798.59, "duration": 2.44}, {"text": "So that's good for\nexamples for now.", "start": 808.08, "duration": 5.79}, {"text": "Let's prove a general theorem\nabout convergent sequences--", "start": 813.87, "duration": 3.69}, {"text": "namely, that if I have a\nconvergent sequence, then", "start": 817.56, "duration": 2.67}, {"text": "it's bounded.", "start": 820.23, "duration": 0.63}, {"text": "And let me just give\nyou a quick reminder", "start": 837.69, "duration": 6.54}, {"text": "about what being bounded\nmeans for a sequence, i.e.,", "start": 844.23, "duration": 6.66}, {"text": "there exists some number\nsuch that for all n,", "start": 850.89, "duration": 5.7}, {"text": "x n is less than or\nequal to capital B. Now,", "start": 856.59, "duration": 5.84}, {"text": "as beginning students of\nmath, what types of questions", "start": 862.43, "duration": 8.08}, {"text": "should we start\nasking ourselves?", "start": 870.51, "duration": 1.62}, {"text": "So one of the types,\nor a type of question", "start": 877.22, "duration": 3.29}, {"text": "that we should ask when we\ncome across the theorem that", "start": 880.51, "duration": 3.54}, {"text": "is a one-way street,\nmeaning if something", "start": 884.05, "duration": 3.78}, {"text": "happens then something\nelse happens,", "start": 887.83, "duration": 2.73}, {"text": "is does the converse hold?", "start": 890.56, "duration": 1.38}, {"text": "Namely, if I assume x sub\nn is a bounded sequence,", "start": 891.94, "duration": 3.51}, {"text": "does this imply that\nx n is convergent?", "start": 895.45, "duration": 2.94}, {"text": "So if you ever hear the words,\n\"does the converse hold,\"", "start": 901.09, "duration": 2.58}, {"text": "that's what they mean.", "start": 903.67, "duration": 1.83}, {"text": "Now, for this statement,\nthis is false.", "start": 905.5, "duration": 2.48}, {"text": "If x n is bounded implies\nx n is convergent,", "start": 907.98, "duration": 2.92}, {"text": "that's false because we\nhave an example right", "start": 910.9, "duration": 2.09}, {"text": "there of a bounded sequence.", "start": 912.99, "duration": 1.71}, {"text": "This sequence here,\nminus 1 to the n,", "start": 914.7, "duration": 1.86}, {"text": "is bounded since the\nsequence in absolute value", "start": 916.56, "duration": 6.03}, {"text": "equals 1 for all n, so\nwe could take B equals 1.", "start": 922.59, "duration": 3.51}, {"text": "So minus 1 to the n\nis a bounded sequence,", "start": 926.1, "duration": 1.95}, {"text": "which is not convergent.", "start": 928.05, "duration": 1.06}, {"text": "So the converse does not hold.", "start": 929.11, "duration": 2.68}, {"text": "So let's prove the theorem.", "start": 931.79, "duration": 3.46}, {"text": "And let me draw a little\npicture to go along", "start": 935.25, "duration": 4.37}, {"text": "with it of what's going on.", "start": 939.62, "duration": 3.74}, {"text": "So here is 0, let's imagine.", "start": 943.36, "duration": 4.08}, {"text": "Let me make this picture\na little bit bigger", "start": 947.44, "duration": 5.22}, {"text": "before I go into the proof.", "start": 952.66, "duration": 1.29}, {"text": "So let x be the limit.", "start": 959.91, "duration": 1.725}, {"text": "So this is not the proof yet.", "start": 964.69, "duration": 1.31}, {"text": "This is just more discussion.", "start": 966.0, "duration": 5.2}, {"text": "So if this x sub n\nis converging to x,", "start": 971.2, "duration": 3.41}, {"text": "then let's say I go\nout distance 1 to x.", "start": 974.61, "duration": 7.17}, {"text": "Then all of the x\nsub n's eventually", "start": 981.78, "duration": 2.25}, {"text": "have to land in this strip here.", "start": 984.03, "duration": 3.18}, {"text": "So x sub n, n bigger than\nor equal to some capital M.", "start": 990.88, "duration": 8.46}, {"text": "So I know that they're all\nwithin distance 1 to x.", "start": 999.34, "duration": 2.49}, {"text": "So in absolute value,\nthey'd be bounded by,", "start": 1001.83, "duration": 3.9}, {"text": "let's say for this\npicture, it'd be x plus 1.", "start": 1005.73, "duration": 2.13}, {"text": "But if this was\nover there, capital,", "start": 1007.86, "duration": 3.66}, {"text": "it would be the absolute\nvalue of x plus 1.", "start": 1011.52, "duration": 3.3}, {"text": "So that takes care of all x\nsub n's with n bigger than", "start": 1014.82, "duration": 2.79}, {"text": "or equal to M. And then\nthere's only finitely many guys", "start": 1017.61, "duration": 6.32}, {"text": "to deal with.", "start": 1023.93, "duration": 1.35}, {"text": "Maybe there's a bigger one\nout here, x sub n minus 1", "start": 1025.28, "duration": 4.799}, {"text": "to deal with, and that's\nhow we define our number,", "start": 1030.079, "duration": 4.02}, {"text": "is by the absolute\nvalue of x plus 1", "start": 1034.099, "duration": 4.47}, {"text": "plus the absolute value\nof these finitely many.", "start": 1038.569, "duration": 3.266}, {"text": "And then that would\nbe something that's", "start": 1041.835, "duration": 1.625}, {"text": "bigger than or equal to every\nelement in the sequence.", "start": 1043.46, "duration": 2.25}, {"text": "That's the picture\nthat goes with it.", "start": 1048.298, "duration": 1.542}, {"text": "How do we take that\npicture to proof?", "start": 1053.83, "duration": 1.8}, {"text": "So suppose x n converges to\nx-- that's our assumption,", "start": 1060.24, "duration": 10.26}, {"text": "and there exists a\nnatural number M such", "start": 1070.5, "duration": 4.53}, {"text": "that for all n bigger than\nor equal to capital m, x sub", "start": 1075.03, "duration": 7.95}, {"text": "n minus x is within distance 1,\nor x sub n is within distance 1", "start": 1082.98, "duration": 4.79}, {"text": "to x, then for all n bigger\nthan or equal to capital M,", "start": 1087.77, "duration": 6.64}, {"text": "if we look at the\nabsolute value of x sub n,", "start": 1094.41, "duration": 5.75}, {"text": "add and subtract x, and now\nuse the triangle inequality,", "start": 1100.16, "duration": 3.81}, {"text": "this is less than or equal\nto absolute value of x sub", "start": 1103.97, "duration": 2.82}, {"text": "n minus x plus the absolute\nvalue of x equals 1 plus x.", "start": 1106.79, "duration": 6.27}, {"text": "So for all entries in the\nsequence past this point", "start": 1113.06, "duration": 4.81}, {"text": "capital M, they're bounded in\nabsolute value by 1 plus x.", "start": 1117.87, "duration": 4.36}, {"text": "That's just a number.", "start": 1122.23, "duration": 0.875}, {"text": "So then, we just have\nto take care of the 1", "start": 1126.3, "duration": 6.36}, {"text": "up to capital M minus 1 guys.", "start": 1132.66, "duration": 2.1}, {"text": "We just have to\nfind a number bigger", "start": 1134.76, "duration": 1.5}, {"text": "than those guys and\nthis number, and we'll", "start": 1136.26, "duration": 3.18}, {"text": "have found a B.\nSo then let's take", "start": 1139.44, "duration": 2.97}, {"text": "B to be x sub 1 plus x\nsub 2 plus x sub capital", "start": 1142.41, "duration": 9.26}, {"text": "M minus 1 plus this.", "start": 1151.67, "duration": 7.2}, {"text": "Then for all n a natural number,\nthe absolute value of x of n", "start": 1169.53, "duration": 8.66}, {"text": "is less than or equal\nto B. Because if n", "start": 1178.19, "duration": 2.4}, {"text": "is less than capital M,\nthen its absolute value", "start": 1180.59, "duration": 3.87}, {"text": "is bounded by one of these.", "start": 1184.46, "duration": 1.36}, {"text": "Its absolute value is\nequal to 1 over these,", "start": 1185.82, "duration": 2.09}, {"text": "and therefore, less than\nor equal to one of these", "start": 1187.91, "duration": 3.0}, {"text": "plus some non-negative numbers.", "start": 1190.91, "duration": 3.11}, {"text": "And if n is bigger than\nor equal to capital M,", "start": 1194.02, "duration": 2.82}, {"text": "then we have this\nbound that we use.", "start": 1196.84, "duration": 3.39}, {"text": "And this number is\ncertainly less than", "start": 1200.23, "duration": 1.74}, {"text": "or equal to capital\nB, again, because it's", "start": 1201.97, "duration": 2.7}, {"text": "this number plus a sum\nof non-negative numbers.", "start": 1204.67, "duration": 2.13}, {"text": "Now in general, there's no\neasily checkable criterion", "start": 1224.23, "duration": 5.64}, {"text": "for a sequence to converge.", "start": 1229.87, "duration": 2.85}, {"text": "All we can do is\nverify the definition.", "start": 1232.72, "duration": 2.22}, {"text": "But there are some sequences\nwhich you can easily", "start": 1238.69, "duration": 2.81}, {"text": "check or figure out\nif they converge just", "start": 1241.5, "duration": 3.54}, {"text": "by telling if they're bounded.", "start": 1245.04, "duration": 3.15}, {"text": "Those are what are called\n\"monotone sequences.\"", "start": 1248.19, "duration": 2.21}, {"text": "So this is, in some\nsense, the class", "start": 1255.71, "duration": 2.04}, {"text": "of sequences that\ndoes kind of satisfy", "start": 1257.75, "duration": 2.28}, {"text": "the converse of that theorem\nI stated a minute ago.", "start": 1260.03, "duration": 4.41}, {"text": "So sequence x sub n\nis monotone increasing", "start": 1264.44, "duration": 18.29}, {"text": "if for all natural numbers\nlittle n, x sub n is less than", "start": 1282.73, "duration": 9.79}, {"text": "or equal to x sub n plus 1.", "start": 1292.52, "duration": 1.83}, {"text": "So this means that\nx sub 1 is less", "start": 1294.35, "duration": 4.53}, {"text": "than or equal to x sub 2 is\nless than or equal to x to 3,", "start": 1298.88, "duration": 4.14}, {"text": "and so on.", "start": 1303.02, "duration": 0.72}, {"text": "Sequence is monotone\ndecreasing if we", "start": 1314.58, "duration": 8.47}, {"text": "have the other inequality.", "start": 1323.05, "duration": 1.45}, {"text": "So things are getting\nsmaller, so x sub n", "start": 1324.5, "duration": 4.58}, {"text": "is bigger than or equal\nto x sub n plus 1.", "start": 1329.08, "duration": 2.205}, {"text": "And if we have a sequence\nwhich is monotone increasing", "start": 1352.24, "duration": 7.59}, {"text": "or monotone decreasing--", "start": 1359.83, "duration": 1.68}, {"text": "so it's one or the other--", "start": 1364.85, "duration": 2.34}, {"text": "we say x n is\nmonotone or monotonic.", "start": 1367.19, "duration": 8.25}, {"text": "So it's simple enough to come\nup with examples of sequences", "start": 1380.14, "duration": 6.12}, {"text": "which are monotone\nincreasing, monotone", "start": 1386.26, "duration": 3.36}, {"text": "decreasing, or neither.", "start": 1389.62, "duration": 1.87}, {"text": "So 1 over n--", "start": 1391.49, "duration": 5.45}, {"text": "this is just 1, 1/2, 1/3, 1/4--", "start": 1396.94, "duration": 5.82}, {"text": "this is monotone decreasing.", "start": 1402.76, "duration": 5.01}, {"text": "And if I take\nminus the sequence,", "start": 1407.77, "duration": 5.01}, {"text": "then that reverses basically\nall of the inequalities,", "start": 1412.78, "duration": 3.48}, {"text": "and I get a monotone\nincreasing one.", "start": 1416.26, "duration": 1.87}, {"text": "So then this is minus\n1, minus 1/2, minus 1/3,", "start": 1418.13, "duration": 3.95}, {"text": "minus 1/4, monotone increasing.", "start": 1422.08, "duration": 4.71}, {"text": "And then something\nthat's not is-- we've", "start": 1426.79, "duration": 1.68}, {"text": "already come across that--\nminus 1 to the n, and so on.", "start": 1428.47, "duration": 8.84}, {"text": "And so the theorem\nabout monotone sequences", "start": 1444.06, "duration": 3.48}, {"text": "is that there is a simpler\ncriterion than the definition", "start": 1447.54, "duration": 5.28}, {"text": "for determining when\nthey're convergent,", "start": 1452.82, "duration": 2.58}, {"text": "or if you like, they satisfy\nthe converse of this theorem up", "start": 1455.4, "duration": 3.6}, {"text": "here.", "start": 1459.0, "duration": 1.18}, {"text": "So a monotonic sequence\nis convergent if and only", "start": 1460.18, "duration": 18.48}, {"text": "if it is bounded.", "start": 1478.66, "duration": 6.19}, {"text": "So we proved a minute ago\nthat convergent sequences", "start": 1484.85, "duration": 2.57}, {"text": "are bounded, so for monotonic\nsequences, the converse holds.", "start": 1487.42, "duration": 3.555}, {"text": "So let's do the proof of this.", "start": 1494.51, "duration": 1.25}, {"text": "I'm going to do the proof for\nmonotone increasing sequences,", "start": 1495.76, "duration": 4.44}, {"text": "and as an exercise,\nI'll leave it to you", "start": 1500.2, "duration": 2.52}, {"text": "to do the proof for monotone\ndecreasing sequences.", "start": 1502.72, "duration": 5.37}, {"text": "So suppose x sub n is a\nmonotone increasing sequence.", "start": 1508.09, "duration": 11.07}, {"text": "So there's two directions to\nprove convergence if and only", "start": 1523.08, "duration": 5.27}, {"text": "if bounded.", "start": 1528.35, "duration": 1.21}, {"text": "So we have one\ndirection, meaning", "start": 1529.56, "duration": 3.71}, {"text": "convergence implies bounded.", "start": 1533.27, "duration": 1.89}, {"text": "This is just the\nprevious theorem,", "start": 1535.16, "duration": 5.41}, {"text": "where we proved that every\nconvergent sequence is bounded,", "start": 1540.57, "duration": 2.58}, {"text": "not just monotonic sequences.", "start": 1543.15, "duration": 3.07}, {"text": "So the meat is in proving\nthe converse direction.", "start": 1546.22, "duration": 3.8}, {"text": "And in fact, we'll be\nable to pick out what", "start": 1550.02, "duration": 3.03}, {"text": "the limit is of this sequence.", "start": 1553.05, "duration": 1.635}, {"text": "So suppose x sub n is bounded.", "start": 1557.2, "duration": 11.41}, {"text": "Then if I look at\nthe set of entries--", "start": 1574.06, "duration": 2.41}, {"text": "so not the sequence but if\nI look at the set of values", "start": 1576.47, "duration": 5.66}, {"text": "that this sequence takes-- so x\nsub n is just a natural number,", "start": 1582.13, "duration": 4.98}, {"text": "this is now a subset\nof the real numbers.", "start": 1587.11, "duration": 3.615}, {"text": "This is a bounded set, meaning\nit's bounded above and below", "start": 1594.88, "duration": 16.18}, {"text": "because there exists some\ncapital B so that x sub", "start": 1611.06, "duration": 2.7}, {"text": "n is, in absolute\nvalue, less than", "start": 1613.76, "duration": 3.63}, {"text": "or equal to capital B for all n.", "start": 1617.39, "duration": 1.53}, {"text": "So that means x sub n is\nbetween B and minus B.", "start": 1618.92, "duration": 7.68}, {"text": "All right, so what's the picture\nagain that goes with this?", "start": 1626.6, "duration": 3.76}, {"text": "So we have these x sub n's,\nx sub 1, x sub 2, x sub 3.", "start": 1630.36, "duration": 5.64}, {"text": "We know that they cannot\ngo past a certain B,", "start": 1636.0, "duration": 5.1}, {"text": "and they're steadily increasing,\nbut they can't keep strictly--", "start": 1641.1, "duration": 7.89}, {"text": "they can't increase\nwithout bound.", "start": 1648.99, "duration": 2.07}, {"text": "Not only that, they're\nbounded by some number", "start": 1654.16, "duration": 8.0}, {"text": "x, which I'll define to be\nthe supremum of this set.", "start": 1662.16, "duration": 4.47}, {"text": "And these guys are just\ngetting, in essence--", "start": 1666.63, "duration": 2.88}, {"text": "so this is a picture,\njust me trying", "start": 1669.51, "duration": 3.06}, {"text": "to explain to you\nwhat's going on.", "start": 1672.57, "duration": 1.68}, {"text": "If x is the sup of this\nset of entries, then", "start": 1674.25, "duration": 8.37}, {"text": "what do I know about x sub n?", "start": 1682.62, "duration": 2.93}, {"text": "I mean x is that if I go a\nlittle bit to the left of x--", "start": 1685.55, "duration": 4.33}, {"text": "so let me draw this again.", "start": 1689.88, "duration": 3.31}, {"text": "So let me back up a minute here.", "start": 1693.19, "duration": 3.71}, {"text": "So since this set is bounded\nin R, it has a supremum in R.", "start": 1701.07, "duration": 12.72}, {"text": "And what I claim is that\nthis supremum is, in fact,", "start": 1713.79, "duration": 2.92}, {"text": "the limit of this sequence.", "start": 1716.71, "duration": 1.88}, {"text": "This is a supremum of a set.", "start": 1718.59, "duration": 2.58}, {"text": "I'm saying it's the\nlimit of the sequence.", "start": 1721.17, "duration": 2.55}, {"text": "So what's the thinking here?", "start": 1723.72, "duration": 3.49}, {"text": "So x is the supremum of\nall the entries in x sub n,", "start": 1727.21, "duration": 2.99}, {"text": "so nothing's ever\ngoing to be bigger", "start": 1730.2, "duration": 1.5}, {"text": "than some tolerance\nx plus epsilon.", "start": 1731.7, "duration": 4.6}, {"text": "So we just need to worry about\nwhat's to the left of it.", "start": 1736.3, "duration": 3.66}, {"text": "And we need to find\na capital number", "start": 1739.96, "duration": 1.5}, {"text": "M so that all the x sub\nn's are in this interval", "start": 1741.46, "duration": 5.1}, {"text": "here, because we're trying\nto show that the x sub", "start": 1746.56, "duration": 17.175}, {"text": "n's converge to x, the\nsupremum of this guy.", "start": 1763.735, "duration": 2.945}, {"text": "So let's walk through\nthis by this picture.", "start": 1766.68, "duration": 4.35}, {"text": "So x is the supremum.", "start": 1771.03, "duration": 2.34}, {"text": "Nothing's bigger than x,\nso all of the x sub n's", "start": 1773.37, "duration": 3.63}, {"text": "are to the left\nof x plus epsilon.", "start": 1777.0, "duration": 2.14}, {"text": "So now we just need to\nworry about x minus epsilon.", "start": 1779.14, "duration": 2.94}, {"text": "Now, since x is the supremum\nof this set, x minus epsilon", "start": 1782.08, "duration": 4.04}, {"text": "can't be an upper bound for\nthe set of entries of x sub n.", "start": 1786.12, "duration": 4.32}, {"text": "So that means there has to\nexist some x sub M so that it's", "start": 1790.44, "duration": 6.39}, {"text": "bigger than x minus epsilon.", "start": 1796.83, "duration": 3.3}, {"text": "But now, this is where we\nuse the fact that this is", "start": 1800.13, "duration": 2.64}, {"text": "a monotone increasing sequence.", "start": 1802.77, "duration": 3.33}, {"text": "Because then, if n is bigger\nthan or equal to capital M,", "start": 1806.1, "duration": 3.03}, {"text": "then x sub n is strictly\nto the right of x sub M,", "start": 1809.13, "duration": 4.23}, {"text": "or it's equal to x sub M\nand still less than or equal", "start": 1813.36, "duration": 6.7}, {"text": "to x because x is the\nsupremum of all the entries.", "start": 1820.06, "duration": 3.31}, {"text": "So then, for n bigger than\nor equal to capital M,", "start": 1823.37, "duration": 2.45}, {"text": "they all have to lie\nin this interval.", "start": 1825.82, "duration": 2.01}, {"text": "And that's why the\nsequence converges", "start": 1827.83, "duration": 2.25}, {"text": "to the supremum of this set.", "start": 1830.08, "duration": 2.13}, {"text": "So we're using,\nin a crucial way,", "start": 1832.21, "duration": 3.63}, {"text": "the fact that this sequence\nis monotonic increasing.", "start": 1835.84, "duration": 5.73}, {"text": "I'll prove this claim.", "start": 1841.57, "duration": 1.18}, {"text": "Let epsilon be positive.", "start": 1842.75, "duration": 3.09}, {"text": "Then since x minus\nepsilon is not", "start": 1845.84, "duration": 11.27}, {"text": "an upper bound for\nthis set, x sub n,", "start": 1857.11, "duration": 7.59}, {"text": "there exists an\nM0, natural number,", "start": 1864.7, "duration": 4.17}, {"text": "so that x sub n minus\nepsilon is less than x", "start": 1868.87, "duration": 5.1}, {"text": "sub M0 is less\nthan or equal to x.", "start": 1873.97, "duration": 3.855}, {"text": "And we'll choose\nM to be this M0.", "start": 1880.38, "duration": 4.26}, {"text": "Then for all n bigger\nthan or equal to M,", "start": 1888.76, "duration": 3.5}, {"text": "we have x sub n minus epsilon,\nwhich is less than x sub M0.", "start": 1892.26, "duration": 6.03}, {"text": "And because n is bigger\nthan or equal to capital M,", "start": 1898.29, "duration": 2.52}, {"text": "and this is a monotone\nincreasing sequence,", "start": 1900.81, "duration": 2.7}, {"text": "x sub M0 is less than\nor equal to x sub n.", "start": 1903.51, "duration": 2.91}, {"text": "And because x is a\nsupremum of the set", "start": 1906.42, "duration": 6.06}, {"text": "of all entries of\nthis sequence, this", "start": 1912.48, "duration": 1.74}, {"text": "is less than or equal to\nx, which is less than x", "start": 1914.22, "duration": 4.35}, {"text": "plus epsilon.", "start": 1918.57, "duration": 1.95}, {"text": "Or to summarize--\nso this should not", "start": 1920.52, "duration": 1.59}, {"text": "be x sub n, that should\nbe x minus epsilon.", "start": 1922.11, "duration": 2.61}, {"text": "Or x minus epsilon is less\nthan x sub n is less than x", "start": 1924.72, "duration": 5.52}, {"text": "plus epsilon.", "start": 1930.24, "duration": 0.72}, {"text": "And that's the same as showing\nthe absolute value of x sub", "start": 1930.96, "duration": 2.76}, {"text": "n minus x is less than epsilon.", "start": 1933.72, "duration": 1.68}, {"text": "Now, what's the change\nfor monotone decreasing", "start": 1939.53, "duration": 2.64}, {"text": "is that we'll be\nable to identify", "start": 1942.17, "duration": 2.79}, {"text": "the limit of the\nsequence as being", "start": 1944.96, "duration": 3.18}, {"text": "the imf, the\ninfimum of this set.", "start": 1948.14, "duration": 4.85}, {"text": "But I'll leave that\nto you as an exercise.", "start": 1952.99, "duration": 1.94}, {"text": "So let's use this real quick to\nprove the limit of a little bit", "start": 1977.86, "duration": 4.55}, {"text": "more interesting sequence\nthan 1 over this one,", "start": 1982.41, "duration": 4.62}, {"text": "which is reasonably\ninteresting, I guess.", "start": 1987.03, "duration": 4.08}, {"text": "But typically in math, we\njust don't prove theorems", "start": 1991.11, "duration": 15.38}, {"text": "for the sake of\nproving theorems.", "start": 2006.49, "duration": 2.22}, {"text": "Typically, there's some\nconcrete reason we do things.", "start": 2008.71, "duration": 6.46}, {"text": "There's some concrete\nsequence we're", "start": 2015.17, "duration": 1.55}, {"text": "trying to prove converges,\nor has a certain property,", "start": 2016.72, "duration": 4.17}, {"text": "or doesn't have a\ncertain property.", "start": 2020.89, "duration": 2.56}, {"text": "And so the basic one is if\nyou like a geometric sequence.", "start": 2023.45, "duration": 9.75}, {"text": "So if c is a positive\nnumber between 0 and 1,", "start": 2033.2, "duration": 11.09}, {"text": "then we'll prove that\nthis sequence c to the n", "start": 2044.29, "duration": 7.349}, {"text": "is convergent and\nit converges to 0.", "start": 2051.639, "duration": 1.891}, {"text": "And we'll prove if\nc is bigger than 1,", "start": 2057.969, "duration": 8.121}, {"text": "then the sequence c\nto the n is unbounded.", "start": 2066.09, "duration": 5.884}, {"text": "In particular, it\ncan't converge.", "start": 2071.974, "duration": 1.375}, {"text": "So let's go in reverse order.", "start": 2080.65, "duration": 3.63}, {"text": "Let's prove that if\nc is bigger than 1,", "start": 2084.28, "duration": 3.129}, {"text": "then this thing is unbounded.", "start": 2087.409, "duration": 1.83}, {"text": "It doesn't require 1,\nbut it's shorter than 1,", "start": 2089.239, "duration": 2.451}, {"text": "and we can go ahead and do it.", "start": 2091.69, "duration": 1.87}, {"text": "And we'll use this\nfact that we proved,", "start": 2093.56, "duration": 2.54}, {"text": "I think, in the first or\nsecond lecture using induction.", "start": 2096.1, "duration": 6.82}, {"text": "So what does it mean to\nshow something is unbounded?", "start": 2102.92, "duration": 3.74}, {"text": "Again, we get to use\nour negation skills.", "start": 2106.66, "duration": 2.76}, {"text": "It means that-- what\ndo we want to show--", "start": 2109.42, "duration": 2.415}, {"text": "for all B bigger\nthan or equal to 0,", "start": 2115.29, "duration": 5.01}, {"text": "there exists an n, a natural\nnumber, so that c to the n", "start": 2120.3, "duration": 5.82}, {"text": "is bigger than B. Now, how\nare we going to find this n?", "start": 2126.12, "duration": 17.86}, {"text": "This seems like a\ncomplicated thing.", "start": 2143.98, "duration": 1.605}, {"text": "So let's replace it\nby something simpler.", "start": 2149.41, "duration": 3.38}, {"text": "Again, this is analysis, which\nmeans we get to use our wits,", "start": 2152.79, "duration": 5.34}, {"text": "and try to replace complicated\nthings by simpler things,", "start": 2158.13, "duration": 2.39}, {"text": "and work with the\nsimpler things.", "start": 2160.52, "duration": 1.375}, {"text": "So let me do a little bit of--", "start": 2164.46, "duration": 1.58}, {"text": "again, this is off\nto the side, how", "start": 2166.04, "duration": 2.82}, {"text": "would one think through this.", "start": 2168.86, "duration": 2.79}, {"text": "If you look at c to\nthe n, remember we", "start": 2171.65, "duration": 1.71}, {"text": "have this inequality\nfrom infinite time", "start": 2173.36, "duration": 3.21}, {"text": "ago that as long as x is bigger\nthan or equal to minus 1,", "start": 2176.57, "duration": 8.19}, {"text": "then I get 1 plus x to\nthe n is bigger than 1", "start": 2184.76, "duration": 3.09}, {"text": "plus n times x for\nall natural numbers n.", "start": 2187.85, "duration": 5.0}, {"text": "So that means c to the n, which\nis equal to 1 plus c minus 1", "start": 2198.45, "duration": 8.6}, {"text": "to the n-- so this is my x--", "start": 2207.05, "duration": 1.8}, {"text": "is bigger than or\nequal to 1 plus n times", "start": 2208.85, "duration": 4.02}, {"text": "c minus 1, which is bigger than\nor equal to n times c minus 1.", "start": 2212.87, "duration": 7.57}, {"text": "You see?", "start": 2220.44, "duration": 0.63}, {"text": "So if I want to make this big,\nit suffices to make this big.", "start": 2221.07, "duration": 7.46}, {"text": "So that's what I'll do.", "start": 2228.53, "duration": 2.25}, {"text": "So let n be a natural\nnumber such that n is", "start": 2230.78, "duration": 11.1}, {"text": "bigger than B over c minus 1.", "start": 2241.88, "duration": 3.24}, {"text": "Then now this we did off\nto the side we'll just", "start": 2255.92, "duration": 2.73}, {"text": "put in the proof now.", "start": 2258.65, "duration": 1.32}, {"text": "And c to the n equals 1\nplus 1 minus c to the n", "start": 2262.51, "duration": 5.34}, {"text": "is bigger than or\nequal to 1 plus n times", "start": 2267.85, "duration": 3.21}, {"text": "c minus 1, which is bigger\nthan or equal to n times", "start": 2271.06, "duration": 3.47}, {"text": "c minus 1, which is bigger\nthan B over c minus 1 times c", "start": 2274.53, "duration": 6.04}, {"text": "minus 1 equals B. So now\nlet's prove the first claim,", "start": 2280.57, "duration": 10.36}, {"text": "that if c is between 0\nand 1, then the limit as n", "start": 2290.93, "duration": 2.94}, {"text": "goes to infinity of\nc to the n equals 0.", "start": 2293.87, "duration": 3.84}, {"text": "So first, I want to prove\nthe following claim--", "start": 2297.71, "duration": 2.76}, {"text": "that we, in fact, have a\ndecreasing sequence, which", "start": 2300.47, "duration": 3.57}, {"text": "is bounded below by 0.", "start": 2304.04, "duration": 4.56}, {"text": "So claim for all n, 0\nis less than c to the n", "start": 2308.6, "duration": 10.61}, {"text": "is less than, I should say, c\nto the n plus 1 is less than c", "start": 2319.21, "duration": 6.3}, {"text": "to the n.", "start": 2325.51, "duration": 0.51}, {"text": "So the proof of this claim is a\nvery simple induction argument,", "start": 2333.72, "duration": 4.27}, {"text": "so we'll do this by induction.", "start": 2337.99, "duration": 1.92}, {"text": "So we have the base\ncase n equals 1.", "start": 2339.91, "duration": 6.11}, {"text": "So we are assuming that\nc is between 0 and 1.", "start": 2346.02, "duration": 11.39}, {"text": "And if I multiply\nthrough by c, I", "start": 2357.41, "duration": 2.28}, {"text": "conclude that 0 is less than\nc squared is less than c.", "start": 2359.69, "duration": 4.52}, {"text": "And that's the n equals 1 case.", "start": 2364.21, "duration": 2.58}, {"text": "And the inductive step is\nessentially the same proof.", "start": 2366.79, "duration": 5.995}, {"text": "Suppose 0 is less than c to\nthe m plus 1 is less than c", "start": 2375.5, "duration": 6.95}, {"text": "to the m.", "start": 2382.45, "duration": 0.64}, {"text": "Then multiplying through by c,\nwe get that 0 is less than c", "start": 2386.17, "duration": 5.11}, {"text": "to the m plus 2 is\nless than c to the m", "start": 2391.28, "duration": 3.74}, {"text": "plus 1, which is\nn equals m plus 1.", "start": 2395.02, "duration": 3.86}, {"text": "Here, we're using\nthis fact here--", "start": 2398.88, "duration": 2.91}, {"text": "that c is positive so it\ndoesn't flip the inequalities,", "start": 2401.79, "duration": 3.61}, {"text": "so I can multiply it through\nand preserve the inequalities.", "start": 2405.4, "duration": 4.61}, {"text": "So this shows that this\nsequence is monotone decreasing", "start": 2410.01, "duration": 10.73}, {"text": "and it's bounded below by\n0, and in fact bounded,", "start": 2420.74, "duration": 4.8}, {"text": "because c to the n and absolute\nvalue, these are all positive,", "start": 2425.54, "duration": 14.99}, {"text": "is equal to c to the n.", "start": 2440.53, "duration": 1.59}, {"text": "And c to the n is less\nthan c to the n minus 1,", "start": 2442.12, "duration": 4.68}, {"text": "which is less than c to the\nn minus 2, so on, and so on,", "start": 2446.8, "duration": 3.84}, {"text": "which is less than c.", "start": 2450.64, "duration": 2.16}, {"text": "So it's bounded.", "start": 2452.8, "duration": 3.54}, {"text": "I guess I could have built\nthat into this inequality", "start": 2456.34, "duration": 6.0}, {"text": "and proved that as\nwell, but that's OK.", "start": 2462.34, "duration": 4.28}, {"text": "So it has a limit.", "start": 2466.62, "duration": 2.13}, {"text": "So by the previous theorem, has\na limit, and I'll call it L.", "start": 2468.75, "duration": 16.34}, {"text": "And now what I want to\nshow is that L is 0.", "start": 2485.09, "duration": 2.145}, {"text": "And how we'll do that is one of\nthese analysis tricks, where--", "start": 2494.35, "duration": 5.55}, {"text": "not really tricks-- but\nrather than show with L", "start": 2499.9, "duration": 2.85}, {"text": "is directly equal\nto 0, we'll show", "start": 2502.75, "duration": 2.22}, {"text": "that the absolute value\nof L is less than epsilon", "start": 2504.97, "duration": 2.79}, {"text": "for every epsilon\npositive, and therefore", "start": 2507.76, "duration": 1.74}, {"text": "it has to be 0 because\nit's just a fixed number.", "start": 2509.5, "duration": 2.0}, {"text": "So let epsilon be positive.", "start": 2518.01, "duration": 1.16}, {"text": "Again, we're going to show that\nthe absolute value of L, which", "start": 2519.17, "duration": 2.583}, {"text": "is just a fixed number,\nis less than epsilon.", "start": 2521.753, "duration": 3.037}, {"text": "And therefore,\ncapital L has to be 0.", "start": 2524.79, "duration": 2.51}, {"text": "Then there exists, since this\nsequence converges, an M,", "start": 2532.43, "duration": 6.06}, {"text": "a natural number, such\nthat for all n bigger", "start": 2538.49, "duration": 5.19}, {"text": "than or equal to capital\nM, c to the n minus L", "start": 2543.68, "duration": 4.11}, {"text": "is less than 1 minus c\ntimes epsilon over 2.", "start": 2547.79, "duration": 8.08}, {"text": "Now, maybe you're wondering why\ndidn't I just use epsilon here?", "start": 2555.87, "duration": 6.61}, {"text": "Well, in the end,\nit's just going", "start": 2562.48, "duration": 3.08}, {"text": "to come out to this being less\nthan epsilon in absolute value.", "start": 2565.56, "duration": 3.818}, {"text": "Otherwise, I would have come\nout with less than epsilon times", "start": 2569.378, "duration": 2.542}, {"text": "some number, and I did\naway with that number", "start": 2571.92, "duration": 2.19}, {"text": "by choosing a\ndifferent number here.", "start": 2574.11, "duration": 3.1}, {"text": "So now we compute that\nif I look at 1 minus c", "start": 2577.21, "duration": 7.61}, {"text": "times the absolute\nvalue of L, this", "start": 2584.82, "duration": 2.28}, {"text": "is equal to L minus\nc times L. And this", "start": 2587.1, "duration": 5.42}, {"text": "is less than or equal to L\nminus c to the capital M plus--", "start": 2592.52, "duration": 8.98}, {"text": "let's put a plus 1, plus c to\nthe M plus 1 minus c to the L.", "start": 2604.89, "duration": 5.1}, {"text": "And by the triangle\ninequality, this", "start": 2609.99, "duration": 2.43}, {"text": "is less than or equal to L\nminus c to the M plus 1 plus--", "start": 2612.42, "duration": 5.46}, {"text": "now c to the M plus\n1 minus c to the L,", "start": 2617.88, "duration": 2.85}, {"text": "so I can pull out a c which\nis positive, so I can get L.", "start": 2620.73, "duration": 7.13}, {"text": "Now, M plus 1 is\nbigger than M. So it", "start": 2627.86, "duration": 4.53}, {"text": "satisfies this inequality,\nand so does this one.", "start": 2632.39, "duration": 5.43}, {"text": "And therefore, this is less\nthan epsilon over 2 times", "start": 2637.82, "duration": 3.75}, {"text": "1 minus c plus c times epsilon\nover 2 times 1 minus c.", "start": 2641.57, "duration": 6.63}, {"text": "And C is less than 1,\nso this whole thing", "start": 2648.2, "duration": 2.94}, {"text": "is less than\nepsilon over 2 times", "start": 2651.14, "duration": 1.83}, {"text": "1 minus c plus another\nepsilon over 2.", "start": 2652.97, "duration": 2.64}, {"text": "So I get epsilon over\n2 times 1 minus c.", "start": 2655.61, "duration": 5.24}, {"text": "And thus, the absolute value\nof L is less than epsilon.", "start": 2666.66, "duration": 8.3}, {"text": "And since epsilon was\narbitrary, that implies", "start": 2674.96, "duration": 2.39}, {"text": "the absolute value of L is equal\nto 0, i.e., L is equal to 0.", "start": 2677.35, "duration": 4.92}, {"text": "So that's a very\nconcrete application", "start": 2691.56, "duration": 4.44}, {"text": "of some of these theorems we've\nbeen proving, which is really--", "start": 2696.0, "duration": 4.98}, {"text": "I think the only real\nreason one proves", "start": 2700.98, "duration": 2.07}, {"text": "theorems is you typically have\na concrete example of something", "start": 2703.05, "duration": 7.163}, {"text": "in mind that you want to study.", "start": 2710.213, "duration": 1.292}, {"text": "But in order to\nstudy it, it often", "start": 2714.3, "duration": 2.06}, {"text": "requires some general\nmachinery, i.e., theorems.", "start": 2716.36, "duration": 6.575}, {"text": "So now we'll talk a\nlittle bit about sequences", "start": 2735.68, "duration": 3.15}, {"text": "obtained from other sequences.", "start": 2738.83, "duration": 3.72}, {"text": "So these are called\nsubsequences,", "start": 2742.55, "duration": 6.39}, {"text": "or sequences obtained\nfrom a single sequence.", "start": 2748.94, "duration": 7.177}, {"text": "So what do I mean by this?", "start": 2756.117, "duration": 1.083}, {"text": "Let me give you the\nprecise definition.", "start": 2757.2, "duration": 2.19}, {"text": "So we started off with a\nsequence and an increasing", "start": 2767.2, "duration": 9.18}, {"text": "sequence of integers, n sub k.", "start": 2776.38, "duration": 6.87}, {"text": "So let me just say what this\nmeans rather than write out", "start": 2791.91, "duration": 3.91}, {"text": "the word and then\nsay what it means.", "start": 2795.82, "duration": 2.29}, {"text": "So this is the sequence\nof natural numbers, which", "start": 2798.11, "duration": 13.82}, {"text": "are strictly increasing.\nn sub 1 is less", "start": 2811.93, "duration": 2.4}, {"text": "than n sub 2 is less\nthan n sub 3, and so on.", "start": 2814.33, "duration": 3.81}, {"text": "The new sequence, x sub n sub\nk-- so now the index is not n,", "start": 2825.62, "duration": 8.51}, {"text": "but the index is k--", "start": 2834.13, "duration": 1.22}, {"text": "is called a subsequence of\nthe original sequence x sub n.", "start": 2840.6, "duration": 12.87}, {"text": "So how should you view\nsubsequences of a sequence?", "start": 2857.4, "duration": 4.92}, {"text": "You should think that I line\nup all the entries of x,", "start": 2862.32, "duration": 5.46}, {"text": "of my original sequence\nx sub n, and then", "start": 2867.78, "duration": 4.05}, {"text": "I just start picking\nentries out of the sequence.", "start": 2871.83, "duration": 5.94}, {"text": "But every time I make a choice,\nI have to move to the right", "start": 2877.77, "duration": 3.48}, {"text": "and make another choice.", "start": 2881.25, "duration": 2.16}, {"text": "That's what I was just saying\nthere, expresses this condition", "start": 2883.41, "duration": 3.57}, {"text": "that this sequence of natural\nnumbers is increasing.", "start": 2886.98, "duration": 3.76}, {"text": "So I pick an entry\nin the sequence.", "start": 2890.74, "duration": 2.69}, {"text": "That's going to be my first\nguy, the first element", "start": 2893.43, "duration": 3.27}, {"text": "of my new subsequence.", "start": 2896.7, "duration": 1.5}, {"text": "And then I move to the right,\nand maybe I pick the next one,", "start": 2898.2, "duration": 4.44}, {"text": "maybe I pick one three down.", "start": 2902.64, "duration": 1.68}, {"text": "I pick that one.", "start": 2904.32, "duration": 1.18}, {"text": "And then I move to the right\nof that one and pick a new one.", "start": 2905.5, "duration": 4.9}, {"text": "And then I move to the right of\nthat one and pick another one.", "start": 2910.4, "duration": 2.91}, {"text": "So if you want to\ngenerate a subsequence", "start": 2913.31, "duration": 2.97}, {"text": "from an original sequence,\nhow do you think about this?", "start": 2916.28, "duration": 4.2}, {"text": "Again, line them all up,\nstart picking entries,", "start": 2920.48, "duration": 2.585}, {"text": "but every time\nyou pick an entry,", "start": 2923.065, "duration": 1.375}, {"text": "you have to move to the right in\norder to pick your next entry.", "start": 2924.44, "duration": 3.435}, {"text": "So let me give you some\nexamples and non-examples.", "start": 2932.6, "duration": 3.58}, {"text": "So for example, if our original\nsequence is 1, 2, 3, 4, 5, 6,", "start": 2936.18, "duration": 10.55}, {"text": "and so on, what would be\nexamples of some subsequences?", "start": 2946.73, "duration": 6.15}, {"text": "The odd numbers 1,\n3, 5, 7, 9 and so on.", "start": 2952.88, "duration": 6.28}, {"text": "You see how I'm taking\nthe original sequence,", "start": 2959.16, "duration": 2.4}, {"text": "so this is x sub n.", "start": 2961.56, "duration": 4.003}, {"text": "I'm just picking entries\nfrom the original sequence,", "start": 2965.563, "duration": 2.167}, {"text": "but every time I pick an\nentry, I move to the right", "start": 2967.73, "duration": 2.208}, {"text": "and pick a new one.", "start": 2969.938, "duration": 0.792}, {"text": "So I picked 1, now I\nget to choose anything", "start": 2970.73, "duration": 2.34}, {"text": "to the right of that.", "start": 2973.07, "duration": 1.24}, {"text": "I pick 3, now I can choose\nanything to the right of that.", "start": 2974.31, "duration": 3.75}, {"text": "So in the language\nof this definition,", "start": 2978.06, "duration": 2.42}, {"text": "the increasing\nsequence of integers", "start": 2980.48, "duration": 3.9}, {"text": "is just the odd\nintegers, so 2k minus 1.", "start": 2984.38, "duration": 3.87}, {"text": "So here, x sub n\nis equal to just n.", "start": 2988.25, "duration": 3.84}, {"text": "I could pick another\nsubsequence would be", "start": 2997.42, "duration": 3.39}, {"text": "the sequence of even numbers--", "start": 3000.81, "duration": 2.07}, {"text": "2, 4, 6, 8, 10.", "start": 3002.88, "duration": 4.74}, {"text": "This is a subsequence of\nthis original sequence.", "start": 3007.62, "duration": 2.43}, {"text": "Here, the increasing\nsequence of integers is 2k.", "start": 3010.05, "duration": 3.54}, {"text": "I could pick the\nsubsequence could be", "start": 3020.16, "duration": 3.09}, {"text": "the sequence of prime numbers--", "start": 3023.25, "duration": 2.79}, {"text": "2, 3, 5, 7, 11, 13.", "start": 3026.04, "duration": 4.5}, {"text": "And the increasing--\nso I don't have", "start": 3034.0, "duration": 4.51}, {"text": "a general formula for that.", "start": 3038.51, "duration": 2.22}, {"text": "Good luck finding\none, but we'll just", "start": 3040.73, "duration": 2.49}, {"text": "write this as k-th prime number.", "start": 3043.22, "duration": 5.905}, {"text": "Now, what would\nnot be an example?", "start": 3057.79, "duration": 2.62}, {"text": "So these are examples\nof subsequences.", "start": 3060.41, "duration": 2.48}, {"text": "Not examples of\nsubsequences would", "start": 3065.65, "duration": 2.13}, {"text": "be, for example, the\nsequence 1, 1, 1, 1, 1.", "start": 3067.78, "duration": 4.5}, {"text": "This is not a subsequence\nof this original sequence.", "start": 3072.28, "duration": 3.84}, {"text": "Remember, I have a\nsequence of natural numbers", "start": 3076.12, "duration": 2.41}, {"text": "which is increasing.", "start": 3078.53, "duration": 1.49}, {"text": "And x sub n sub k means\nthat this new sequence is", "start": 3080.02, "duration": 7.05}, {"text": "obtained from the old\none by picking entries--", "start": 3087.07, "duration": 3.09}, {"text": "by picking an entry,\nmoving to the right,", "start": 3090.16, "duration": 2.067}, {"text": "picking the next entry,\nmoving to the right,", "start": 3092.227, "duration": 1.833}, {"text": "and picking the next entry.", "start": 3094.06, "duration": 1.65}, {"text": "Here, in order to get this\nsequence from this one,", "start": 3095.71, "duration": 3.345}, {"text": "I just stay on the first\nentry and keep picking it.", "start": 3099.055, "duration": 3.585}, {"text": "But that's not how a\nsubsequence is defined.", "start": 3102.64, "duration": 3.945}, {"text": "So here, if you like, this would\nmean n sub k is equal to 1.", "start": 3109.43, "duration": 7.07}, {"text": "And this is not a strictly\nincreasing sequence", "start": 3116.5, "duration": 2.163}, {"text": "of natural numbers.", "start": 3118.663, "duration": 0.792}, {"text": "Or 1, 1 3, 3, 5, 5, and\nso on-- that's also not.", "start": 3123.04, "duration": 6.79}, {"text": "So these are not subsequences\nof the original sequence", "start": 3129.83, "duration": 10.22}, {"text": "given here.", "start": 3140.05, "duration": 0.75}, {"text": "So I hope that's clear.", "start": 3147.527, "duration": 0.958}, {"text": "Now, I want to emphasize this--", "start": 3154.0, "duration": 6.04}, {"text": "I'm not saying that the\nnumber you pick each time", "start": 3160.04, "duration": 6.39}, {"text": "has to be different from the\nnumber you picked before,", "start": 3166.43, "duration": 3.48}, {"text": "meaning the value.", "start": 3169.91, "duration": 2.31}, {"text": "You couldn't have 1, 1, 1, 1\nas a subsequence of this guy", "start": 3172.22, "duration": 4.53}, {"text": "because 1 is only\nin the first entry.", "start": 3176.75, "duration": 2.49}, {"text": "You can't keep\npicking one entry.", "start": 3179.24, "duration": 2.16}, {"text": "You have to pick\nan entry and move", "start": 3181.4, "duration": 1.71}, {"text": "to the next entry or\nthe one after that", "start": 3183.11, "duration": 2.97}, {"text": "and pick that entry to\nobtain your sequence.", "start": 3186.08, "duration": 3.72}, {"text": "But that doesn't\nnecessarily mean", "start": 3189.8, "duration": 1.77}, {"text": "that those actual\nnumbers in those entries", "start": 3191.57, "duration": 4.56}, {"text": "have to be different.", "start": 3196.13, "duration": 1.12}, {"text": "So for example, if I look\nat the sequence minus 1,", "start": 3197.25, "duration": 6.86}, {"text": "1, 1, minus 1, 1, one meaning\nx sub n is equal to minus 1", "start": 3204.11, "duration": 7.16}, {"text": "to the n-th then a perfectly\ngood subsequence is given", "start": 3211.27, "duration": 5.58}, {"text": "by minus 1, minus\n1, minus 1, minus 1,", "start": 3216.85, "duration": 5.22}, {"text": "where it looks like I'm just\npicking one of the entries,", "start": 3222.07, "duration": 3.06}, {"text": "but really I'm not.", "start": 3225.13, "duration": 0.99}, {"text": "Here, the increasing\nsequence-- so", "start": 3226.12, "duration": 2.91}, {"text": "what I'm doing here to\nget this new sequence,", "start": 3229.03, "duration": 2.43}, {"text": "I picked the first entry,\nthen I picked the third,", "start": 3231.46, "duration": 2.7}, {"text": "then I picked the fifth,\nthen I picked the seventh.", "start": 3234.16, "duration": 2.25}, {"text": "So n sub k equals 2k minus 1.", "start": 3236.41, "duration": 5.1}, {"text": "So all the actual numbers\nappearing in the sequence", "start": 3241.51, "duration": 3.09}, {"text": "are the same, but I'm actually\npicking different entries", "start": 3244.6, "duration": 5.37}, {"text": "from the original sequence,\nwhere they appear.", "start": 3249.97, "duration": 4.25}, {"text": "So this is also fine.", "start": 3254.22, "duration": 1.41}, {"text": "So is 1, 1, 1, 1, 1.", "start": 3255.63, "duration": 2.795}, {"text": "This is n k equals 2 times k.", "start": 3262.08, "duration": 2.35}, {"text": "So these are all\ngood subsequences.", "start": 3264.43, "duration": 3.0}, {"text": "So these were subsequences\nof the original sequence.", "start": 3267.43, "duration": 6.76}, {"text": "These are subsequences\nof this sequence.", "start": 3274.19, "duration": 2.88}, {"text": "So from a given sequence,\nwe can obtain new sequences", "start": 3309.12, "duration": 5.58}, {"text": "from this original\nsequence in this way.", "start": 3314.7, "duration": 4.91}, {"text": "So a natural question\nis to ask, how", "start": 3319.61, "duration": 2.45}, {"text": "does going from a\nsequence to a subsequence", "start": 3322.06, "duration": 11.29}, {"text": "behave with respect to limits?", "start": 3333.35, "duration": 2.22}, {"text": "If the original sequence\nconverges to a limit,", "start": 3335.57, "duration": 2.76}, {"text": "does a subsequence or\ndoes every subsequence", "start": 3338.33, "duration": 2.73}, {"text": "converge to that limit as well?", "start": 3341.06, "duration": 2.38}, {"text": "And this shouldn't come as\na surprise that it does.", "start": 3343.44, "duration": 2.48}, {"text": "Because all of the elements of\nthe sequence are getting close,", "start": 3345.92, "duration": 4.728}, {"text": "all the entries of the\nsequence are getting", "start": 3350.648, "duration": 1.792}, {"text": "close to a given number.", "start": 3352.44, "duration": 1.5}, {"text": "And a subsequence is just, if\nyou like, picking certain ones", "start": 3353.94, "duration": 3.57}, {"text": "along the way.", "start": 3357.51, "duration": 1.12}, {"text": "So certainly if I'm\nin a subsequence,", "start": 3358.63, "duration": 2.99}, {"text": "then as long as I'm far\nenough out in the subsequence,", "start": 3361.62, "duration": 2.85}, {"text": "I'll be close to\nthat limit as well.", "start": 3364.47, "duration": 2.07}, {"text": "So if you didn't get\nall that rambling,", "start": 3366.54, "duration": 3.0}, {"text": "I'll go ahead and state\nthe theorem and prove it.", "start": 3369.54, "duration": 3.87}, {"text": "x n is a sequence\nwhich converges to x.", "start": 3389.27, "duration": 4.305}, {"text": "And x sub n sub k is a\nsubsequence of x sub n.", "start": 3403.15, "duration": 15.37}, {"text": "Then the subsequence\nconverges to x, as well.", "start": 3418.52, "duration": 4.33}, {"text": "Limit as k goes to infinity\nof x sub n sub k equals x.", "start": 3427.2, "duration": 5.31}, {"text": "So let me just start off by\nmaking a simple observation", "start": 3454.25, "duration": 4.44}, {"text": "based on the fact that\nthis increasing sequence--", "start": 3458.69, "duration": 3.3}, {"text": "that the n sub k's\nare an increasing", "start": 3461.99, "duration": 1.728}, {"text": "sequence of natural numbers.", "start": 3463.718, "duration": 1.167}, {"text": "Since n sub 1 is bigger than\nor equal to 1 is less than n", "start": 3467.42, "duration": 8.02}, {"text": "sub 2 is less than\nn sub 3 and so on,", "start": 3475.44, "duration": 4.56}, {"text": "this implies that for\nall k, a natural number,", "start": 3480.0, "duration": 5.37}, {"text": "n sub k is bigger\nthan or equal to k.", "start": 3485.37, "duration": 2.82}, {"text": "So this is not so hard to\nbelieve because n sub 1 has", "start": 3493.92, "duration": 2.67}, {"text": "to be at least bigger than 1.", "start": 3496.59, "duration": 1.56}, {"text": "And n sub 2 has to be at\nleast bigger than n sub 1,", "start": 3498.15, "duration": 2.73}, {"text": "so it's either n sub\n1 plus 1 or bigger.", "start": 3500.88, "duration": 3.39}, {"text": "And therefore, by a\nsimple induction argument,", "start": 3504.27, "duration": 3.787}, {"text": "which I'll leave to\nyou, you can prove", "start": 3508.057, "duration": 1.583}, {"text": "that for all k,\na natural number,", "start": 3509.64, "duration": 1.485}, {"text": "n sub k is bigger than or\nequal to k, induct on k.", "start": 3511.125, "duration": 4.58}, {"text": "So we want to prove that\nthis sequence converges to x.", "start": 3521.34, "duration": 3.6}, {"text": "And we're going to do that\nusing the only means we have.", "start": 3524.94, "duration": 2.55}, {"text": "So this is just some arbitrary\nsubsequence of x sub n.", "start": 3527.49, "duration": 4.29}, {"text": "All we have is the\ndefinition to use.", "start": 3531.78, "duration": 1.695}, {"text": "So we're going to\nnow prove this.", "start": 3536.22, "duration": 1.43}, {"text": "So we have to show\nfor all epsilon", "start": 3537.65, "duration": 1.68}, {"text": "positive, blah, blah, blah, so\nthe first thing we have to do", "start": 3539.33, "duration": 3.12}, {"text": "is let epsilon be positive.", "start": 3542.45, "duration": 2.14}, {"text": "Now we'll use the fact that\nx sub n converges to x.", "start": 3544.59, "duration": 2.54}, {"text": "So x sub n converges to x.", "start": 3550.655, "duration": 3.855}, {"text": "There exists a natural\nnumber M0 such that", "start": 3554.51, "duration": 5.28}, {"text": "for all n bigger than\nor equal to M0, x", "start": 3559.79, "duration": 10.57}, {"text": "sub n minus x absolute\nvalue is less than epsilon.", "start": 3570.36, "duration": 5.36}, {"text": "This M sub 0 we'll choose\nfor our subsequence.", "start": 3575.72, "duration": 2.691}, {"text": "Choose M to equal\nM sub 0, so now we", "start": 3581.06, "duration": 4.64}, {"text": "need to show that this capital\nM works for our subsequence,", "start": 3585.7, "duration": 2.55}, {"text": "meaning if k is bigger\nthan or equal to capital M,", "start": 3588.25, "duration": 2.82}, {"text": "then x sub n sub k minus x\nis an absolute value less", "start": 3591.07, "duration": 5.28}, {"text": "than epsilon.", "start": 3596.35, "duration": 0.82}, {"text": "But this just follows\nfrom this fact.", "start": 3597.17, "duration": 4.67}, {"text": "And if k is bigger than or\nequal to M, this implies,", "start": 3601.84, "duration": 4.95}, {"text": "by this inequality, n\nsub k is bigger than", "start": 3606.79, "duration": 5.07}, {"text": "or equal to M, which\nremember, is equal to M sub 0.", "start": 3611.86, "duration": 3.33}, {"text": "So n sub k is some\ninteger, so natural number", "start": 3615.19, "duration": 3.18}, {"text": "bigger than or equal to M sub 0.", "start": 3618.37, "duration": 3.12}, {"text": "So that implies\nby this inequality", "start": 3621.49, "duration": 2.22}, {"text": "that x sub n sub k minus\nx in absolute value", "start": 3623.71, "duration": 4.38}, {"text": "is less than epsilon.", "start": 3628.09, "duration": 2.015}] |