transcripts/0_w-R_g5lRA.json

[{"text": "[SQUEAKING]", "start": 0.0, "duration": 2.495}, {"text": "[RUSTLING]", "start": 2.495, "duration": 1.497}, {"text": "[CLICKING]", "start": 3.992, "duration": 1.497}, {"text": "CASEY RODRIGUEZ: OK, so let's\ncontinue our study of sequences", "start": 13.99, "duration": 3.173}, {"text": "of real numbers.", "start": 17.163, "duration": 0.667}, {"text": "So we've seen special types of\nsequences, monotone sequences,", "start": 24.24, "duration": 3.9}, {"text": "before.", "start": 28.14, "duration": 0.51}, {"text": "And then in the\nprevious lecture,", "start": 28.65, "duration": 4.12}, {"text": "we looked at sequences\nobtained from sequences,", "start": 32.77, "duration": 6.45}, {"text": "namely the sequences that give\nyou the lim sup and lim inf.", "start": 39.22, "duration": 2.958}, {"text": "And we showed that\nthese are actually", "start": 42.178, "duration": 1.542}, {"text": "limits of subsequences.", "start": 43.72, "duration": 3.453}, {"text": "Now I'm going to\ndefine what looks", "start": 47.173, "duration": 1.417}, {"text": "like a new class of sequences.", "start": 48.59, "duration": 5.88}, {"text": "But we'll see it's actually not.", "start": 54.47, "duration": 1.74}, {"text": "These are called\nCauchy sequences.", "start": 56.21, "duration": 1.83}, {"text": "So \"coe-shee\"-- not\n\"couch-ee,\" not \"cawt-shee\"--", "start": 62.45, "duration": 7.33}, {"text": "Cauchy, so a French guy.", "start": 69.78, "duration": 3.15}, {"text": "So it's pronounced Cauchy and\nprobably not even pronounced", "start": 72.93, "duration": 3.03}, {"text": "like that.", "start": 75.96, "duration": 0.5}, {"text": "It's probably got a\ndifferent pronunciation", "start": 76.46, "duration": 8.04}, {"text": "by people who\nactually speak French.", "start": 84.5, "duration": 1.98}, {"text": "So what is the definition\nof a Cauchy sequence?", "start": 86.48, "duration": 2.655}, {"text": "A Cauchy sequence,\nintuitively, it's", "start": 89.135, "duration": 3.945}, {"text": "a sequence so that if you go\nfar enough out in the sequence,", "start": 93.08, "duration": 5.28}, {"text": "any two entries in that\nsequence are close together.", "start": 98.36, "duration": 4.35}, {"text": "So convergent sequences\nhad the property", "start": 102.71, "duration": 2.61}, {"text": "that if you go far enough out,\nthe entries in the sequence", "start": 105.32, "duration": 3.66}, {"text": "are getting close\nto a real number.", "start": 108.98, "duration": 2.76}, {"text": "Cauchy sequence is\nthat any two entries", "start": 111.74, "duration": 2.58}, {"text": "are close to each other.", "start": 114.32, "duration": 1.305}, {"text": "So a Cauchy sequence, so\nwe say a sequence is Cauchy", "start": 118.72, "duration": 12.87}, {"text": "if we're all epsilon\npositive, there", "start": 131.59, "duration": 6.9}, {"text": "exists an M,\nnatural number, such", "start": 138.49, "duration": 3.54}, {"text": "that if n is bigger\nthan or equal to M and k", "start": 142.03, "duration": 7.54}, {"text": "is bigger than or equal\nto M, then xn minus xk", "start": 149.57, "duration": 6.895}, {"text": "is less than epsilon.", "start": 156.465, "duration": 0.875}, {"text": "So maybe not write if.", "start": 161.05, "duration": 2.33}, {"text": "I mean, it's the same\nstatement, but since it looks--", "start": 163.38, "duration": 2.49}, {"text": "so it'll look a little more\nlike previous statements", "start": 165.87, "duration": 4.25}, {"text": "when we put a \"for all\" there.", "start": 170.12, "duration": 1.25}, {"text": "So you have a definition here.", "start": 174.07, "duration": 3.345}, {"text": "It's the definition\nof a new thing.", "start": 180.99, "duration": 3.43}, {"text": "You should now try\nto look at an example", "start": 184.42, "duration": 2.88}, {"text": "and then possibly negate the\ndefinition to see if you really", "start": 187.3, "duration": 2.82}, {"text": "understand it.", "start": 190.12, "duration": 1.15}, {"text": "So an example of\na Cauchy sequence", "start": 191.27, "duration": 3.02}, {"text": "is x of n equals 1 over\nn, our favorite sequence.", "start": 194.29, "duration": 6.37}, {"text": "So let's prove this.", "start": 200.66, "duration": 2.27}, {"text": "So all we have is\nthe definition.", "start": 202.93, "duration": 1.96}, {"text": "So we have to verify that\nx of n equals 1 over n", "start": 204.89, "duration": 3.41}, {"text": "verifies the definition\nof being Cauchy.", "start": 208.3, "duration": 3.78}, {"text": "So just like when we try to\nprove something is convergent,", "start": 212.08, "duration": 2.7}, {"text": "which is a \"for all\" epsilon\nstatement, the first thing", "start": 214.78, "duration": 3.51}, {"text": "you have to do is let\nepsilon be positive.", "start": 218.29, "duration": 3.43}, {"text": "And then I have to choose M\nand show that that capital", "start": 221.72, "duration": 2.48}, {"text": "M produces this statement here.", "start": 224.2, "duration": 3.76}, {"text": "So choose M, a natural\nnumber, so that 1 over M", "start": 227.96, "duration": 9.17}, {"text": "is less than epsilon over 2.", "start": 237.13, "duration": 2.13}, {"text": "So I could phrase that as\ncapital M being bigger than 2", "start": 239.26, "duration": 3.33}, {"text": "over epsilon, but I'm going\nto phrase it this way.", "start": 242.59, "duration": 4.05}, {"text": "Now we have to show\nthat it works-- namely,", "start": 246.64, "duration": 2.34}, {"text": "if I take n bigger than\nor equal to capital M", "start": 248.98, "duration": 2.16}, {"text": "and k bigger than or\nequal to a capital M,", "start": 251.14, "duration": 2.13}, {"text": "then this difference\nis less than epsilon.", "start": 253.27, "duration": 2.67}, {"text": "Then if n is bigger\nthan or equal to M,", "start": 255.94, "duration": 5.97}, {"text": "k is bigger than or equal to M,\nand I look at 1 over n minus 1", "start": 261.91, "duration": 4.71}, {"text": "over k, this is less than\nor equal to, by the triangle", "start": 266.62, "duration": 4.47}, {"text": "inequality, the\nabsolute value of each", "start": 271.09, "duration": 3.45}, {"text": "of these added together, which\nis just 1 over n plus 1 over k.", "start": 274.54, "duration": 5.01}, {"text": "And since these are both\nbigger than or equal to M,", "start": 279.55, "duration": 3.06}, {"text": "each 1 is less than\nor equal to 1 over M 1", "start": 282.61, "duration": 3.705}, {"text": "over M, so I get 2 over M,\nwhich, by our choice of M,", "start": 286.315, "duration": 4.635}, {"text": "is less than epsilon.", "start": 290.95, "duration": 3.41}, {"text": "So x of n equals 1 over n is an\nexample of a Cauchy sequence.", "start": 294.36, "duration": 7.33}, {"text": "So let's negate the\ndefinition, and then we'll", "start": 301.69, "duration": 8.68}, {"text": "look at an example of a\nsequence which is not Cauchy.", "start": 310.37, "duration": 2.23}, {"text": "And as you'll probably guess, if\nthis is our favorite sequence,", "start": 312.6, "duration": 2.625}, {"text": "which converges, our favorite\nsequence which doesn't converge", "start": 315.225, "duration": 4.535}, {"text": "will be an example of a\nsequence which is not Cauchy.", "start": 319.76, "duration": 3.885}, {"text": "And this should shouldn't\ncome as a surprise.", "start": 323.645, "duration": 1.875}, {"text": "Because, again, a sequence\nwhich is Cauchy, if you", "start": 325.52, "duration": 4.56}, {"text": "go far enough out,\nany two entries", "start": 330.08, "duration": 3.33}, {"text": "are close to each other.", "start": 333.41, "duration": 1.74}, {"text": "But if we look at, for\nexample, the sequence minus 1", "start": 335.15, "duration": 2.22}, {"text": "to the n, which is just\nminus 1 plus 1 minus 1,", "start": 337.37, "duration": 2.64}, {"text": "any two entries will differ by--", "start": 340.01, "duration": 4.987}, {"text": "or you can always\nchoose two entries--", "start": 344.997, "duration": 1.583}, {"text": "that differ by 2 in distance.", "start": 346.58, "duration": 5.83}, {"text": "So let's negate this\ndefinition to get what it means", "start": 352.41, "duration": 8.47}, {"text": "for something not to be Cauchy.", "start": 360.88, "duration": 2.4}, {"text": "So we'll not write all that out.", "start": 363.28, "duration": 3.42}, {"text": "So x of n is not Cauchy if--", "start": 373.45, "duration": 4.65}, {"text": "so every time we\nsee a \"for all,\"", "start": 378.1, "duration": 2.19}, {"text": "it becomes \"there exists.\"", "start": 380.29, "duration": 3.75}, {"text": "If there exists a bad\nepsilon 0 positive", "start": 384.04, "duration": 5.1}, {"text": "such that for all\nM, a natural number,", "start": 389.14, "duration": 4.89}, {"text": "you can find two entries\nfurther out than M", "start": 394.03, "duration": 4.68}, {"text": "that are greater than epsilon\n0 distance to each other.", "start": 398.71, "duration": 3.28}, {"text": "So there exists n bigger\nthan or equal to M", "start": 401.99, "duration": 3.44}, {"text": "and k bigger than or equal to\nM such that x of n minus x of k", "start": 405.43, "duration": 7.68}, {"text": "is bigger than or equal\nto this bad epsilon.", "start": 413.11, "duration": 1.89}, {"text": "OK", "start": 417.72, "duration": 2.53}, {"text": "Again, the definition\nof Cauchy means", "start": 420.25, "duration": 3.03}, {"text": "that, as long as I go far\nenough out in the sequence,", "start": 423.28, "duration": 8.33}, {"text": "this distance is supposed\nto be less than epsilon.", "start": 431.61, "duration": 2.72}, {"text": "So for all epsilon positive,\nthere exists a capital M", "start": 434.33, "duration": 3.13}, {"text": "so that I have this picture.", "start": 437.46, "duration": 1.62}, {"text": "If I choose x sub k plus\n1, then it should also", "start": 439.08, "duration": 4.53}, {"text": "be within distance epsilon\nto x sub n or x sub k.", "start": 443.61, "duration": 5.22}, {"text": "So they're getting closer\nand closer together.", "start": 448.83, "duration": 2.58}, {"text": "The negation means that they're\nnot getting closer and closer", "start": 451.41, "duration": 3.27}, {"text": "together to each other.", "start": 454.68, "duration": 1.15}, {"text": "So there exists\nsome small distance", "start": 455.83, "duration": 3.27}, {"text": "so that you can always go as\nfar out as you want and find", "start": 459.1, "duration": 4.13}, {"text": "two entries that are greater\nthan epsilon 0 distance", "start": 463.23, "duration": 4.47}, {"text": "to each other.", "start": 467.7, "duration": 0.78}, {"text": "So what's an example of that?", "start": 471.51, "duration": 2.06}, {"text": "Like I said, the sequence\nminus 1 to the n is not Cauchy.", "start": 473.57, "duration": 13.18}, {"text": "That just doesn't look right.", "start": 497.04, "duration": 1.32}, {"text": "There we go.", "start": 498.36, "duration": 0.63}, {"text": "Now it is.", "start": 503.98, "duration": 0.78}, {"text": "So this is not Cauchy.", "start": 507.94, "duration": 1.35}, {"text": "So that means there should\nexist some bad epsilon 0.", "start": 509.29, "duration": 4.769}, {"text": "So I can go as far\nout as I want and find", "start": 514.059, "duration": 2.881}, {"text": "two entries in the sequence\ndiffering from each other", "start": 516.94, "duration": 3.239}, {"text": "by epsilon 0 in distance.", "start": 520.179, "duration": 2.201}, {"text": "So basically, I can\nalways find two entries", "start": 522.38, "duration": 6.56}, {"text": "in the sequence which\ndiffer from each other by 2.", "start": 528.94, "duration": 3.6}, {"text": "So that'll be my bad epsilon 0.", "start": 532.54, "duration": 3.51}, {"text": "So if you like, here's a proof.", "start": 536.05, "duration": 3.96}, {"text": "Choose epsilon 0 equals 2.", "start": 540.01, "duration": 3.855}, {"text": "Let M be a natural number.", "start": 546.83, "duration": 1.42}, {"text": "So now we have to find\nan element of entries", "start": 548.25, "duration": 3.98}, {"text": "in the sequence further out than\nM whose distance to each other", "start": 552.23, "duration": 3.412}, {"text": "is bigger than or equal to 2.", "start": 555.642, "duration": 1.208}, {"text": "We can just take M plus\n1 and capital M. Choose n", "start": 556.85, "duration": 8.24}, {"text": "equals M and k equals M plus 1.", "start": 565.09, "duration": 4.59}, {"text": "So these are both bigger\nthan or equal to M.", "start": 569.68, "duration": 3.87}, {"text": "Then minus 1 to the\nn minus 1 to the k,", "start": 573.55, "duration": 5.52}, {"text": "this is equal to 1 minus minus\n1 after I factor out a minus 1", "start": 579.07, "duration": 8.15}, {"text": "to the capital M,\nwhich equals 2.", "start": 587.22, "duration": 2.19}, {"text": "So minus 1 to the\nn is not Cauchy.", "start": 593.95, "duration": 4.095}, {"text": "So I, at the beginning,\nsaid that this", "start": 605.75, "duration": 3.69}, {"text": "will look like a definition\nof a new type of sequence,", "start": 609.44, "duration": 4.71}, {"text": "but it's not, really.", "start": 614.15, "duration": 1.59}, {"text": "So as it turns out, the\nelements of the sequence", "start": 615.74, "duration": 11.9}, {"text": "are getting closer\nand closer together", "start": 627.64, "duration": 2.37}, {"text": "as you go far enough out.", "start": 630.01, "duration": 3.01}, {"text": "So", "start": 633.02, "duration": 0.5}, {"text": "They're all kind of clustering\nnear each other, which kind of", "start": 633.52, "duration": 2.94}, {"text": "makes you think they're all\nclustering near something", "start": 636.46, "duration": 2.31}, {"text": "in the real number line.", "start": 638.77, "duration": 2.64}, {"text": "And therefore, maybe, the\nsequence is convergent.", "start": 641.41, "duration": 3.21}, {"text": "Now, this is true--\nand we'll prove this--", "start": 644.62, "duration": 5.75}, {"text": "that a sequence is Cauchy if\nand only if it's convergent.", "start": 650.37, "duration": 3.63}, {"text": "Now, this is true only\nfor the real numbers.", "start": 654.0, "duration": 2.43}, {"text": "And I'll say a little bit\nabout this in a minute--", "start": 656.43, "duration": 2.947}, {"text": "or not only true for\nthe real numbers,", "start": 659.377, "duration": 1.583}, {"text": "but it's not true for\nthe rational numbers.", "start": 660.96, "duration": 3.73}, {"text": "And I'll get to that\nin just a second.", "start": 664.69, "duration": 1.84}, {"text": "So what we're going to prove\nis that a sequence is Cauchy if", "start": 666.53, "duration": 3.56}, {"text": "and only if it is convergent.", "start": 670.09, "duration": 2.98}, {"text": "So the first thing\nI want to show", "start": 673.07, "duration": 5.85}, {"text": "is that Cauchy\nsequences are bounded.", "start": 678.92, "duration": 6.775}, {"text": "So the proof of this\nstatement is essentially", "start": 695.36, "duration": 3.45}, {"text": "the same as the proof\nthat convergent sequences", "start": 698.81, "duration": 5.35}, {"text": "are bounded.", "start": 704.16, "duration": 0.5}, {"text": "So let me draw a picture that\ngoes along with this proof.", "start": 704.66, "duration": 3.97}, {"text": "So as long as I\ngo far enough out,", "start": 708.63, "duration": 2.51}, {"text": "there exists an M so\nthat for all n bigger", "start": 711.14, "duration": 6.39}, {"text": "than or equal to capital\nM, I can say this.", "start": 717.53, "duration": 2.67}, {"text": "Let's look at this entry x\nof M. Then for all n bigger", "start": 720.2, "duration": 4.2}, {"text": "than or equal to capital\nM, all of the other entries", "start": 724.4, "duration": 2.49}, {"text": "have to be within a certain\ndistance to x sub n,", "start": 726.89, "duration": 3.6}, {"text": "based on the\ndefinition of Cauchy.", "start": 730.49, "duration": 2.67}, {"text": "So let's say I make\nthat distance 1.", "start": 733.16, "duration": 3.39}, {"text": "And let's say 0's over\nhere, just for this picture.", "start": 740.51, "duration": 3.39}, {"text": "So then for all n\nfor all n bigger", "start": 743.9, "duration": 4.5}, {"text": "than or equal to capital M,\nx sub n lies in this interval", "start": 748.4, "duration": 2.815}, {"text": "here.", "start": 751.215, "duration": 0.5}, {"text": "And therefore, we'll get\nthat x of n is bounded.", "start": 754.45, "duration": 9.11}, {"text": "So the way this picture looks,\nI'm going to write it this way.", "start": 763.56, "duration": 3.14}, {"text": "It's 1 plus 1.", "start": 766.7, "duration": 1.08}, {"text": "Now, that handles all n bigger\nthan or equal to capital M.", "start": 770.66, "duration": 3.45}, {"text": "So we just need to deal\nwith the first capital", "start": 774.11, "duration": 2.495}, {"text": "M minus 1 other guy.", "start": 776.605, "duration": 3.355}, {"text": "So maybe there's capital\nM minus 1 is over here.", "start": 779.96, "duration": 4.74}, {"text": "Capital X sub 1 is over there.", "start": 784.7, "duration": 1.59}, {"text": "Capital X sub 2 is over here.", "start": 786.29, "duration": 1.81}, {"text": "So then our bound\nwill just be this one,", "start": 788.1, "duration": 3.05}, {"text": "which handles all of the n\nbigger than or equal to M", "start": 791.15, "duration": 2.88}, {"text": "plus the absolute values of\nthese guys that we missed.", "start": 794.03, "duration": 6.105}, {"text": "So since xn is\nCauchy, there exists", "start": 804.81, "duration": 10.33}, {"text": "and M, a natural number,\nsuch that for all n", "start": 815.14, "duration": 4.91}, {"text": "bigger than or equal\nto M, and k bigger", "start": 820.05, "duration": 4.92}, {"text": "than or equal to M, x sub n\nminus x sub k is less than 1", "start": 824.97, "duration": 9.16}, {"text": "in distance.", "start": 834.13, "duration": 2.38}, {"text": "So this is certainly\ntrue for k equals", "start": 836.51, "duration": 3.41}, {"text": "capital M. So this\nimplies for all n bigger", "start": 839.92, "duration": 4.83}, {"text": "than or equal to M, x\nsub n minus x sub capital", "start": 844.75, "duration": 4.29}, {"text": "M is less than 1.", "start": 849.04, "duration": 2.16}, {"text": "So now, if I use the\ntriangle inequality,", "start": 851.2, "duration": 4.74}, {"text": "I can show that the\nprevious implies", "start": 855.94, "duration": 2.85}, {"text": "that for all n bigger\nthan or equal to M,", "start": 858.79, "duration": 2.31}, {"text": "if I look at the absolute\nvalue of x sub n,", "start": 861.1, "duration": 2.43}, {"text": "this is equal to x sub n minus\nx sub capital M plus capital M.", "start": 863.53, "duration": 8.49}, {"text": "And this is less than or\nequal to the absolute value", "start": 872.02, "duration": 2.31}, {"text": "of this guy plus the\nabsolute value of this guy.", "start": 874.33, "duration": 5.97}, {"text": "And this is bounded by 1.", "start": 880.3, "duration": 1.95}, {"text": "So in summary, I've shown\nthat for all n bigger", "start": 886.69, "duration": 5.84}, {"text": "than or equal to this\nfixed integer, capital M, x", "start": 892.53, "duration": 5.02}, {"text": "sub n is less than or\nequal to x sub capital", "start": 897.55, "duration": 5.9}, {"text": "M in absolute value plus 1.", "start": 903.45, "duration": 4.537}, {"text": "So that's for all\nlittle n bigger than", "start": 907.987, "duration": 1.583}, {"text": "or equal to capital\nM. So now I just", "start": 909.57, "duration": 1.5}, {"text": "need to pick a big enough number\nthat bounds the first capital M", "start": 911.07, "duration": 7.21}, {"text": "entries that are not\ncovered by this inequality.", "start": 918.28, "duration": 3.24}, {"text": "Capital M is fixed.", "start": 921.52, "duration": 1.005}, {"text": "So let B be the absolute\nvalue of x sub 1", "start": 926.83, "duration": 5.52}, {"text": "plus absolute value of x sub\n2 plus this fixed number now.", "start": 932.35, "duration": 11.42}, {"text": "Then for all n bigger than\nor equal to capital M,", "start": 947.49, "duration": 4.4}, {"text": "I have, by this\ninequality up here--", "start": 951.89, "duration": 3.28}, {"text": "this is a sum of\nnon-negative numbers,", "start": 958.56, "duration": 1.945}, {"text": "so this number is\ncertainly bigger than", "start": 960.505, "duration": 1.625}, {"text": "or equal to just this part.", "start": 962.13, "duration": 1.425}, {"text": "And if I have n bigger than\nequal to 1 and less than M,", "start": 969.39, "duration": 9.22}, {"text": "then x of n, the absolute\nvalue of this guy", "start": 978.61, "duration": 5.51}, {"text": "is going to be one of these\nthat appears here, which", "start": 984.12, "duration": 2.73}, {"text": "is certainly less than\nor equal to if I add", "start": 986.85, "duration": 3.54}, {"text": "on this number and the\nothers, which is less than", "start": 990.39, "duration": 3.63}, {"text": "or equal to B. So now I've found\na B which is non-negative which", "start": 994.02, "duration": 5.08}, {"text": "bounds all the absolute values.", "start": 999.1, "duration": 1.59}, {"text": "And therefore, this proves\nthat the sequence is bounded.", "start": 1000.69, "duration": 4.17}, {"text": "So we've shown that a\nCauchy sequence is bounded.", "start": 1014.16, "duration": 4.65}, {"text": "And so what I'm now going\nto show is the following.", "start": 1021.47, "duration": 10.879}, {"text": "So again, all of the entries\nare getting close to each other.", "start": 1032.349, "duration": 3.75}, {"text": "They're kind of clustering\nnear each other.", "start": 1036.099, "duration": 2.161}, {"text": "So it kind of feels like\nthey want to converge.", "start": 1038.26, "duration": 5.349}, {"text": "And this next theorem\nsays that, well,", "start": 1043.609, "duration": 2.341}, {"text": "if you've identified a\nlimit along a subsequence,", "start": 1045.95, "duration": 3.93}, {"text": "then, in fact, the entire\nsequence converges.", "start": 1049.88, "duration": 2.95}, {"text": "So of course, this is not true\nfor an arbitrary sequence.", "start": 1052.83, "duration": 3.45}, {"text": "If a subsequence converges--", "start": 1056.28, "duration": 2.99}, {"text": "or I should say, for\nan arbitrary sequence,", "start": 1059.27, "duration": 3.27}, {"text": "it's not true that a\nsubsequence converging implies", "start": 1062.54, "duration": 2.64}, {"text": "a full sequence converging.", "start": 1065.18, "duration": 1.92}, {"text": "We have minus 1 to the n for\nwhich a subsequence converges,", "start": 1067.1, "duration": 4.56}, {"text": "but the whole sequence\ndoes not converge.", "start": 1071.66, "duration": 2.13}, {"text": "But if we make the\nadditional hypothesis", "start": 1073.79, "duration": 2.4}, {"text": "that the sequence is\nCauchy, then the sequence", "start": 1076.19, "duration": 3.39}, {"text": "converges if and only if\nthat subsequence converges.", "start": 1079.58, "duration": 3.76}, {"text": "So the statement of the\ntheorem is following.", "start": 1083.34, "duration": 2.72}, {"text": "If x sub n is Cauchy and there\nexists a subsequence which", "start": 1092.38, "duration": 22.24}, {"text": "is converging to some number--", "start": 1114.62, "duration": 3.06}, {"text": "call it x-- then the whole\nsequence converges to x.", "start": 1117.68, "duration": 12.355}, {"text": "So what I was saying right\nbefore I stated this theorem is", "start": 1133.5, "duration": 2.79}, {"text": "that if I hide this\npart and just say,", "start": 1136.29, "duration": 1.83}, {"text": "there exists a subsequence\nwhich is converging to x,", "start": 1138.12, "duration": 3.48}, {"text": "this does not imply that the\nfull sequence converges to x.", "start": 1141.6, "duration": 2.67}, {"text": "Because we had this example\nof minus 1 to the n.", "start": 1144.27, "duration": 2.79}, {"text": "But if I also assume\nthe sequence is Cauchy,", "start": 1147.06, "duration": 3.84}, {"text": "then it does follow that\nCauchy plus subsequence", "start": 1150.9, "duration": 3.21}, {"text": "converging implies the\nfull sequence converges.", "start": 1154.11, "duration": 2.61}, {"text": "So I want to show that\nxn converges to x.", "start": 1161.96, "duration": 4.56}, {"text": "So want to show--", "start": 1166.52, "duration": 1.275}, {"text": "and we're going to do this\njust by using the definition,", "start": 1176.18, "duration": 5.13}, {"text": "by verifying this\nthrough the definition,", "start": 1181.31, "duration": 1.89}, {"text": "not using the squeeze theorem\nor anything like that.", "start": 1183.2, "duration": 3.818}, {"text": "So let epsilon be positive.", "start": 1187.018, "duration": 4.102}, {"text": "Since xn is Cauchy, there\nexist M0, a natural number", "start": 1198.26, "duration": 17.53}, {"text": "such that for all n\nbigger than or equal to M0", "start": 1215.79, "duration": 4.53}, {"text": "and k bigger than\nor equal to M0, x", "start": 1220.32, "duration": 6.825}, {"text": "sub n minus x sub k is\nless than epsilon over 2.", "start": 1227.145, "duration": 5.055}, {"text": "Why this epsilon over 2?", "start": 1232.2, "duration": 3.0}, {"text": "Or why should you\nnot be surprised?", "start": 1235.2, "duration": 1.53}, {"text": "Well, we have two\nassumptions here.", "start": 1236.73, "duration": 1.99}, {"text": "So like we did when we did\nconvergence of products", "start": 1238.72, "duration": 5.51}, {"text": "of sequences and so on,\nwhich had two assumptions,", "start": 1244.23, "duration": 4.86}, {"text": "namely two sequences converged\nto something, typically,", "start": 1249.09, "duration": 3.27}, {"text": "that means we'll have\ntwo integers coming.", "start": 1252.36, "duration": 1.8}, {"text": "We'll choose a bigger integer\nand then some inequalities", "start": 1254.16, "duration": 4.14}, {"text": "to get an epsilon.", "start": 1258.3, "duration": 0.94}, {"text": "So that's a little bit\nof a rambling answer", "start": 1259.24, "duration": 3.08}, {"text": "to why we get an epsilon over 2\nhere, or why we put one there.", "start": 1262.32, "duration": 4.92}, {"text": "Since the subsequence-- so this\nsubsequence converges to x--", "start": 1272.69, "duration": 7.23}, {"text": "there exists another\ninteger, M sub 1 such", "start": 1288.67, "duration": 4.92}, {"text": "that if k is bigger than\nor equal to M sub 1,", "start": 1293.59, "duration": 7.95}, {"text": "then x sub n sub k minus x\nis less than epsilon over 2.", "start": 1301.54, "duration": 9.3}, {"text": "So maybe I should have used\na different letter here.", "start": 1310.84, "duration": 2.97}, {"text": "Let's use a little m.", "start": 1313.81, "duration": 1.865}, {"text": "Because I don't\nwant you to think", "start": 1315.675, "duration": 1.375}, {"text": "these have to be the same k.", "start": 1317.05, "duration": 1.38}, {"text": "So now we'll choose an integer\nbigger than both M1 and M2", "start": 1327.95, "duration": 7.95}, {"text": "and show that it works.", "start": 1335.9, "duration": 1.42}, {"text": "Choose M to be M0 plus M1.", "start": 1337.32, "duration": 5.87}, {"text": "Now we need to show this works.", "start": 1359.47, "duration": 2.04}, {"text": "And if n is bigger than\nor equal to M, so let me,", "start": 1361.51, "duration": 11.62}, {"text": "actually, make a\nfirst observation", "start": 1373.13, "duration": 6.02}, {"text": "before I go to the n bigger\nthan or equal to capital M.", "start": 1379.15, "duration": 3.57}, {"text": "Then, since n sub k is bigger\nthan or equal to k for all", "start": 1382.72, "duration": 10.63}, {"text": "of k, a natural number-- just\nbecause the n sub k is there", "start": 1393.35, "duration": 2.94}, {"text": "in increasing\nsequence of integers,", "start": 1396.29, "duration": 2.84}, {"text": "which starts at least at 1--", "start": 1399.13, "duration": 1.68}, {"text": "and since n sub k is bigger\nthan or equal to k for all k,", "start": 1404.43, "duration": 3.69}, {"text": "this implies that the integer\nn sub capital M is bigger", "start": 1408.12, "duration": 5.4}, {"text": "than or equal to\nM, which, remember,", "start": 1413.52, "duration": 1.59}, {"text": "is M0 plus M1, which\nimplies that n sub M is", "start": 1415.11, "duration": 5.97}, {"text": "bigger than or equal\nto M0 and n sub M is", "start": 1421.08, "duration": 4.11}, {"text": "bigger than or equal to M1.", "start": 1425.19, "duration": 2.82}, {"text": "So I just wanted to make\nthis preliminary observation.", "start": 1428.01, "duration": 6.3}, {"text": "And now we'll go to showing\nthat this capital M works.", "start": 1434.31, "duration": 2.835}, {"text": "So now, if n is bigger\nthan or equal to capital", "start": 1440.36, "duration": 2.34}, {"text": "M, and I look at x sub n\nminus x, an absolute value,", "start": 1442.7, "duration": 5.47}, {"text": "and add and subtract x sub n\ncapital M capital M minus x", "start": 1448.17, "duration": 12.44}, {"text": "and use the triangle\ninequality, then--", "start": 1460.61, "duration": 13.75}, {"text": "so since n is bigger than\nor equal to capital M, which", "start": 1474.36, "duration": 3.66}, {"text": "is bigger than or\nequal to M0, that", "start": 1478.02, "duration": 2.34}, {"text": "means n is bigger\nthan or equal to M0.", "start": 1480.36, "duration": 2.85}, {"text": "And then n sub M we just showed\nis bigger than or equal to 0.", "start": 1483.21, "duration": 3.75}, {"text": "So by this inequality, I\nget that the first term", "start": 1486.96, "duration": 5.88}, {"text": "is less than epsilon over 2.", "start": 1492.84, "duration": 2.94}, {"text": "And now, so M is\ncertainly bigger than", "start": 1495.78, "duration": 6.06}, {"text": "or equal to M sub 1.", "start": 1501.84, "duration": 2.04}, {"text": "And therefore, I will get that\nthis part is less than epsilon", "start": 1503.88, "duration": 6.06}, {"text": "over 2 because of\nthis inequality.", "start": 1509.94, "duration": 4.24}, {"text": "So that choice of\ncapital M works.", "start": 1514.18, "duration": 8.47}, {"text": "And now, we'll\nprove the following,", "start": 1527.78, "duration": 8.02}, {"text": "that a sequence is convergent\nif and only if it's Cauchy.", "start": 1535.8, "duration": 3.75}, {"text": "So this is a two-way street.", "start": 1567.14, "duration": 1.75}, {"text": "So we need to show the\nleft implies the right", "start": 1568.89, "duration": 3.38}, {"text": "and then the right\nimplies the left.", "start": 1572.27, "duration": 1.665}, {"text": "So this direction\nis, in fact, easy.", "start": 1580.8, "duration": 9.757}, {"text": "Based on what we've done--", "start": 1590.557, "duration": 1.083}, {"text": "I shouldn't say it's easy--", "start": 1591.64, "duration": 1.14}, {"text": "but what we've done so\nfar, it quickly follows.", "start": 1592.78, "duration": 2.97}, {"text": "So we're assuming\nx sub n is Cauchy.", "start": 1602.31, "duration": 3.14}, {"text": "I'm trying to show\nit's convergent.", "start": 1605.45, "duration": 1.47}, {"text": "So if x sub n is\nCauchy, this implies", "start": 1612.25, "duration": 2.91}, {"text": "that x sub n is bounded,\nthe sequence is bounded,", "start": 1615.16, "duration": 7.08}, {"text": "which implies by the\nBolzano-Weierstrass theorem", "start": 1622.24, "duration": 3.48}, {"text": "that x sub n has a\nconvergent subsequence.", "start": 1625.72, "duration": 15.11}, {"text": "And by the theorem\nwe just proved,", "start": 1640.83, "duration": 1.89}, {"text": "if a Cauchy sequence has\na convergent subsequence,", "start": 1642.72, "duration": 3.81}, {"text": "it must be convergent.", "start": 1646.53, "duration": 1.32}, {"text": "Now, for the converse\ndirection, that xn", "start": 1656.05, "duration": 3.39}, {"text": "is convergent implies xn\nis Cauchy, well, so this", "start": 1659.44, "duration": 5.13}, {"text": "should not come as a surprise.", "start": 1664.57, "duration": 1.26}, {"text": "Let me draw a picture.", "start": 1665.83, "duration": 0.917}, {"text": "Let's suppose x sub\nn is converging to x,", "start": 1670.51, "duration": 5.35}, {"text": "and epsilon is positive.", "start": 1675.86, "duration": 1.77}, {"text": "Then, since the\nxn's are converging", "start": 1680.55, "duration": 3.0}, {"text": "to x, if I draw a\nlittle interval around x", "start": 1683.55, "duration": 9.3}, {"text": "of total length epsilon--", "start": 1692.85, "duration": 3.07}, {"text": "so x minus epsilon over 2\nand x plus epsilon over 2--", "start": 1695.92, "duration": 3.38}, {"text": "then I will find, as long\nas so then there exists M", "start": 1699.3, "duration": 5.61}, {"text": "so that, for all n bigger\nthan or equal to capital M,", "start": 1704.91, "duration": 4.89}, {"text": "all of the x sub n's\nlie in this interval.", "start": 1709.8, "duration": 5.67}, {"text": "They all lie in this\ninterval because they", "start": 1721.54, "duration": 1.75}, {"text": "have to be within\ndistance epsilon over 2", "start": 1723.29, "duration": 1.8}, {"text": "to x if the xn's\nare converging to x.", "start": 1725.09, "duration": 5.46}, {"text": "And since they lie\nin this interval,", "start": 1730.55, "duration": 1.86}, {"text": "the distance between\nany two of them", "start": 1732.41, "duration": 2.37}, {"text": "can only be as big as the\nlength of the interval, which", "start": 1734.78, "duration": 2.7}, {"text": "is epsilon.", "start": 1737.48, "duration": 2.79}, {"text": "So this is essentially\nthe picture", "start": 1740.27, "duration": 3.78}, {"text": "of why a convergence\nsequence has to be Cauchy.", "start": 1744.05, "duration": 2.67}, {"text": "So now let's turn this\npicture into math.", "start": 1746.72, "duration": 1.8}, {"text": "We have to verify xn is\nCauchy through the definition.", "start": 1753.2, "duration": 2.34}, {"text": "That's all we have.", "start": 1755.54, "duration": 1.02}, {"text": "So let epsilon be positive.", "start": 1756.56, "duration": 3.48}, {"text": "Since the xn's\nconverge to x, there", "start": 1760.04, "duration": 4.32}, {"text": "exists an integer M sub\n0, a natural number,", "start": 1764.36, "duration": 3.84}, {"text": "such that for all n bigger\nthan or equal to M sub 0, x", "start": 1768.2, "duration": 5.64}, {"text": "sub n minus x is less\nthan epsilon over 2.", "start": 1773.84, "duration": 3.42}, {"text": "And so we'll choose the M\nfor our definition of Cauchy", "start": 1782.86, "duration": 4.65}, {"text": "to be this M sub 0.", "start": 1787.51, "duration": 1.41}, {"text": "And if n is bigger\nthan or equal to M", "start": 1792.52, "duration": 3.66}, {"text": "and k is bigger\nthan or equal to M", "start": 1796.18, "duration": 3.51}, {"text": "and I look at the absolute\nvalue of x sub n minus x sub k", "start": 1799.69, "duration": 4.62}, {"text": "and add and subtract x and\nuse the triangle inequality,", "start": 1804.31, "duration": 4.14}, {"text": "this is less than\nor equal to x sub", "start": 1808.45, "duration": 1.65}, {"text": "n minus x, an absolute\nvalue, plus x minus x sub k.", "start": 1810.1, "duration": 6.93}, {"text": "Each of these is less\nthan epsilon over 2", "start": 1817.03, "duration": 5.1}, {"text": "since n is bigger than\nor equal to capital M,", "start": 1822.13, "duration": 1.988}, {"text": "and k is bigger than\nor equal to capital M.", "start": 1824.118, "duration": 1.792}, {"text": "So this is less than epsilon\nover 2 plus epsilon over 2", "start": 1825.91, "duration": 4.2}, {"text": "equals epsilon.", "start": 1830.11, "duration": 1.29}, {"text": "And therefore, xn is Cauchy.", "start": 1831.4, "duration": 1.5}, {"text": "Now I want to make\na brief remark", "start": 1837.23, "duration": 4.5}, {"text": "about the previous theorem.", "start": 1841.73, "duration": 3.24}, {"text": "So remember how this\nwhole story started off?", "start": 1847.94, "duration": 3.18}, {"text": "There was something wrong with\nthe rational numbers, namely,", "start": 1856.07, "duration": 5.06}, {"text": "they didn't contain\nthe square root of 2.", "start": 1861.13, "duration": 2.76}, {"text": "So we couldn't solve\nthe algebraic equation", "start": 1863.89, "duration": 2.34}, {"text": "x squared minus 2 equals 0.", "start": 1866.23, "duration": 2.79}, {"text": "But this also, then,\nturned into the rationals", "start": 1869.02, "duration": 4.74}, {"text": "not being complete in\nthe sense of order.", "start": 1873.76, "duration": 3.72}, {"text": "Not every non-empty\nbounded set had a supremum.", "start": 1877.48, "duration": 3.348}, {"text": "It didn't have the least\nupper bound property.", "start": 1880.828, "duration": 1.917}, {"text": "But you can also interpret this\nlack of having square root of 2", "start": 1885.7, "duration": 5.16}, {"text": "as somehow saying\nthat the rationals are", "start": 1890.86, "duration": 6.24}, {"text": "incomplete in this sense.", "start": 1897.1, "duration": 1.93}, {"text": "So hopefully, at the\nend of this class,", "start": 1899.03, "duration": 3.747}, {"text": "we'll be able to get\nto metrics basis.", "start": 1902.777, "duration": 1.583}, {"text": "But so what do I mean by that?", "start": 1904.36, "duration": 4.03}, {"text": "Let's say I look\nat this statement", "start": 1908.39, "duration": 1.82}, {"text": "now within the universe\nof rational numbers.", "start": 1910.21, "duration": 4.87}, {"text": "So now, if this sequence\nis rational numbers--", "start": 1919.26, "duration": 19.03}, {"text": "meaning sequences are only\nsequences of rational numbers,", "start": 1938.29, "duration": 4.07}, {"text": "limits are only elements\nof the rational numbers,", "start": 1942.36, "duration": 5.01}, {"text": "epsilon is only a rational\nnumber, and so on-- then", "start": 1947.37, "duration": 5.1}, {"text": "we still have many\nof the same theorems", "start": 1952.47, "duration": 2.01}, {"text": "that we proved--\nnot all of them,", "start": 1954.48, "duration": 2.22}, {"text": "and I'll indicate\nwhich ones don't hold.", "start": 1956.7, "duration": 3.28}, {"text": "But if we only\nwork in rationals,", "start": 1959.98, "duration": 1.55}, {"text": "then we always do have a\nconvergence implies Cauchy,", "start": 1961.53, "duration": 11.38}, {"text": "meaning convergent\nsequences are Cauchy.", "start": 1972.91, "duration": 4.23}, {"text": "But Cauchy sequences are\nnot necessarily convergent.", "start": 1977.14, "duration": 13.74}, {"text": "Again, what's the\nexample here, or what's", "start": 1990.88, "duration": 8.44}, {"text": "the intuitive example?", "start": 1999.32, "duration": 1.17}, {"text": "Take x sub n so\nthat x of n is in Q.", "start": 2000.49, "duration": 11.05}, {"text": "And now viewed in the\nuniverse of real numbers,", "start": 2011.54, "duration": 6.62}, {"text": "x sub n's converge to root 2.", "start": 2018.16, "duration": 1.815}, {"text": "Then such a sequence would\nbe a Cauchy sequence.", "start": 2026.59, "duration": 4.38}, {"text": "We just proved that, basically.", "start": 2030.97, "duration": 2.34}, {"text": "So such a sequence would\nbe a Cauchy sequence", "start": 2033.31, "duration": 2.19}, {"text": "of rational numbers.", "start": 2035.5, "duration": 1.71}, {"text": "However, it would not converge\nin the set of rational numbers.", "start": 2037.21, "duration": 3.95}, {"text": "It would converge to\nthe square root of 2,", "start": 2041.16, "duration": 1.75}, {"text": "which is not a rational number.", "start": 2042.91, "duration": 1.87}, {"text": "So because the square root of\n2 is not a rational number,", "start": 2044.78, "duration": 5.04}, {"text": "this shows that the\nrational numbers do not", "start": 2049.82, "duration": 1.919}, {"text": "have this completeness property\nthat Cauchy sequences converge.", "start": 2051.739, "duration": 4.68}, {"text": "So there's a whole,\nstill, to this day, kind", "start": 2062.34, "duration": 4.87}, {"text": "of industry of studying\nspaces for which Cauchy", "start": 2067.21, "duration": 4.35}, {"text": "is equivalent to convergent.", "start": 2071.56, "duration": 2.55}, {"text": "These are called\ncomplete metric spaces.", "start": 2074.11, "duration": 1.799}, {"text": "And then if you add a\nlittle more structure,", "start": 2075.909, "duration": 1.833}, {"text": "they're called Banach\nspaces and so on,", "start": 2077.742, "duration": 4.147}, {"text": "which are very important,\nnot just in math", "start": 2081.889, "duration": 1.75}, {"text": "but also for\nformulating rigorously", "start": 2083.639, "duration": 6.27}, {"text": "a lot of the\nunderlying assumptions", "start": 2089.909, "duration": 6.421}, {"text": "for mathematical physics.", "start": 2096.33, "duration": 1.86}, {"text": "So if we're just looking\ninside the rationals,", "start": 2101.96, "duration": 2.76}, {"text": "it does not follow that Cauchy\nsequences always converge.", "start": 2104.72, "duration": 3.48}, {"text": "And now let's just\nstop for a minute", "start": 2108.2, "duration": 1.8}, {"text": "and take stock of why this\nwas true for the real numbers.", "start": 2110.0, "duration": 3.43}, {"text": "What did we use going back?", "start": 2113.43, "duration": 1.756}, {"text": "So if you really go\nback to the proof", "start": 2115.186, "duration": 8.374}, {"text": "of the Bolzano-Weierstrass--\nso that's what we used here", "start": 2123.56, "duration": 3.57}, {"text": "to show that Cauchy\nsequences converge--", "start": 2127.13, "duration": 2.25}, {"text": "we use the fact that the lim sup\nand the lim inf always exists.", "start": 2132.11, "duration": 5.37}, {"text": "And lim sup and lim\ninf, first off, they're", "start": 2137.48, "duration": 3.63}, {"text": "defined to be sups and\ninfs, which may not always", "start": 2141.11, "duration": 2.16}, {"text": "exist as rational numbers,\nas we've already shown.", "start": 2143.27, "duration": 2.88}, {"text": "So that's definitely a\nproblem already there.", "start": 2146.15, "duration": 5.28}, {"text": "But even more so, when we prove\nthat every bounded monotone", "start": 2151.43, "duration": 4.02}, {"text": "sequence converges,\nwhat we showed", "start": 2155.45, "duration": 2.76}, {"text": "was that this limit is\nactually a sup of a certain set", "start": 2158.21, "duration": 3.27}, {"text": "or an inf of a certain set,\nwhich, again, may or may not", "start": 2161.48, "duration": 2.843}, {"text": "exist if we're just looking\nin the rational numbers.", "start": 2164.323, "duration": 2.167}, {"text": "Because the rational numbers do\nnot have the least upper bound", "start": 2166.49, "duration": 2.7}, {"text": "property.", "start": 2169.19, "duration": 1.23}, {"text": "So it really is the least\nupper bound property", "start": 2170.42, "duration": 4.17}, {"text": "that gives us\nconvergence equivalent", "start": 2174.59, "duration": 3.66}, {"text": "to Cauchy for the real numbers.", "start": 2178.25, "duration": 2.17}, {"text": "So for R, the least\nupper bound property is--", "start": 2180.42, "duration": 12.492}, {"text": "it has to be because\nthat's the main thing that", "start": 2192.912, "duration": 1.958}, {"text": "separates the two fields,\nbut I'm just reiterating this", "start": 2194.87, "duration": 4.89}, {"text": "here--", "start": 2199.76, "duration": 1.56}, {"text": "is the reason why convergent\nis equivalent to Cauchy.", "start": 2201.32, "duration": 12.1}, {"text": "Now that I've proved that Cauchy\nis equivalent to convergence,", "start": 2227.75, "duration": 5.49}, {"text": "maybe you'll ask, then why\ndid we introduce it at all?", "start": 2233.24, "duration": 2.73}, {"text": "If these two notions\nare the same,", "start": 2238.48, "duration": 4.96}, {"text": "why even introduce\nthem if they're just", "start": 2243.44, "duration": 2.19}, {"text": "convergent sequences already?", "start": 2245.63, "duration": 2.23}, {"text": "And the reason is because to\nshow that a sequence converges,", "start": 2247.86, "duration": 6.59}, {"text": "you have to somehow\nhave your hands", "start": 2254.45, "duration": 1.86}, {"text": "on a candidate for the limit.", "start": 2256.31, "duration": 4.47}, {"text": "If you want to prove\nthat xn converges,", "start": 2260.78, "duration": 4.71}, {"text": "you have to somehow come up\nwith an x that it converges to.", "start": 2265.49, "duration": 3.39}, {"text": "And it's not always\nclear how to find that x.", "start": 2268.88, "duration": 6.234}, {"text": "But Cauchy, although it's\nequivalent to a convergent", "start": 2275.114, "duration": 3.306}, {"text": "in the set of real\nnumbers, doesn't", "start": 2278.42, "duration": 3.45}, {"text": "require you to find a\ncandidate for convergence.", "start": 2281.87, "duration": 5.28}, {"text": "All it requires\nyou to do is show", "start": 2287.15, "duration": 2.67}, {"text": "that, as long as you\ngo far enough out,", "start": 2289.82, "duration": 3.61}, {"text": "any two entries in the\nsequence are close together", "start": 2293.43, "duration": 4.98}, {"text": "without requiring you\nto come up with a limit.", "start": 2298.41, "duration": 2.62}, {"text": "See, computing limits\nis quite difficult.", "start": 2301.03, "duration": 2.15}, {"text": "We're about to do series.", "start": 2303.18, "duration": 1.71}, {"text": "And there's maybe, I don't\nknow, five series people", "start": 2304.89, "duration": 3.15}, {"text": "can compute explicitly.", "start": 2308.04, "duration": 1.937}, {"text": "But you do know that there's\na ton of other series", "start": 2309.977, "duration": 2.083}, {"text": "that are actually convergent,\neven though you don't know what", "start": 2312.06, "duration": 2.7}, {"text": "the limit as.", "start": 2314.76, "duration": 1.32}, {"text": "And why do you know that?", "start": 2316.08, "duration": 1.65}, {"text": "This is exactly\nbecause and exactly", "start": 2317.73, "duration": 2.16}, {"text": "why people thought\nof Cauchy sequences", "start": 2319.89, "duration": 3.96}, {"text": "to begin with in\nmuch of analysis.", "start": 2323.85, "duration": 2.59}, {"text": "So again, just to\nsummarize this,", "start": 2326.44, "duration": 5.33}, {"text": "convergent sequences are nice.", "start": 2331.77, "duration": 1.47}, {"text": "But in practice, it's\ndifficult to get your hands", "start": 2333.24, "duration": 2.94}, {"text": "on what could be a\nlimit of a sequence,", "start": 2336.18, "duration": 2.1}, {"text": "especially if that sequence\nis pretty complicated.", "start": 2338.28, "duration": 3.37}, {"text": "So if you're trying to show\na certain sequence converges,", "start": 2341.65, "duration": 3.5}, {"text": "it suffices, by what we've done\nhere, to show that it's Cauchy.", "start": 2345.15, "duration": 4.18}, {"text": "And that's a little bit\neasier to do because that just", "start": 2349.33, "duration": 3.51}, {"text": "requires you to work with\nthe original sequence.", "start": 2352.84, "duration": 2.485}, {"text": "You don't have to\ncome up with a limit.", "start": 2355.325, "duration": 1.625}, {"text": "You can just take your sequence\nand start playing directly", "start": 2356.95, "duration": 4.41}, {"text": "with the entries rather than\ntry to come up with a limit", "start": 2361.36, "duration": 6.01}, {"text": "explicitly.", "start": 2367.37, "duration": 0.93}, {"text": "So that's what we're\ngoing to move on to", "start": 2372.02, "duration": 2.46}, {"text": "is series now, which,\nas I said a minute ago,", "start": 2374.48, "duration": 5.46}, {"text": "is original reason why\npeople started developing", "start": 2379.94, "duration": 8.81}, {"text": "the foundations of\nanalysis, what we're talking", "start": 2388.75, "duration": 2.248}, {"text": "about right now, to begin with.", "start": 2390.998, "duration": 1.292}, {"text": "Because they were just kind\nof doing very formal things", "start": 2395.64, "duration": 6.33}, {"text": "that ended up not\nmaking sense, like they", "start": 2401.97, "duration": 5.07}, {"text": "were adding infinitely\nmany positive numbers", "start": 2407.04, "duration": 2.94}, {"text": "and coming up with\na negative number.", "start": 2409.98, "duration": 3.048}, {"text": "Well, that can't be right.", "start": 2413.028, "duration": 1.252}, {"text": "So all of this was\ncreated, discovered--", "start": 2414.28, "duration": 7.31}, {"text": "however you want to phrase it--", "start": 2421.59, "duration": 1.56}, {"text": "to put on rigorous\nfoundations this next topic,", "start": 2423.15, "duration": 6.143}, {"text": "which is series.", "start": 2429.293, "duration": 0.667}, {"text": "And you dealt with\nseries in calculus,", "start": 2435.2, "duration": 3.51}, {"text": "so you know what a series is.", "start": 2438.71, "duration": 1.885}, {"text": "Maybe you don't\nremember all the proofs", "start": 2440.595, "duration": 1.625}, {"text": "of the properties of series.", "start": 2442.22, "duration": 1.35}, {"text": "But suffice it to say, series\nis a pretty good motivation", "start": 2443.57, "duration": 6.16}, {"text": "since it's one of the most\nuseful things that comes out", "start": 2449.73, "duration": 4.375}, {"text": "of math.", "start": 2454.105, "duration": 0.5}, {"text": "Series expansions are\nhow you solve ODEs, PDEs,", "start": 2458.79, "duration": 6.21}, {"text": "Taylor expansions.", "start": 2465.0, "duration": 1.02}, {"text": "All these things are, in\nsome sense, a form of series.", "start": 2466.02, "duration": 3.33}, {"text": "So being able to justify\nthem as being real things", "start": 2469.35, "duration": 5.25}, {"text": "is a necessity.", "start": 2474.6, "duration": 2.56}, {"text": "So the definition of\na series, for now,", "start": 2477.16, "duration": 8.66}, {"text": "really is just this symbol\nI'm about to write down.", "start": 2485.82, "duration": 2.7}, {"text": "So given a sequence\nx sub n, the symbol--", "start": 2488.52, "duration": 8.945}, {"text": "or maybe I'll just sometimes\nwrite just the sum x sub n--", "start": 2505.13, "duration": 3.105}, {"text": "is what's called the series\nassociated to the sequence", "start": 2510.8, "duration": 9.31}, {"text": "x sub n.", "start": 2520.11, "duration": 0.56}, {"text": "So right now, that's\njust a symbol.", "start": 2523.607, "duration": 1.458}, {"text": "We're going to interpret\nthis as a real number", "start": 2527.6, "duration": 2.46}, {"text": "in the following situation.", "start": 2530.06, "duration": 1.53}, {"text": "We say that the\nseries converges,", "start": 2535.04, "duration": 13.34}, {"text": "if the following sequence\ngiven by s sub m equals--", "start": 2548.38, "duration": 20.51}, {"text": "so s sub m, this is the\nelement of the sequence.", "start": 2574.06, "duration": 3.06}, {"text": "And what is it?", "start": 2577.12, "duration": 0.84}, {"text": "It is the actual sum.", "start": 2577.96, "duration": 1.93}, {"text": "So this is not a formal thing.", "start": 2579.89, "duration": 1.38}, {"text": "This is just a finite\nsum from n equals 1 to m.", "start": 2581.27, "duration": 3.56}, {"text": "So this is-- so this sequence\nnow, m equals 1 to infinity.", "start": 2584.83, "duration": 10.24}, {"text": "And these guys we call\npartial sums converges.", "start": 2600.0, "duration": 14.07}, {"text": "So right now, if we just\nhave a sequence x sub n,", "start": 2616.87, "duration": 5.04}, {"text": "the series associated to that\nsequence is just a symbol.", "start": 2621.91, "duration": 4.5}, {"text": "We say that this\nseries converges", "start": 2626.41, "duration": 1.8}, {"text": "if this sequence of\npartial sums converge", "start": 2628.21, "duration": 4.72}, {"text": "and if s is this limit.", "start": 2632.93, "duration": 9.43}, {"text": "And we write s is\nequal to the series", "start": 2651.81, "duration": 13.26}, {"text": "and treat the series\nnow as a number.", "start": 2665.07, "duration": 1.8}, {"text": "So in general, if\nI have a sequence,", "start": 2677.03, "duration": 1.98}, {"text": "I just have this formal\nsymbol, which I'm writing down,", "start": 2679.01, "duration": 3.73}, {"text": "which I call a series\nassociated to it.", "start": 2682.74, "duration": 2.96}, {"text": "In the case that the sequence\nof partial sums converges,", "start": 2685.7, "duration": 3.85}, {"text": "then I actually identify this\nseries with a real number", "start": 2689.55, "duration": 3.585}, {"text": "and treat it as a real number.", "start": 2693.135, "duration": 1.25}, {"text": "And so the way\nI've written this,", "start": 2699.12, "duration": 3.96}, {"text": "the series is starting at 1.", "start": 2703.08, "duration": 1.44}, {"text": "But it doesn't\nnecessarily have to.", "start": 2704.52, "duration": 2.01}, {"text": "So just by shifting the index--", "start": 2706.53, "duration": 4.905}, {"text": "so let me just say here\nthat we don't necessarily", "start": 2716.08, "duration": 9.69}, {"text": "have to start a\nseries at n equals 1.", "start": 2725.77, "duration": 9.63}, {"text": "So this could be\nsum from n equals 0", "start": 2735.4, "duration": 2.25}, {"text": "to infinity, in which case, we\nhave a sequence starting now", "start": 2737.65, "duration": 3.0}, {"text": "at x sub 0.", "start": 2740.65, "duration": 1.53}, {"text": "Or this could be starting\nat 2, in which place", "start": 2742.18, "duration": 3.24}, {"text": "the sequence of x sub\nn starts at n equals 2.", "start": 2745.42, "duration": 3.12}, {"text": "And then the sequence\nof partial sums", "start": 2748.54, "duration": 2.43}, {"text": "would start at not m\nequals 1 but m equals 0", "start": 2750.97, "duration": 3.3}, {"text": "or m equals 2, depending\non where the series starts.", "start": 2754.27, "duration": 3.57}, {"text": "So some examples.", "start": 2760.86, "duration": 0.84}, {"text": "The series sum from n equals\n1 to infinity 1 over n", "start": 2765.18, "duration": 8.93}, {"text": "plus 1 times n, this\nis a convergent series.", "start": 2774.11, "duration": 7.47}, {"text": "So why is this?", "start": 2786.01, "duration": 1.935}, {"text": "So let's look at the proof.", "start": 2787.945, "duration": 1.125}, {"text": "So we look at the\nm-th partial sum--", "start": 2789.07, "duration": 2.7}, {"text": "this is the sum from n equals\n1 to m of 1 over n plus 1n.", "start": 2791.77, "duration": 6.93}, {"text": "And this is equal to--", "start": 2798.7, "duration": 2.73}, {"text": "now, if I write 1 over n plus\n1 times n as 1 over n minus 1", "start": 2801.43, "duration": 15.11}, {"text": "over n plus 1, this is\nnow the sum of 1 over n", "start": 2816.54, "duration": 4.98}, {"text": "plus-- these are finite sums,\nso I can always split them up.", "start": 2821.52, "duration": 3.87}, {"text": "This should be a minus.", "start": 2827.747, "duration": 0.958}, {"text": "And so now, this is\nequal to 1 plus 1/2", "start": 2831.64, "duration": 2.91}, {"text": "plus 1 over m minus 1/2\nplus 1/3 plus 1 over m", "start": 2834.55, "duration": 12.0}, {"text": "plus 1 over n plus 1.", "start": 2846.55, "duration": 4.41}, {"text": "And you see all of these cancel.", "start": 2850.96, "duration": 4.02}, {"text": "And all that's left is\n1 minus 1 over m plus 1.", "start": 2854.98, "duration": 5.82}, {"text": "So the m-th partial sum is equal\nto 1 minus 1 over m plus 1.", "start": 2860.8, "duration": 5.01}, {"text": "And therefore, the\nsequence of partial sums", "start": 2865.81, "duration": 7.48}, {"text": "is the limit as m\ngoes to infinity", "start": 2873.29, "duration": 1.62}, {"text": "of this, which is just 1.", "start": 2874.91, "duration": 2.2}, {"text": "And therefore, this\nseries converges.", "start": 2877.11, "duration": 2.84}, {"text": "Now, our favorite sequence,\nwhich does not converge,", "start": 2889.61, "duration": 4.44}, {"text": "will give us a series\nwhich does not converge.", "start": 2894.05, "duration": 2.64}, {"text": "So let's look at sum from n\nequals 1 to infinity minus 1", "start": 2896.69, "duration": 5.55}, {"text": "to the n.", "start": 2902.24, "duration": 1.2}, {"text": "This does not converge.", "start": 2903.44, "duration": 3.06}, {"text": "So what's the proof?", "start": 2909.532, "duration": 0.833}, {"text": "The m-th partial sum,\nthis is equal to minus 1", "start": 2913.1, "duration": 6.24}, {"text": "plus 1 plus minus 1 up until\nI get minus 1 to the m.", "start": 2919.34, "duration": 6.14}, {"text": "And therefore, this is always\nequal to one of two things.", "start": 2928.61, "duration": 6.45}, {"text": "If m is odd, then I have an\nodd number of these guys.", "start": 2935.06, "duration": 7.61}, {"text": "And therefore, the\nminuses and pluses", "start": 2942.67, "duration": 4.38}, {"text": "cancel, just leaving\na minus 1 in the end--", "start": 2947.05, "duration": 3.75}, {"text": "this last one, the odd one.", "start": 2950.8, "duration": 1.44}, {"text": "And m is odd and 0 if m is even.", "start": 2955.9, "duration": 5.86}, {"text": "If I add up an even\nnumber of these terms,", "start": 2961.76, "duration": 4.2}, {"text": "then all of the minus\n1's and 1's cancel out,", "start": 2965.96, "duration": 2.23}, {"text": "so I just get 0.", "start": 2968.19, "duration": 1.49}, {"text": "And therefore, this sequence,\nwhich is just minus 1", "start": 2969.68, "duration": 10.25}, {"text": "for m odd, 0 for m\neven, does not converge.", "start": 2979.93, "duration": 11.487}, {"text": "And therefore, the\nseries does not converge.", "start": 2991.417, "duration": 1.833}, {"text": "So when I write this,\nthis is just a symbol.", "start": 2993.25, "duration": 3.0}, {"text": "This is just chalk\non a chalkboard.", "start": 2996.25, "duration": 1.838}, {"text": "It doesn't mean anything.", "start": 2998.088, "duration": 1.042}, {"text": "So let's go to another\nseries, which does converge.", "start": 3006.32, "duration": 6.0}, {"text": "And this is kind of\nthe one to which we", "start": 3012.32, "duration": 5.22}, {"text": "compare all other series,\nessentially, as you'll see.", "start": 3017.54, "duration": 4.98}, {"text": "You have all these series tests\nthat you remember, hopefully,", "start": 3022.52, "duration": 3.18}, {"text": "from calculus that tell you\nwhen a series converges.", "start": 3025.7, "duration": 7.0}, {"text": "But maybe, if you\nremember the proof", "start": 3032.7, "duration": 2.58}, {"text": "or don't, how you do\nthat is you converge it", "start": 3035.28, "duration": 2.46}, {"text": "to one series, which\nyou do know how to sum.", "start": 3037.74, "duration": 2.58}, {"text": "So it was just by pure luck we\nwere able to compute the sum,", "start": 3040.32, "duration": 5.34}, {"text": "or compute the explicit\nseries for this guy.", "start": 3045.66, "duration": 4.71}, {"text": "Another one which we can do\nthat for is geometric series.", "start": 3050.37, "duration": 3.64}, {"text": "So the theorem is if\nI have a real number", "start": 3054.01, "duration": 4.22}, {"text": "with absolute value less\nthan 1, then the series", "start": 3058.23, "duration": 7.77}, {"text": "starting now at 0 R\nto the n converges.", "start": 3066.0, "duration": 7.23}, {"text": "And I can actually compute\nthe sum of this series.", "start": 3073.23, "duration": 6.57}, {"text": "And this is 1 over 1 minus r.", "start": 3079.8, "duration": 3.09}, {"text": "So what's the proof?", "start": 3088.775, "duration": 2.595}, {"text": "Let's look at the partial sums.", "start": 3091.37, "duration": 2.46}, {"text": "And we can actually\ncompute these, as well,", "start": 3093.83, "duration": 4.23}, {"text": "just as we were able to\ndo for the first example.", "start": 3098.06, "duration": 2.55}, {"text": "We compute that the sum from n\nequals 0 to m of r to the m--", "start": 3105.63, "duration": 6.44}, {"text": "now, you can prove\nthis by induction.", "start": 3112.07, "duration": 2.94}, {"text": "I cannot exactly\nremember if I did this--", "start": 3115.01, "duration": 2.52}, {"text": "I believe I did-- in\nthe second lecture", "start": 3120.44, "duration": 1.8}, {"text": "on induction, first or second\nlecture on the induction.", "start": 3122.24, "duration": 2.52}, {"text": "But if I add up some\nnumber raised to the n-th--", "start": 3124.76, "duration": 4.41}, {"text": "so this should be to the n--", "start": 3129.17, "duration": 2.91}, {"text": "power from 0 to m, this\nis equal to 1 minus r", "start": 3132.08, "duration": 3.36}, {"text": "to the m plus 1 over 1 minus r.", "start": 3135.44, "duration": 5.23}, {"text": "And I guess two lectures\nago, we proved that", "start": 3140.67, "duration": 5.46}, {"text": "if the absolute value--", "start": 3146.13, "duration": 1.96}, {"text": "so let me state this now.", "start": 3148.09, "duration": 2.585}, {"text": "Two lectures ago, we proved\nthat if the absolute value of r", "start": 3158.49, "duration": 7.8}, {"text": "is less than 1, then limit as,\nlet's make it m, of r to the m", "start": 3166.29, "duration": 9.12}, {"text": "equals 0, which implies that--", "start": 3175.41, "duration": 7.37}, {"text": "so this was the\nm-th partial sum--", "start": 3182.78, "duration": 4.33}, {"text": "which implies that the limit as\nm goes to infinity of s sub m", "start": 3187.11, "duration": 4.82}, {"text": "equals the limit as m goes to\ninfinity of this thing, which", "start": 3191.93, "duration": 4.29}, {"text": "is, if you like--", "start": 3196.22, "duration": 1.05}, {"text": "so that plus 1 there just\nmultiplies r to the m by r.", "start": 3200.25, "duration": 4.62}, {"text": "And using the algebraic\nfacts we proved", "start": 3204.87, "duration": 5.04}, {"text": "about limits is 1\nminus r time 0 over 1", "start": 3209.91, "duration": 2.7}, {"text": "minus r, which equals\n1 over 1 minus r.", "start": 3212.61, "duration": 3.93}, {"text": "So now you ask about the other.", "start": 3222.756, "duration": 3.534}, {"text": "What about r bigger than 1?", "start": 3226.29, "duration": 2.43}, {"text": "Well, when r equals\nminus 1, then we", "start": 3232.57, "duration": 2.73}, {"text": "get the second\nexample we looked at.", "start": 3235.3, "duration": 4.14}, {"text": "If we get r equals 1, then\nthat's just summing up 1,", "start": 3239.44, "duration": 4.89}, {"text": "and I'll leave it to you\nto check that that does not", "start": 3244.33, "duration": 3.24}, {"text": "converge, that the sequence of\npartial sums, if I just sum up", "start": 3247.57, "duration": 3.72}, {"text": "1 from n equals 0 to\nm, is equal 2n plus 1,", "start": 3251.29, "duration": 5.53}, {"text": "which does not converge.", "start": 3256.82, "duration": 1.0}, {"text": "And let me make just a\nkind of silly comment.", "start": 3272.67, "duration": 3.33}, {"text": "And maybe I didn't\nexplicitly make", "start": 3276.0, "duration": 2.16}, {"text": "this comment about sequences.", "start": 3278.16, "duration": 2.01}, {"text": "Maybe I forgot to\ndo that, as well.", "start": 3280.17, "duration": 1.81}, {"text": "So for a sequence, you could\nstart your sequence not", "start": 3281.98, "duration": 8.24}, {"text": "necessarily at the first entry.", "start": 3290.22, "duration": 1.95}, {"text": "Maybe you look at a, if you\nlike, new sequence where--", "start": 3292.17, "duration": 3.78}, {"text": "well, so we know from\nsequences that subsequences", "start": 3300.05, "duration": 4.24}, {"text": "of convergent\nsequences converge.", "start": 3304.29, "duration": 1.68}, {"text": "So if I, instead of looking at\nthe whole sequence, start at,", "start": 3305.97, "duration": 6.01}, {"text": "let's say, x100 and\nthen go x101, x102,", "start": 3311.98, "duration": 3.98}, {"text": "and that's the\nsequence I look at,", "start": 3315.96, "duration": 1.74}, {"text": "well, that's a subsequence\nof the original one, which,", "start": 3317.7, "duration": 2.79}, {"text": "if it converges, implies\nthat subsequence converges.", "start": 3320.49, "duration": 2.92}, {"text": "So all that is to say--", "start": 3323.41, "duration": 3.35}, {"text": "and for this simple way of\nobtaining this new sequence--", "start": 3326.76, "duration": 4.74}, {"text": "all of that is to say\nthat to understand", "start": 3331.5, "duration": 5.04}, {"text": "if a sequence\nconverges, I don't have", "start": 3336.54, "duration": 1.86}, {"text": "to consider what happens for\nthe first finitely many terms", "start": 3338.4, "duration": 5.58}, {"text": "in the sequence,\nmeaning a sequence x", "start": 3343.98, "duration": 5.33}, {"text": "sub n converges if and only if\na sequence starting now at say,", "start": 3349.31, "duration": 5.57}, {"text": "n equals 100-- and 101, 102,\n103 and so on-- converges.", "start": 3354.88, "duration": 5.49}, {"text": "And the same is true for series,\nthat a series converges if", "start": 3360.37, "duration": 6.73}, {"text": "and only if a series\nconverges, now,", "start": 3367.1, "duration": 4.68}, {"text": "starting at a different\npoint along the sequence.", "start": 3371.78, "duration": 2.95}, {"text": "So this is the\nfollowing theorem.", "start": 3374.73, "duration": 4.52}, {"text": "Let xn be a sequence, and let\ncapital M be a natural number.", "start": 3383.34, "duration": 15.29}, {"text": "Then n equals 1 to\ninfinity of x sub n.", "start": 3398.63, "duration": 8.01}, {"text": "This converges if and only\nif the sequence, now starting", "start": 3406.64, "duration": 5.25}, {"text": "at capital M,\nconverges, meaning when", "start": 3411.89, "duration": 6.29}, {"text": "I have to decide whether\na series converges or not,", "start": 3418.18, "duration": 3.03}, {"text": "it doesn't matter\nwhat's going on", "start": 3421.21, "duration": 2.46}, {"text": "for the first\nfinitely many terms.", "start": 3423.67, "duration": 3.01}, {"text": "What matters is\nwhat's going on as I", "start": 3426.68, "duration": 1.79}, {"text": "keep adding terms from\nfurther and further out", "start": 3428.47, "duration": 4.467}, {"text": "of this sequence.", "start": 3432.937, "duration": 0.708}, {"text": "And what's the proof?", "start": 3436.57, "duration": 0.93}, {"text": "The proof is just\nexpressing the partial sums", "start": 3437.5, "duration": 2.93}, {"text": "for this guy in terms of the\npartial sums for this guy.", "start": 3440.43, "duration": 3.48}, {"text": "So a partial sum satisfied\nfor all m, sum from n", "start": 3443.91, "duration": 18.46}, {"text": "equals 1 to m, and now\nx sub n as sum from n", "start": 3462.37, "duration": 9.67}, {"text": "equals capital M to m\nx sub n plus sum from n", "start": 3472.04, "duration": 6.63}, {"text": "equals 1 to capital\nM minus 1 x sub n.", "start": 3478.67, "duration": 6.35}, {"text": "So this is now just\na fixed number.", "start": 3485.02, "duration": 2.32}, {"text": "So this is a sequence of\npartial sums corresponding", "start": 3487.34, "duration": 4.94}, {"text": "to this series.", "start": 3492.28, "duration": 1.92}, {"text": "This is a sequence of\npartial sums corresponding", "start": 3494.2, "duration": 2.28}, {"text": "to this series.", "start": 3496.48, "duration": 0.99}, {"text": "And this is just a fixed number.", "start": 3497.47, "duration": 2.01}, {"text": "Therefore, if this\nconverges, then this side", "start": 3499.48, "duration": 3.78}, {"text": "converges to this plus this--", "start": 3503.26, "duration": 4.86}, {"text": "so maybe I'm going\na little quick.", "start": 3508.12, "duration": 2.25}, {"text": "But so if this\nconverges, then this", "start": 3510.37, "duration": 4.29}, {"text": "converges to this\nminus this number.", "start": 3514.66, "duration": 3.31}, {"text": "And if this converges, then\nthis sequence of partial sums", "start": 3517.97, "duration": 5.36}, {"text": "converges to this limit\nplus this fixed number.", "start": 3523.33, "duration": 5.67}, {"text": "And that's all I'm\ngoing to write.", "start": 3529.0, "duration": 1.65}, {"text": "Now, coming back to the\nusefulness of Cauchy sequences,", "start": 3538.72, "duration": 15.74}, {"text": "this is kind of\nwhere they really", "start": 3554.46, "duration": 5.0}, {"text": "become useful is in\nthe study of series.", "start": 3559.46, "duration": 2.91}, {"text": "Because again, it's\ndifficult to sum.", "start": 3565.58, "duration": 2.41}, {"text": "So when I keep talking\nsaying the word sum a series,", "start": 3567.99, "duration": 2.81}, {"text": "I'm talking about find\nthe limit of partial sums.", "start": 3570.8, "duration": 5.055}, {"text": "But because we have this\nequivalence between Cauchy", "start": 3578.45, "duration": 2.79}, {"text": "sequences and\nconvergent sequences,", "start": 3581.24, "duration": 3.63}, {"text": "to decide if a series\nis convergent or not,", "start": 3584.87, "duration": 3.96}, {"text": "we can just decide if, in some\nsense, it's Cauchy or not.", "start": 3588.83, "duration": 5.02}, {"text": "So let me make this definition.", "start": 3593.85, "duration": 5.39}, {"text": "We say that a series x sub\nn is Cauchy if the sequence", "start": 3599.24, "duration": 10.41}, {"text": "of partial sums--", "start": 3609.65, "duration": 1.17}, {"text": "again, I'm just going to\nput an m up top because this", "start": 3610.82, "duration": 8.66}, {"text": "may start at 0 or n equals 1\nor something-- so the sequence", "start": 3619.48, "duration": 4.11}, {"text": "of partial sums is Cauchy.", "start": 3623.59, "duration": 7.73}, {"text": "And so let me just restate\nwhat we proved for sequences", "start": 3636.84, "duration": 5.67}, {"text": "in terms of series.", "start": 3642.51, "duration": 1.62}, {"text": "So we proved that every\nCauchy sequence is convergent.", "start": 3644.13, "duration": 4.15}, {"text": "So a series is Cauchy means that\nthe sequence of partial sums", "start": 3648.28, "duration": 7.73}, {"text": "is Cauchy.", "start": 3656.01, "duration": 0.78}, {"text": "But we've proven that Cauchy\nsequences are convergent.", "start": 3656.79, "duration": 3.79}, {"text": "So if this is Cauchy, then\nit's convergent and vice versa.", "start": 3660.58, "duration": 4.32}, {"text": "So based on what we've\nproven already for sequences,", "start": 3664.9, "duration": 6.99}, {"text": "it follows that--", "start": 3671.89, "duration": 1.985}, {"text": "and this just\nfollows immediately", "start": 3673.875, "duration": 1.375}, {"text": "from what we've proven\nalready, so I'm not even", "start": 3675.25, "duration": 1.958}, {"text": "going to write a proof--", "start": 3677.208, "duration": 1.672}, {"text": "a series is Cauchy if and only\nif the series is convergent--", "start": 3678.88, "duration": 7.11}, {"text": "again, because both\nare defined in terms", "start": 3688.57, "duration": 2.04}, {"text": "of the sequence of partial\nsums associated to the series.", "start": 3690.61, "duration": 3.03}, {"text": "And we've already proven the\nequivalence between Cauchy", "start": 3693.64, "duration": 2.85}, {"text": "and convergence for sequences.", "start": 3696.49, "duration": 3.54}, {"text": "Now, let me write what\nit means to be Cauchy", "start": 3710.74, "duration": 3.0}, {"text": "in a slightly different way.", "start": 3713.74, "duration": 3.435}, {"text": "And it's the following.", "start": 3723.66, "duration": 1.29}, {"text": "So before, we had that\na sequence is Cauchy,", "start": 3731.86, "duration": 5.13}, {"text": "intuitively, if the\nelements of the sequence", "start": 3736.99, "duration": 3.63}, {"text": "are getting close to each other.", "start": 3740.62, "duration": 2.07}, {"text": "Now, for a series to be\nCauchy, the intuitive way", "start": 3742.69, "duration": 3.87}, {"text": "to think about it is\nthat the tail of the sum", "start": 3746.56, "duration": 3.9}, {"text": "is getting small, is\ngetting arbitrarily", "start": 3750.46, "duration": 3.72}, {"text": "small, the tail of the\nsum being if I add up", "start": 3754.18, "duration": 3.93}, {"text": "finitely many numbers\nfar enough out.", "start": 3758.11, "duration": 2.88}, {"text": "So a series is\nCauchy if and only", "start": 3760.99, "duration": 4.48}, {"text": "if all epsilon positive there\nexist to M, a natural number,", "start": 3765.47, "duration": 7.86}, {"text": "such that for all integers\nl bigger than m, which", "start": 3773.33, "duration": 7.35}, {"text": "is bigger than or equal to\ncapital M, the sum from n", "start": 3780.68, "duration": 7.92}, {"text": "equals n plus 1\nto l of x sub n--", "start": 3788.6, "duration": 4.71}, {"text": "so this is a sum involving\nterms that are pretty far out", "start": 3793.31, "duration": 4.53}, {"text": "there, at least all\nindexed by something bigger", "start": 3797.84, "duration": 4.44}, {"text": "than or equal to capital M--", "start": 3802.28, "duration": 1.65}, {"text": "is less than epsilon.", "start": 3803.93, "duration": 1.86}, {"text": "So a series is Cauchy\nif you're adding up", "start": 3805.79, "duration": 3.39}, {"text": "smaller and smaller pieces, not\nindividual pieces but actual", "start": 3809.18, "duration": 4.29}, {"text": "adding those up.", "start": 3813.47, "duration": 1.29}, {"text": "So I'll leave it to you to do--", "start": 3824.74, "duration": 5.16}, {"text": "they're both pretty\neasy-- but this direction,", "start": 3829.9, "duration": 3.25}, {"text": "I'll leave it to\nyou as an exercise.", "start": 3833.15, "duration": 1.85}, {"text": "And it'll follow\nimmediately from what", "start": 3835.0, "duration": 4.29}, {"text": "I'm going to write for,\nessentially, this direction.", "start": 3839.29, "duration": 3.57}, {"text": "So let's suppose-- and\nlet's make things concrete,", "start": 3846.45, "duration": 8.2}, {"text": "starting somewhere-- let's\nsuppose this sum is Cauchy.", "start": 3854.65, "duration": 2.535}, {"text": "And we want to prove now\nthat it has this property.", "start": 3861.08, "duration": 4.23}, {"text": "So let epsilon be positive.", "start": 3865.31, "duration": 5.3}, {"text": "We now want to produce\nsome capital number", "start": 3870.61, "duration": 1.75}, {"text": "M so that this holds.", "start": 3872.36, "duration": 2.01}, {"text": "So since the sequence of\npartial sums s sub m is Cauchy--", "start": 3883.73, "duration": 8.49}, {"text": "that's what it means for\nthe series to be Cauchy--", "start": 3892.22, "duration": 3.99}, {"text": "there exists a natural\nnumber, capital M,", "start": 3896.21, "duration": 4.14}, {"text": "such that for all M\nbigger than or equal to M0", "start": 3900.35, "duration": 9.75}, {"text": "and l bigger than or equal\nto M0, s sub m minus s", "start": 3910.1, "duration": 13.5}, {"text": "sub l is less than epsilon.", "start": 3923.6, "duration": 2.04}, {"text": "So we're actually going to\ntake capital M to be this M0.", "start": 3931.72, "duration": 4.47}, {"text": "Choose M to be M0.", "start": 3936.19, "duration": 8.65}, {"text": "Then, if l is\nbigger than m, which", "start": 3944.84, "duration": 6.77}, {"text": "is bigger than or equal to\nM, which is equal to M0,", "start": 3951.61, "duration": 6.12}, {"text": "if I look at this sum from n\nequals m plus 1 to l x sub n,", "start": 3957.73, "duration": 7.98}, {"text": "I can write--", "start": 3965.71, "duration": 2.568}, {"text": "so this is absolute value.", "start": 3968.278, "duration": 1.272}, {"text": "So in fact, let me remove\nthe absolute value so", "start": 3973.73, "duration": 2.09}, {"text": "that this becomes pretty clear.", "start": 3975.82, "duration": 1.71}, {"text": "I can write this sum\nas the l-th partial sum", "start": 3980.54, "duration": 4.29}, {"text": "minus the n-th partial sum.", "start": 3984.83, "duration": 3.42}, {"text": "Because the l-th partial sum\nsums from n equals 1 up to l.", "start": 3988.25, "duration": 4.32}, {"text": "The m-th partial sums\nfrom n equals 1 up to m.", "start": 3992.57, "duration": 3.27}, {"text": "So the sum containing only the\nterms between n plus 1 and l", "start": 3995.84, "duration": 4.05}, {"text": "is the difference of these guys.", "start": 3999.89, "duration": 1.395}, {"text": "So now I'll put it\non absolute values.", "start": 4003.91, "duration": 3.79}, {"text": "And this thing,\nbecause l and m are", "start": 4007.7, "duration": 2.46}, {"text": "bigger than or equal to capital\nM, which is equal to M0,", "start": 4010.16, "duration": 3.09}, {"text": "and because I have\nthis inequality,", "start": 4013.25, "duration": 2.4}, {"text": "this is less than epsilon.", "start": 4015.65, "duration": 1.83}, {"text": "And the converse\ndirection, again, it", "start": 4017.48, "duration": 1.74}, {"text": "follows immediately,\nessentially,", "start": 4019.22, "duration": 1.74}, {"text": "from this equality here.", "start": 4020.96, "duration": 3.54}, {"text": "So to check whether\na series converges,", "start": 4028.57, "duration": 2.97}, {"text": "I don't have to somehow come up\nwith a limit for this series,", "start": 4031.54, "duration": 6.69}, {"text": "a sum for this series.", "start": 4038.23, "duration": 3.21}, {"text": "I can just prove\nthat the tail can", "start": 4041.44, "duration": 3.72}, {"text": "be made arbitrarily\nsmall, as long", "start": 4045.16, "duration": 1.95}, {"text": "as I go far enough\nout in the series.", "start": 4047.11, "duration": 3.105}, {"text": "And from this, we get a pretty\nsimple elementary property,", "start": 4065.97, "duration": 7.98}, {"text": "so a theorem.", "start": 4073.95, "duration": 0.78}, {"text": "If a series converges, then\nthis implies that the limit as n", "start": 4080.19, "duration": 14.98}, {"text": "goes to infinity of x sub\nn of the sequence you used", "start": 4095.17, "duration": 5.88}, {"text": "to obtain the series equals 0.", "start": 4101.05, "duration": 3.36}, {"text": "So this should fall in line\nwith a series being convergent", "start": 4104.41, "duration": 6.9}, {"text": "if and only if it satisfies this\nproperty here, which is somehow", "start": 4111.31, "duration": 3.87}, {"text": "saying the tail of the sum is\ngetting smaller and smaller,", "start": 4115.18, "duration": 4.719}, {"text": "which means you can't be adding\nup big things as you go on", "start": 4119.899, "duration": 5.75}, {"text": "out in the series.", "start": 4125.649, "duration": 1.516}, {"text": "So proof.", "start": 4133.399, "duration": 0.541}, {"text": "So we'll show this by a\nsimple epsilon M definition.", "start": 4139.02, "duration": 10.549}, {"text": "So suppose xn converges.", "start": 4149.569, "duration": 6.96}, {"text": "Then xn the series is Cauchy.", "start": 4159.56, "duration": 5.945}, {"text": "And now I'm going to verify this\nusing the epsilon M definition.", "start": 4168.87, "duration": 4.59}, {"text": "Let epsilon be positive.", "start": 4176.76, "duration": 1.44}, {"text": "Since xn is Cauchy,\nthis implies that there", "start": 4180.76, "duration": 6.36}, {"text": "exists a natural number M0\nsuch that for all l bigger", "start": 4187.12, "duration": 7.38}, {"text": "than or equal to m\nbigger than or equal M0,", "start": 4194.5, "duration": 4.11}, {"text": "I have that condition there,\nsum from n equals m plus 1", "start": 4198.61, "duration": 9.4}, {"text": "to l x sub n is\nless than epsilon.", "start": 4208.01, "duration": 3.66}, {"text": "Choose M to be M0 plus 1.", "start": 4220.74, "duration": 5.06}, {"text": "So why M0 plus 1\nbut not exactly M0?", "start": 4229.16, "duration": 2.76}, {"text": "Just because, basically,\nwhat I'm going to do", "start": 4231.92, "duration": 2.76}, {"text": "is I'm going to take l\nto be equal to m plus 1.", "start": 4234.68, "duration": 2.85}, {"text": "And the index gets shifted by 1.", "start": 4237.53, "duration": 2.91}, {"text": "Then if m is bigger\nthan or equal to M,", "start": 4243.32, "duration": 5.52}, {"text": "I get that the absolute\nvalue of x sub m--", "start": 4248.84, "duration": 3.3}, {"text": "maybe instead of\nsaying limit as n", "start": 4255.38, "duration": 3.57}, {"text": "goes to infinity, I'll write\nlimit as m goes to infinity,", "start": 4258.95, "duration": 2.68}, {"text": "it's just a change in\nthe dummy variable-- this", "start": 4261.63, "duration": 2.21}, {"text": "is equal to the sum from\nn equals m to m x sub n.", "start": 4263.84, "duration": 11.88}, {"text": "And so I've shifted this.", "start": 4275.72, "duration": 2.11}, {"text": "So now little m is bigger than\nor equal to capital M0 plus 1.", "start": 4277.83, "duration": 5.6}, {"text": "So you could write this as,\nif you like, m minus 1 plus 1.", "start": 4283.43, "duration": 4.17}, {"text": "So little m minus 1 is\nbigger than or equal to M0.", "start": 4287.6, "duration": 4.56}, {"text": "So by this inequality,\nthis is less than epsilon.", "start": 4292.16, "duration": 2.64}, {"text": "So we see that if a series\nconverges the terms,", "start": 4312.96, "duration": 5.19}, {"text": "the individual terms, x\nsub n must converge to 0.", "start": 4318.15, "duration": 6.03}, {"text": "So there's another reason why\nthis series minus 1 to the n", "start": 4324.18, "duration": 4.5}, {"text": "does not converge.", "start": 4328.68, "duration": 1.98}, {"text": "Because those terms\ndo not converge.", "start": 4330.66, "duration": 1.53}, {"text": "And this also tells us that\nfor this geometric series, when", "start": 4335.64, "duration": 4.08}, {"text": "r is greater than or equal\nto 1 in absolute value,", "start": 4339.72, "duration": 3.802}, {"text": "the series does not converge.", "start": 4343.522, "duration": 1.208}, {"text": "It does not converge.", "start": 4354.25, "duration": 2.64}, {"text": "And the proof is--", "start": 4356.89, "duration": 0.78}, {"text": "and we proved this, in fact,\nI think, a few lectures ago,", "start": 4361.24, "duration": 6.71}, {"text": "as well, that if the absolute\nvalue of r is bigger than 1,", "start": 4367.95, "duration": 5.79}, {"text": "then the limit as n goes\nto infinity of r to the n,", "start": 4373.74, "duration": 5.1}, {"text": "this limit does not exist.", "start": 4378.84, "duration": 3.56}, {"text": "We showed it's, in\nfact, unbounded.", "start": 4385.55, "duration": 3.39}, {"text": "r to the n is an\nunbounded sequence.", "start": 4388.94, "duration": 6.23}, {"text": "So it does not convert.", "start": 4395.17, "duration": 6.71}, {"text": "So I'm using that\ntheorem over there", "start": 4401.88, "duration": 3.21}, {"text": "in a little bit roundabout way.", "start": 4405.09, "duration": 1.35}, {"text": "Let me restate this\ntheorem over here.", "start": 4417.89, "duration": 1.68}, {"text": "This theorem says if\nthe series converges,", "start": 4419.57, "duration": 6.55}, {"text": "then this limit equals 0.", "start": 4426.12, "duration": 4.49}, {"text": "Now, this statement is logically\nequivalent to the negation", "start": 4430.61, "duration": 6.15}, {"text": "of the converse, or there's\nan actual word for that,", "start": 4436.76, "duration": 5.76}, {"text": "but I can't remember--", "start": 4442.52, "duration": 1.95}, {"text": "namely that the negation of this\nimplies the negation of this.", "start": 4444.47, "duration": 8.43}, {"text": "So a logically equivalent way\nto rewrite that statement over", "start": 4452.9, "duration": 6.06}, {"text": "there is that if this\nlimit does not equal to 0--", "start": 4458.96, "duration": 12.94}, {"text": "so if it doesn't exist at\nall, that's also fine--", "start": 4471.9, "duration": 4.68}, {"text": "this implies that\ndoes not converge.", "start": 4476.58, "duration": 6.57}, {"text": "So this restatement of\nthe theorem over there", "start": 4483.15, "duration": 4.92}, {"text": "is really what I used here.", "start": 4488.07, "duration": 3.84}, {"text": "All right.", "start": 4496.35, "duration": 0.5}, {"text": "And I think I'll stop there.", "start": 4496.85, "duration": 2.59}, {"text": "Next time, we'll see\nthat this theorem here", "start": 4499.44, "duration": 4.7}, {"text": "is a one-way street.", "start": 4504.14, "duration": 1.39}, {"text": "And I think you covered\nthis example in, probably,", "start": 4505.53, "duration": 2.93}, {"text": "calculus, namely that\none-way street in the sense", "start": 4508.46, "duration": 3.72}, {"text": "that if x then converges,\nthen this limit is 0.", "start": 4512.18, "duration": 3.63}, {"text": "But the converse does not hold.", "start": 4515.81, "duration": 2.22}, {"text": "Namely, it is not true\nthat if this limit", "start": 4518.03, "duration": 3.93}, {"text": "is 0, then this converges.", "start": 4521.96, "duration": 3.48}, {"text": "And we'll see the famed\nharmonic series next time.", "start": 4525.44, "duration": 6.23}]