testsolII - 1 Cauchy Convergence Criterion A sequence(xn is...

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1) Cauchy Convergence Criterion: A sequence ( x n ) is Cauchy if and only if it is convergent. Proof. Suppose ( x n ) is a convergent sequence, and lim( x n ) = x . Let ± > 0. We can find N N such that for all n N , | x n - x | < ±/ 2. Therefore, by the triangle inequality, for all m,n N , | x m - x n | ≤ | x m - x | + | x - x n | < ±/ 2 + ±/ 2 = ± . So ( x n ) is Cauchy. Conversely, suppose ( x n ) is Cauchy. Let ± > 0. By a result proved in class, ( x n ) is bounded. By Bolzano-Weierstrass, it has a convergent subsequence ( x n k ) with lim( x n k ) = x for some x . We can find K N such that for all k K , | x n k - x | < ±/ 2. We can also find M such that for all m,n M , | x m - x n | < ±/ 2. Let N = sup { K,M } . Then since n k k for all k , if k N , we have that k,n k M and n k K . Therefore, for all k N , | x n - x | ≤ | x n - x n k | + | x n k - x | < ±/ 2 + ±/ 2 = ± by the Triangle Inequality. Therefore, (
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This note was uploaded on 01/31/2011 for the course AMS 319 taught by Professor Mark during the Fall '10 term at SUNY Stony Brook.

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testsolII - 1 Cauchy Convergence Criterion A sequence(xn is...

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