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79 Pages

### hw6_320

Course: MAT 320, Fall 2008
School: SUNY Stony Brook
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Word Count: 222

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320 MAT Introduction to Analysis Problem Set 6 due Friday, October 27 1. (a) Give an example of a sequence (xn ) which has subsequences (yn ), (zn ), and (wn ), such that lim yn = , lim zn = 0, and lim wn = 3. n n n Can the sequence (xn ) be convergent? (b) Give examples of two sequences (xn ) and (yn ) with n lim xn = lim yn = + n (and yn = 0 for all n), such that (i) limn (xn /yn ) = +; (ii) limn (xn /yn...

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320 MAT Introduction to Analysis Problem Set 6 due Friday, October 27 1. (a) Give an example of a sequence (xn ) which has subsequences (yn ), (zn ), and (wn ), such that lim yn = , lim zn = 0, and lim wn = 3. n n n Can the sequence (xn ) be convergent? (b) Give examples of two sequences (xn ) and (yn ) with n lim xn = lim yn = + n (and yn = 0 for all n), such that (i) limn (xn /yn ) = +; (ii) limn (xn /yn ) = 0; (iii) limn (xn /yn ) = 2; (iv) the sequence (xn /yn ) is bounded but divergent. 2. Suppose that xn 0 for all n N, and ((1)xn ) is a convergent sequence. Show that the sequence (xn ) is convergent. Whats its limit? Let 3. (xn ) be a bounded sequence, and let s = sup{xn : n N}. Also, suppose that s = xn for any n. Show that there is a subsequence of (xn ) which converges to s. What if xn = s for some n? 1 1 1 4. (a) Let xn = 1 + 2! + 3! + + n! . Show directly from the denition that (xn ) is a Cauchy sequence. 1 1 (b) Consider the sequence yn = 1 + 2 + + n . Show that (yn ) diver...

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SUNY Stony Brook - MAT - 320
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