MATH_3333_HW6_solns - MATH 3333 Homework Assignment#5 Due date Monday October 28 NAME(Print 1 TRUEFALSE If your answer is True justify by quoting a

MATH_3333_HW6_solns - MATH 3333 Homework Assignment#5 Due...

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MATH 3333 Homework Assignment #5 Due date: Monday, October 28 NAME (Print): 1. TRUE–FALSE. If your answer is “True,” justify by quoting a definition or theorem, or by giving a proof. If your answer is “False,” give a counter-example. (a) If ( s n ) is a Cauchy sequence, then ( s n ) is monotone. (b) If ( s n ) is a convergent sequence and ( t n ) is bounded, then ( s n · t n ) is a convergent sequence. 1
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(c) If ( s n ) and ( t n ) are sequences such that s n > t n > 1 for all n , and ( s n · t n ) converges, then both ( s n ) and ( t n ) converge. (d) If ( s n ) is an unbounded sequence, then every subsequence of ( s n ) is unbounded. (e) If ( s n ) is a bounded, monotone sequence, then ( s n ) is a Cauchy sequence. 2
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(f) If ( s n ) is a bounded sequence, then ( s n ) has a Cauchy subsequence. (g) Every oscillating sequence has a convergent subsequence. (h) Every oscillating sequence diverges. 3
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(i) If ( s n ) is an unbounded sequence, then either s n + or s n → -∞ . (j) If ( s n ) is a bounded sequence and α = sup { s n } , then ( s n ) has a subsequence which converges to α .
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  • Fall '08
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  • Math, Mathematical analysis, Limit of a sequence, Limit superior and limit inferior, subsequence, Cauchy subsequence

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