JhalyssaWilliams.Analysis.HW5 - Homework 5 Section 4.4 2a False By theorem 4.4.8 Every unbounded sequence contains a monotone subsequence that has

# JhalyssaWilliams.Analysis.HW5 - Homework 5 Section 4.4 2a...

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Homework 5 Section 4.4 2a) False; By theorem 4.4.8, “ Every unbounded sequence contains a monotone subsequence that has either +∞ ?? − ∞ as a limit.” Every sequence does not have a convergent subsequence. 2b) True; By theorem 4.4.7, “ Every bounded sequence has a convergent subsequence .” So by definitions 4.4.9, the subsequential limit must not be empty. 2c) False; By definition 4.4.9, if S n converges to s, then the lim inf S n = lim sup S n . This is true only if s ∈ ℝ 2d) True 2e)False: S n =n for every natural number n. It is not bounded about but the lim Inf S n = 1. . The n. +∞

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• Fall '08
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• Limits, Limit of a sequence, Limit superior and limit inferior, lim sup, subsequence, Lim Sup Vn