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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 Snconverges to s, then the lim inf Sn = lim sup Sn. This is true only if s ∈ ℝ2d) True 2e)False: Sn=n for every natural number n. It is not bounded about but the lim Inf Sn= 1. . The n. +∞
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Limits, Limit of a sequence, Limit superior and limit inferior, lim sup, subsequence, Lim Sup Vn