HW-8_solutions - Homework 8. Solutions December 1, 2008...

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Homework 8. Solutions December 1, 2008 12.2 Show: lim sup | s n | = 0 if and only if lim s n =0 Proof Suppose lim sup | s n | =0. Let ± > 0, so there is N such that k > N implies sup {| s n | : n > k } < ± . This implies that n > N we have | s n | < ± , which is just the definition of lim s n = 0. Now suppose lim | s n | = 0. Let ± > 0, there exist N so that n > N implies | s n | < ± . So for all k > N sup {| s n | : n > k } < ± , which implies that lim sup | s n | = 0. 12.4 Show that: lim sup s n + t n lim sup s n + lim sup t n for bounded sequences s n and t n . Give an example of two sequences for which the equality does not hold. Proof Let ( s n ) and ( t n ) be bounded sequences. Then, the sets { s n : n N } , { t n : n N } and { s j + t k : j,k N } are bounded. Moreover, for each N N the subsets { s n : n > N } , { t n : n > N } and { s n + t n : n > N } are bounded (and are nonempty), by Completeness Axiom the subsets have a supremum. Let s N = sup { s n : n > N
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HW-8_solutions - Homework 8. Solutions December 1, 2008...

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