hw5 - a 2 n implies the convergence of X a n n 5 If ∑ a n...

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Homework 5 – Math 118A, Fall 2009 Due on Thursday, November 5th, 2009 1. Consider a sequence { s n } n N R . Prove the following: (a) lim sup n →∞ s n = inf n (sup k n s k ) . (b) lim inf n →∞ s n = sup n (inf k n s k ) . 2. If s n t n for n N , where N is Fxed, then lim inf n →∞ s n lim inf n →∞ t n , (1) lim sup n →∞ s n lim sup n →∞ t n . (2) 3. Investigate the behavior (convergence or divergence) of a n if (a) a n = n + 1 - n . (b) a n = n +1 - n n . (c) a n = ( n n - 1) n . (d) a n = 1 1+ z n for complex values of z . 4. Prove that the convergence of
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Unformatted text preview: a 2 n implies the convergence of X a n n . 5. If ∑ a n converges, and if b n is monotonic and bounded, prove that ∑ a n b n converges. 6. If { E n } n ∈ N is a sequence of closed nonempty and bounded sets in a complete metric space X , if E n +1 ⊂ E n for all n ∈ N , and if lim n →∞ diam E n = 0 , then ∩ ∞ n =1 E n consists of exactly one point. 1...
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This note was uploaded on 12/27/2011 for the course MATH 118a taught by Professor Stopple,j during the Fall '08 term at UCSB.

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