k For any given sequence s n s n has a convergent subsequence s n k 4 l For any

K for any given sequence s n s n has a convergent

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(k) For any given sequence ( s n ), ( s n ) has a convergent subsequence ( s n k ). 4
(l) For any bounded sequence ( s n ), lim sup s n = sup { s n } . (m) It α is a subsequential limit of a bounded sequence ( s n ), then α lim sup s n (n) If every subsequence of a sequence ( s n ) is convergent, then ( s n ) itself must be convergent. 5
(o) If ( s n ) is a divergent sequence, then some subsequence of ( s n ) must diverge. (p) If ( s n ) is unbounded above, then ( s n ) has an increasing subsequence ( s n k ) which diverges to + . 6
2. Let ( s n ) be a positive sequence such that lim n →∞ s n +1 s n = L > 1 . Prove that s n + . Hint: lim n →∞ x n = + for any number x such that x > 1. 7
3. Limits: Determine the convergence or divergence of each of the following sequences. If the limit exists, state what it is. (Hint: Use the previous problem, and Theorem 3 from Section 17.) (a) s n = n 3 3 n (b) s n = n ! n 10 (c) s n = n n n ! 8
4. For each of the following sequences, determine: (1) Whether the sequence con- verges; if it does, give the limit. (2) If the sequence does not converge, find the set of subsequential limits (if any) and give lim sup and lim inf. (a) s n = 1 n + 1 , n odd 3 2 n - 1 , n even (b) s n = 1 n , n = 3 , 6 , 9 , · · · n n + 1 , n = 1 , 4 , 7 , 10 , · · · n 2 , n = 2 , 5 , 8 , 11 , · · · (c) s k = k sin 2 4 (d) s n = 1 2 n + cos 6 9

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• Fall '08
• Staff
• Math, Mathematical analysis, Limit of a sequence, Limit superior and limit inferior, subsequence, Cauchy subsequence