ps5 - s if and only if every subsequence has a subsequence...

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18.100B and 18.100C Fall 2011 Problem Set 5 Due October 20th at 4 pm in room 2-108. Hand in parts 1, 2 and 3 separately. Put your name and whether you are registered for 18.100B or 18.100C on each part. Part 1 1. (a) Prove that if ( a n ) n N is a bounded sequence of real numbers, then lim sup n →∞ a n = lim n →∞ (sup { a m : m n } ) . (b) Prove that if ( a n ) n N and ( b n ) n N are bounded sequences of real numbers, then lim sup n →∞ ( a n + b n ) lim sup n →∞ a n + lim sup n →∞ b n , and that equality holds if ( a n ) n N converges. Give an example of two sequences for which equality does not hold. 2. (a) If a n b n c n for all n N , and if ( a n ) n N and ( c n ) n N converge and have the same limit, prove that ( b n ) n N also converges and also has the same limit. (b) Let k N and let x 1 x 2 ≥ ··· ≥ x k 0. Evaluate lim n →∞ ( x n 1 + ··· + x n k ) 1 /n . Part 2 3. Prove that a sequence in a metric space converges to a point
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Unformatted text preview: s if and only if every subsequence has a subsequence which converges to s . Students registered for 18.100C should write this problem up in LaTeX. 4. Problem 24 from page 82. Part 3 5. Problem 6 from page 78. 1 6. Let ( a n ) n ∈ N be a sequence of positive numbers which tends to zero but such that ∑ ∞ n =1 a n diverges. Let ( A n ) n ∈ N be the sequence of partial sums A n = n X k =1 a k , and let b n +1 = √ A n +1-√ A n . Show that lim n →∞ b n a n = 0 , but that ∑ ∞ n =1 b n is still divergent. In this sense there is no ‘smallest’ divergent series, and one can similarly show that there is no ‘largest’ convergent one. 2...
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ps5 - s if and only if every subsequence has a subsequence...

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