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

This preview shows pages 1–2. Sign up to view the full content.

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

This preview has intentionally blurred sections. Sign up to view the full version.

View Full Document
This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

### Page1 / 2

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

This preview shows document pages 1 - 2. Sign up to view the full document.

View Full Document
Ask a homework question - tutors are online