Paper 1, Section I, 3F3 F

Analysis I | Part IA, 2011

(a) State, without proof, the Bolzano-Weierstrass Theorem.

(b) Give an example of a sequence that does not have a convergent subsequence.

(c) Give an example of an unbounded sequence having a convergent subsequence.

(d) Let an=1+(1)n/2(1+1/n)a_{n}=1+(-1)^{\lfloor n / 2\rfloor}(1+1 / n), where x\lfloor x\rfloor denotes the integer part of xx. Find all values cc such that the sequence {an}\left\{a_{n}\right\} has a subsequence converging to cc. For each such value, provide a subsequence converging to it.

Typos? Please submit corrections to this page on GitHub.