Past exam of the mathematics course of the University of Oxford / 2015 / Prelims / Mathematics II / 1 / c / i

Choose $N$ beyond every peak of a sequence of both $(a_n)$ and $(-a_n)$. Every term with index at least $N$ has a strictly larger later term and a strictly smaller later term.

Suppose for a contradiction that $a_n\to L$. Starting at $a_N$, recursively select a strictly increasing subsequence and a strictly decreasing subsequence. Both must have limit of a sequence $L$. If their second terms are $u>a_N$ and $v<a_N$, respectively, passage to their limits of sequences givesa contradiction. Hence $(a_n)$ does not converge. This argument includes the possibility that either set of peaks of a sequence is empty.