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

Choose the sequenceEvery even index is a peak of a sequence $(a_n)$, since its value is $1$ and all later values are at most $1$. Every odd index is a peak of a sequence $(-a_n)$ for the same reason. Both sets of indices are infinite.

Choose the sequenceAfter every index there are positive terms arbitrarily large and negative terms arbitrarily small. Thus every term has both a larger later term and a smaller later term. The former excludes a peak of a sequence $(b_n)$, and the latter excludes a peak of a sequence $(-b_n)$.