Past exam of the mathematics course of the University of Cambridge / 2014 / ia / Paper 1 / 3D (Analysis I)

Call an index $n$ a peak if $a_n\ge a_m$ for every $m>n$. If there are infinitely many peaks, list their indices increasingly. Each selected value is at least every later selected value, so this gives a nonincreasing subsequence.

If there are only finitely many peaks, choose an index beyond the last one. Every index from then onward has a later index with strictly larger value. Select these recursively: $n_{j+1}>n_j$ with $a_{n_{j+1}}>a_{n_j}$. This gives an increasing subsequence. Neither complex argument requires the original sequence to be bounded. Every real sequence therefore has a monotone subsequence, as asserted by the monotone subsequence theorem.