To remind, there is a Calculus forum on this website...

So I understand you need to prove the second claim, that every bounded sequence has a monotone subsequence. Suppose there are infinitely many peaks. Then corresponding a{n}'s form a non-increasing subsequence. Suppose now that there are no peaks after some point, i.e., each index is not a peak. Write what it means that n is not a peak; this should give you an idea how to form a non-decreasing subsequence.