First, use the Monotone Convergence Theorem to show that every bounded sequence with a monotone subsequence has a convergent subsequence.

Then, prove that every bounded sequence has a monotone subsequence.

A hint is given for the second part that says: In a sequence <a>, call an index n a peak if a{m} < a{n} for m > n. Consider two cases, accordingly to whether <a> has infinitely many peaks.

My professor says: How can you get a non-increasing subsequence if you have infinitely many peaks? If you have finitely many peaks, what happens after the last peak. Can you get a non-decreasing subsequence?