The original post asks: My idea is: Assume $a_n$ is monotone and not converges and then show that it is not bounded. But: my problem is that I fail to prove it is not bounded. Please can you help me?
If you want to prove the statement, if a sequence is monotone and bounded then it converges, the logically equivalent contrapositive would be, if a sequence is divergent then either it is not monotone or it is not bounded. So, your idea would only get you halfway there. You would also need to prove that divergent bounded sequences cannot be monotone.
To the question asked though, you are seeking a direct proof of monotone and divergent $\implies$ unbounded. This will be a bit awkward as divergent and unbounded are defined in the negative. Without loss of generality assume the sequence $(a_n)$ is divergent and increasing. A divergent sequence in the reals is not a Cauchy sequence, which means there is a small distance $\epsilon>0$ such that there is an infinite subsequence $(b_n)$ whose members are at least $\epsilon$ apart from each other. Then, since $(a_n)$ and hence $(b_n)$ are increasing, $b_n \ge b_0 + n \epsilon$ grows without bound.