Given that $\displaystyle \sum{a_n}$ converges and $\displaystyle [b_n]$ is monotonic and bounded, prove $\displaystyle \sum{a_nb_n}$ converges.

So I know $\displaystyle a_n{\rightarrow{0}}$ and $\displaystyle b_n$ converges, so $\displaystyle a_nb_n{\rightarrow{0}}$

Where do I go from here?