Hello. I am having some difficulty proving the following:

If $\displaystyle \sum_{n=1}^\infty a_n$ converges and $\displaystyle \{b_n\}$ is monotonic and bounded, then $\displaystyle \sum_{n=1}^\infty a_nb_n$ converges.

I tried using the Cauchy Criterion and partial summation to handle this, but I haven't had any luck. Your insight would be appreciated.