If $\sum a_n$ is convergent and $b_n$ is a monotone bounded sequence then $\sum a_nb_n$ is convergent.
What I did is
If $b_n$ is a monotone bounded sequence then $b_n$ is convergent and suppose that $b_n\rightarrow B$ when $b\rightarrow\infty$.
The algebraic limit theorems says that $\sum a_n B$ is convergent. Since $0\leq a_n b_n\leq a_n B$ then by the comparison test $\sum a_n b_n$ is also convergent.
It's a wrong use of comparison test?