real analysis - Show that $sum a_n b_n$ is convergent

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?

Answer

It is enough to apply Dirichlet's test. If $\sum_{n\geq 1}a_n$ is convergent to $A$ then its partial sums are bounded, and if $\{b_n\}_{n\geq 1}$ is monotonic and bounded it is convergent to some $B$ and $B-b_n$ is decreasing to zero. It follows that

$$ \sum_{n\geq 1}a_n b_n = A B - \underbrace{\sum_{n\geq 1} a_n (B-b_n)}_{\text{convergent by Dirichlet}} $$

is convergent as well.