Convergence of term-wise product of convergent series

Question

Let $\sum a_n$ and $\sum b_n$ be two convergent series. It is easy to prove that their term-wise product $\sum a_n b_n$ converges if $a_n,b_n \geq 0$, but $\sum a_n b_n$ does not necessarily converge otherwise.

My question is, must $\sum a_n b_n$ converge if $a_n \geq 0$? Having thought about it some, it seems that there should be a counterexample, but I haven't been able to find one.

Answer 1

$\sum |a_n b_n| \leq M\sum a_n < \infty$ where $M=\sup_n |b_n|$. Note that $b_n \to 0$ so $\{b_n\}$ is a bounded sequence. Hence $M <\infty$ and the series $\sum a_n b_n$ is absolutely convergent.