# Convergence of term-wise product of convergent series

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.

$$\sum |a_n b_n| \leq M\sum a_n < \infty$$ 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.