If series of terms $a_j, b_j$ are convergent, must $\sum_j^\infty a_j b_j$ be convergent?

Question

If $\sum_j^\infty a_j$ and $\sum_j^\infty b_j$ are convergent, show that $\sum_j^\infty a_j \cdot b_j$ is convergent or provide a counter example where it is not convergent.

If it can be assumed that one of the given series is absolutely convergent, then I can see a proof that the resulting series is convergent. But I don't see a proof or a counter example when both given series are conditionally convergent.

Answer 1

No necessarily: Take

$a_j=b_j=(-1)^j\frac{1}{\sqrt{j}}$

Answer 2

Hint If $a >0$ then $$\sum_{n=1}^\infty \frac{(-1)^n}{n^a}$$ is convergent by AST.

Can you multiply two such series to get a divergent one?