Divergence of series - Counter example

Question

I am attempting to disprove following statement with counter example.

If $\sum_{n=1}^{\infty} a_{n}$ converges and lim $b_{n}$ = 0, then $\sum_{n=1}^{\infty} a_{n}b_{n}$ converges

My work: If a series converges then lim $ a_{n} $ = 0. All of the examples I am trying to construct are turning out to be convergent. Can I get some help here?

Answer 1

[Counter-Example] : $a_{n}=b_{n}=\frac{(-1)^n}{\sqrt{n}}$