Let $\{a_{n}\}$ be a positive decreasing sequence such that $\lim_{n\to \infty}na_{n}$ goes to 0. Does it follow that $\sum a_{n}$ converges?

Let $\{a_{n}\}$ be a positive decreasing sequence such that $\lim_{n\to \infty}na_{n}$ goes to 0. Does it follow that $\sum a_{n}$ converges?

Thoughts

I managed to solve the converse of it. But am not able to do it, or find a counter example. Help.