Prove or provide a counterexample:
Let ${r_{n}}$ be a sequence such that $r_{n} \rightarrow 0$ as $n \rightarrow \infty$. Then, the sum $\sum\limits_{n=1}^{\infty} \frac{1}{n} r_{n}$ converges.
This has me stumped... I can't think of any counterexamples, but it just doesn't seem true.