# Proof regarding series

I've encountered a recommended practice proof that I'd like some assistance in starting.

Suppose that $\sum_{i=1}^\infty an$ and $\sum_{i=1}^\infty bn$ are both series with all positive terms and $\sum_{i=1}^\infty bn$ is convergent. Prove that if $lim_{n\to \infty} an/bn$ = 0, then $an$ converges.

This appears to be the limit comparison test for series, but the requirement for 0 < c < inf is not met. Any ideas would be appreciated.

Hint: $$\lim_{n \to \infty} \dfrac{a_n}{b_n} = 0 \qquad \Longrightarrow \qquad \dfrac{a_n}{b_n} < 1$$ for large $n$.