Problem: Suppose that $a_n$, $b_n$ are sequences of positive numbers such that $\lim_{n \rightarrow \infty} a_n = \infty$ and $\lim_{n \rightarrow \infty} \frac {a_n}{b_n} = \alpha$ for some $\alpha \in \Bbb R$, show that $b_n \rightarrow \infty$.
Thoughts: It seems that I could start with a proof by contradiction of 3 different cases where $b_n$ converges to a real number, or that it converges to negative infinity, and finally that it diverges. This seems like the wrong approach. Any hints much appreciated.