Suppose that $a_n$, $b_n$ are such that $\lim_{n \rightarrow \infty} a_n = \infty$ and $\lim_{n \rightarrow \infty} \frac {a_n}{b_n} = \alpha$

Question

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.

Answer 1

if $b_n $ is bounded, given any arbitrary large $\epsilon$ we can find an $N$ such that $\frac{a_n}{b_n} \ge \epsilon$ for some $n \ge N$. after ruling out $b_n$ being bounded, Only case remains is $b_n$ diverging. In this case we can always find a subsequnce of $b_n$ that is bounded, so $\frac{a_n}{b_n}$ contains a subsequence that is going to $\infty$, which is again a contradiction.