# Prove $(a_n,b_n >0) \land \sum a_n$ converges $\land \sum b_n$ diverges$\implies \liminf\limits_{n\rightarrow \infty} \frac{a_n}{b_n}=0$

$$(a_n)_{n \in \mathbb{N}},(b_n)_{n \in \mathbb{N}}$$ are positive, real sequences!

Since $$\sum a_n$$ converges, we know $$\lim\limits_{n\rightarrow \infty}a_n =0$$.

If $$\lim\limits_{n\rightarrow \infty}b_n \ne 0$$ then $$\liminf\limits_{n\rightarrow \infty} \frac{a_n}{b_n}=0$$ is rather easy to show.

So let's assume $$b_n$$ converges to $$0$$. Since $$\sum a_n$$ converges but $$\sum b_n$$ does not, $$a_n$$ must somehow "converge faster" to $$0$$ than $$b_n$$ does, thus causing $$\liminf\limits_{n\rightarrow \infty} \frac{a_n}{b_n}=0$$, but I have a hard time to express that formally.

I'd be very grateful for a push in the right direction.

thanks for helping :)

Hint: If $$c = \liminf_{n\to \infty} \frac{a_n}{b_n} > 0$$ then $$a_n \ge \frac c2 b_n \ge 0$$ for all sufficiently large $$n$$. What does that tell about the convergence of the series $$\sum a_n$$ and $$\sum b_n$$?