$(a_n)_{n \in \mathbb{N}},(b_n)_{n \in \mathbb{N}}$ 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 :)