If the positive series $sum a_n$ diverges and $s_n=sumlimits_{kleqslant n}a_k$ then $sum frac{a_n}{s_n}$ diverges as well

So I've been trying to figure out how to prove the following.

Let $(a_n)$ be a sequence of positive numbers such that $\sum\limits_{n=1}^\infty a_n =\infty$, and define $s_n=\sum\limits_{i=1}^n a_i$. Then $\sum\limits_{n=1}^\infty\frac{a_n}{s_n} =\infty$ as well.

I can prove it by comparing it to $\int_1^\infty \frac{1}{x} \, dx$ if the sequence $a_n$ is bounded by some $M$, but that's as far as I've been able to get.

Answer

Since $s_n\to+\infty$ when $n\to\infty$, for each $n\geqslant1$ there exists some finite $m\gt n$ such that $s_m\geqslant2s_n$. In particular, $\sum\limits_{k=n+1}^m\frac{a_k}{s_k}\geqslant\sum\limits_{k=n+1}^m\frac{a_k}{s_m}=1-\frac{s_n}{s_m}\geqslant1/2$. Thus, the rests of the series $\sum\limits_k\frac{a_k}{s_k}$ do not converge to zero, QED.