Let $\sum a_n$ be a convergent series of positive numbers and set $r_n=\sum_{k=n}^\infty a_n$. Prove that the series $\sum \frac{a_n}{r_n}$ diverges.
Solution I have tried
By definition, $r_n=a_n+r_{n+1}$. Hence,
$\frac{r_{n+1}}{r_n}=1-\frac{a_n}{r_n}$,
$\frac{r_{n+2}}{r_n}=\big(1-\frac{a_n}{r_n}\big)\big(1-\frac{a_{n+1}}{r_{n+1}}\big)>1-\frac{a_n}{r_n}-\frac{a_{n+1}}{r_{n+1}}$.
By induction we have $\frac{r_{n+p}}{r_n}>1-\frac{a_n}{r_n}-\frac{a_{n+1}}{r_{n+1}}- \dots \frac{a_{n+p}}{r_{n+p}}$.
As the series $\sum a_n$ is conv, we have that $r_{n+p} \to 0$ as $p \to \infty$.
Now what? I am stuck here.