I am posting for a friend: he has a strong feeling that the following should be true but can't prove it:
"Let $(a_n)$ be a real sequence. If $a_n \searrow 0$ and $\sum a_n$ converges, then $n a_n \to 0$."
He has checked all the theorems and exercises in Rudin and Apostol but hasn't found an answer to this question. He thinks the question reduces to showing the existence of a series with positive integers, strictly increasing such that
$$ \frac{1-\frac{n_k}{n_{k+1}}}{1-\frac{n_{k-1}}{n_k}} < \frac{1}{M} $$
for some fixed $M > 1$.
