Let $\sum_{n=0}^{\infty} a_n$ be a convergent series such that $a_n \downarrow 0$. Prove that, $ \lim_{n\to \infty} na_n = 0 $
If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum a_n<\infty$, then $\lim{n a_n} = 0$