I'm studying for qualifying exams and ran into this problem.
Show that if $\{a_n\}$ is a nonincreasing sequence of positive real numbers such that $\sum_n a_n$ converges, then $\lim\limits_{n \rightarrow \infty} n a_n = 0$.
Using the definition of the limit, this is equivalent to showing
\begin{equation} \forall \varepsilon > 0, \; \exists n_0\;\text{such that}\; |n a_n| < \varepsilon,\; \forall n > n_0 \end{equation}
or
\begin{equation} \forall \varepsilon > 0, \; \exists n_0\;\text{such that}\; a_n < \frac{\varepsilon}{n},\; \forall n > n_0 \end{equation}
Basically, the terms must be bounded by the harmonic series. Thanks, I'm really stuck on this seemingly simple problem!