If $(a_n)\subset[0,\infty)$ is non-increasing and $\sum_{n=1}^\infty a_n

I'm studying for qualifying exams and ran into this problem.

Show that if $\{a_n\}$ is a nonincreasing sequence of positive realnumbers 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!