If a series is divergent, then series of magnitude of its terms is also divergent.

Question

Is it the case that if $\sum_{n=1}^{\infty}{a_n}$ diverges then $\sum_{n=1}^{\infty}{|a_n|}$ also diverges?

It seems obvious that if $\sum_{n=1}^{\infty}{a_n}$ is unbounded (i.e., $\lim_{n\rightarrow\infty}{S_n}\rightarrow \infty \text{ or }-\infty$) then $\sum_{n=1}^{\infty}{|a_n|}$ is also unbounded and the claim follows for this case.

But for other cases, I find it a bit tricky.

Note: $S_n$ denotes $n$th partial sum.