Suppose that $\sum a_n$ is absolutely convergent, i.e., that $\sum \vert a_n \vert$ is convergent. First of all, notice that $$ 0\le a_n + \vert a_n \vert \le 2 \vert a_n\vert, $$ and since the series $\sum (a_n+\vert a_n\vert)$ has non-negative terms it can be compared with $\sum 2\vert a_n\vert=2\sum \vert a_n\vert$ and hence converges.

On the other hand $$ \sum_{n=1}^N a_n = \sum_{n=1}^N (a_n+\vert a_n\vert) - \sum_{n=1}^N \vert a_n\vert. $$ Since both the partial sums on the right hand side are convergent, the partial sum on the left hand side is also convergent. So, the series $\sum a_n$ is convergent.