proof of absolute convergence theorem

Suppose that $\sum a_n$ is absolutely convergent, i.e., that $\sum |a_n|$ is convergent.First of all, notice that

$$0\le a_n+|a_n|\le 2|a_n|,$$

and since the series $\sum (a_n+|a_n|)$ has non-negative terms it can be compared with $\sum 2|a_n|=2\sum |a_n|$ and hence converges.

On the other hand

$$\sum _{n=1}^N{a}_n=\sum _{n=1}^N(a_n+|a_n|)-\sum _{n=1}^N|a_n|.$$

Since both the partial sums on the right hand side are convergent, the partialsum on the left hand side is also convergent. So, the series $\sum a_n$ is convergent.