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\\leq a_{n}+|a_{n}|\\leq 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.