Absolutely Convergent Series is Convergent/Real Numbers

Theorem
Let $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ be an absolutely convergent series in $\R$.

Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent.

Proof
By the definition of absolute convergence, we have that $\displaystyle \sum_{n \mathop = 1}^\infty \size {a_n}$ is convergent.

From the Comparison Test we have that $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ converges :
 * $\forall n \in \N_{>0}: \size {a_n} \le b_n$

where $\displaystyle \sum_{n \mathop = 1}^\infty b_n$ is convergent.

So substituting $\size {a_n}$ for $b_n$ in the above, the result follows.