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

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

is convergent.

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