Absolutely Convergent Series is Convergent/Real Numbers

Theorem

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

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

Proof

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

From the Comparison Test we have that $\ds \sum_{n \mathop = 1}^\infty a_n$ converges if and only if:

$\forall n \in \N_{>0}: \size {a_n} \le b_n$

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

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

$\blacksquare$