Absolutely Convergent Series is Convergent

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

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

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

From the Comparison Test we have that $$\sum_{n=1}^\infty a_n$$ converges if $$\forall n \in \N^*: \left|{a_n}\right| \le b_n$$ where $$\sum_{n=1}^\infty b_n$$ is convergent.

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