Absolutely Convergent Series is Convergent

Theorem
Let $V$ be a Banach space with norm $\left\Vert{\cdot}\right\Vert$.

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

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

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

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

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