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.