Absolutely Convergent Series in Normed Vector Space is Convergent iff Space is Banach

Theorem
Let $\struct {X, \norm {\, \cdot \,}}$ be a normed vector space.

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

Then $\displaystyle \sum_{n \mathop = 1}^\infty a_n$ is convergent $X$ is a Banach space.