Absolutely Convergent Generalized Sum Converges

Theorem
Let $V$ be a Banach space, and let $\left\Vert{\cdot}\right\Vert$ denote its norm.

Let $\left({v_i}\right)_{i\in I}$ be an indexed subset of $V$ such that $\displaystyle \sum \left\{{v_i: i \in I}\right\}$ converges absolutely.

Then $\displaystyle \sum \left\{{v_i: i \in I}\right\}$ converges.