Generalized Sum is Monotone

Theorem
Let $\left({a_i}\right)_{i \in I}$ be an $I$-indexed family of real numbers.

Then, for every finite subset $F$ of $I$:


 * $\displaystyle \sum_{i \in F} \left\vert{a_i}\right\vert \le \sum \left\{{\left\vert{a_i}\right\vert : i \in I}\right\}$