Generalized Sum is Monotone

Theorem
Let $I$ be an indexing set.

Let $\family {a_i}_{i \mathop \in I}$ be an $I$-indexed family of positive real numbers.

That is, let $a_i \in \R_{\ge 0}$ for all $i \in I$.

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


 * $\ds \sum_{i \mathop \in F} a_i \le \sum_{i \mathop \in I} a_i$

provided the generalized sum on the right converges.