Generalized Sum is Monotone

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Theorem

Let $\left({a_i}\right)_{i \in I}$ 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 $I$:

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

provided the generalized sum on the right converges.


Proof