Generalized Sum Preserves Inequality

Theorem
Let $\left({a_i}\right)_{i \in I}, \left({b_i}\right)_{i \in I}$ be $I$-indexed families of positive real numbers.

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

Suppose that for all $i \in I$, $a_i \le b_i$.

Furthermore, suppose that $\displaystyle \sum \left\{{ b_i: i \in I }\right\}$ converges.

Then $\displaystyle \sum \left\{{ a_i: i \in I }\right\} \le \sum \left\{{ b_i: i \in I }\right\}$.

In particular, $\displaystyle \sum \left\{{ a_i: i \in I }\right\}$ converges.