Square-Summable Indexed Sets Closed Under Addition

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

Suppose that:


 * $\displaystyle \sum \left\{{ a_i^2: i \in I }\right\} < \infty$
 * $\displaystyle \sum \left\{{ b_i^2: i \in I }\right\} < \infty$

where the $\sum$ denote generalized sums.

Then $\displaystyle \sum \left\{{ \left({a_i + b_i}\right)^2: i \in I }\right\} < \infty$.