Square-Summable Indexed Sets Closed Under Addition

Theorem
Let $\family {a_i}_{i \mathop \in I}, \family {b_i}_{i \mathop \in I}$ be $I$-indexed families of real numbers.

Let:


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

where $\displaystyle \sum$ denotes the generalized sums.

Then:
 * $\displaystyle \sum \set {\paren {a_i + b_i}^2: i \in I} < \infty$