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:

$\ds \sum \set {a_i^2: i \in I} < \infty$

$\ds \sum \set {b_i^2: i \in I} < \infty$

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

Then:

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

Proof

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