Generalized Sum is Linear

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 $a_i, b_i \in \R$ for all $i \in I$.

Suppose that $\displaystyle \sum \left\{{ a_i: i \in I }\right\}$ and $\displaystyle \sum \left\{{ b_i: i \in I }\right\}$ converge.

Then:


 * $(1): \displaystyle \sum \left\{{ a_i + b_i: i \in I }\right\}$ converges;
 * $(2): \displaystyle \sum \left\{{ a_i + b_i: i \in I }\right\} = \sum \left\{{ a_i: i \in I }\right\} + \sum \left\{{ b_i: i \in I }\right\}$
 * $(3): \forall \lambda \in \R: \displaystyle \sum \left\{{ \lambda a_i: i \in I }\right\}$ converges;
 * $(4): \forall \lambda \in \R: \displaystyle \sum \left\{{ \lambda a_i: i \in I }\right\} = \lambda \sum \left\{{ a_i: i \in I }\right\}$