Generalized Sum is Linear

Theorem
Let $\left({z_i}\right)_{i \in I}, \left({w_i}\right)_{i \in I}$ be $I$-indexed families of complex numbers.

That is, let $z_i, w_i \in \C$ for all $i \in I$.

Suppose that $\displaystyle \sum \left\{{ z_i: i \in I }\right\}, \sum \left\{{ w_i: i \in I }\right\}$ converge to $z, w \in \C$, respectively.

Then:


 * $(1): \displaystyle \sum \left\{{ z_i + w_i: i \in I }\right\}$ converges to $z+w$;
 * $(2): \forall \lambda \in \C: \displaystyle \sum \left\{{ \lambda z_i: i \in I }\right\}$ converges to $\lambda z$;