Convergence of Generalized Sum of Complex Numbers

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

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

Denote by $\operatorname{Re} z_i$, resp. $\operatorname{Im} z_i$ the families of real, resp. imaginary parts of the family $z_i$.

Then the following are equivalent:


 * $(1): \qquad \displaystyle \sum \left\{{z_i : i \in I}\right\}$ converges to $z \in \C$
 * $(2): \qquad \displaystyle \sum \left\{{\operatorname{Re} z_i : i \in I}\right\}, \sum \left\{{\operatorname{Im} z_i : i \in I}\right\}$ converge to $\operatorname{Re} z, \operatorname{Im} z \in \R$, respectively

$(2)$ implies $(1)$
By Generalized Sum is Linear, the stated convergences lead to:

$(1)$ implies $(2)$
Statement $(1)$, according to the definition of convergence, amounts to the following:

For every $\epsilon > 0$, there exists a finite $G \subseteq I$ such that:


 * For every finite $F \subseteq I$ with $G \subseteq F$: $\left\vert{z - \displaystyle \sum_{i \in F} z_i}\right\vert < \epsilon$

Now suppose that for $\epsilon > 0$, $G$ and $F$ are as above. Then observe that:

Hence, by Even Powers are Positive, both of the terms on the right hand side are smaller than $\epsilon^2$.

It follows that, taking square roots, $G$ satisfies, for any finite $F \supseteq G$:


 * $\displaystyle \left\vert{\operatorname{Re} z - \displaystyle \sum_{i \in F} \operatorname{Re} z_i}\right\vert < \epsilon$
 * $\displaystyle \left\vert{\operatorname{Im} z - \displaystyle \sum_{i \in F} \operatorname{Im} z_i}\right\vert < \epsilon$

As $\epsilon > 0$ was arbitrary, using the definition of convergence, this implies precisely that:


 * $\displaystyle \sum \left\{{\operatorname{Re} z_i : i \in I}\right\}, \sum \left\{{\operatorname{Im} z_i : i \in I}\right\}$ converge to $\operatorname{Re} z, \operatorname{Im} z \in \R$, respectively.

Hence, $(1)$ is shown to imply $(2)$.