Convergence of Generalized Sum of Complex Numbers

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

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

Let $\map \Re {z_i}$ and $\map \Im {z_i}$ denote the families of real and imaginary parts of the family $z_i$.

Then the following are equivalent:


 * $(1): \quad \ds \sum \set {z_i : i \in I}$ converges to $z \in \C$
 * $(2): \quad \ds \sum \set {\map \Re {z_i} : i \in I}, \sum \set{\map \Im {z_i} : i \in I}$ converge to $\map \Re z, \map \Im z \in \R$, 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 $G \subseteq F$:
 * $\displaystyle \left\vert{z - \sum_{i \mathop \in F} z_i}\right\vert < \epsilon$

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

Hence, by Square of Real Number is Non-Negative, 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 $F \supseteq G$:


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

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 $(2)$.

Also see

 * Generalized Sum is Linear, of which this is a partial converse.