Convergence of Generalized Sum of Complex Numbers

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

That is, let $z_j \in \C$ for all $j \in I$.

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

Then the following are equivalent:


 * $(1): \quad \ds \sum_{j \mathop \in I} z_j$ converges to $z \in \C$
 * $(2): \quad \ds \sum_{j \mathop \in I} \map \Re {z_j}, \sum_{j \mathop \in I} \map \Im {z_j}$ 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 $G \subseteq F$:
 * $\ds \cmod {z - \sum_{j \mathop \in F} z_j} < \epsilon$

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 are smaller than $\epsilon^2$.

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


 * $\ds \size {\map \Re z - \sum_{j \mathop \in F} \map \Re {z_j} } < \epsilon$
 * $\ds \size {\map \Im z - \sum_{j \mathop \in F} \map \Im {z_j} } < \epsilon$

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


 * $\ds \sum_{j \mathop \in I} \map \Re {z_j}, \sum_{j \mathop \in I} \map \Im {z_j}$ converge to $\map \Re z, \map \Im z \in \R$, respectively.

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

Also see

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