Convergence of Generalized Sum of Complex Numbers/Corollary

Corollary to Convergence of Generalized Sum of Complex Numbers
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 $i \in I$.

Suppose that $\ds \sum \set {z_i: i \in I}$ converges to $z \in \C$.

Then $\ds \sum \set {\overline {z_i}: i \in I}$ converges to $\overline z$, where $\overline z$ denotes the complex conjugate of $z$. Here, the $\sum$ denote generalized sums.

Proof
Using Convergence of Generalized Sum of Complex Numbers, one has:


 * $\ds \sum \set {\map \Re {z_i} : i \in I} = \map \Re z$
 * $\ds \sum \set {\map \Im {z_i} : i \in I} = \map \Im z$

Now, observe that, from the definition of complex conjugate: