Convergence of Generalized Sum of Complex Numbers/Corollary

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

Suppose that $\displaystyle \sum \left\{{ z_i: i \in I }\right\}$ converges to $z \in \C$.

Then $\displaystyle \sum \left\{{ \overline{z_i}: i \in I }\right\}$ 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:


 * $\displaystyle \sum \left\{{\operatorname{Re} z_i : i \in I}\right\} = \operatorname{Re} z$
 * $\displaystyle \sum \left\{{\operatorname{Im} z_i : i \in I}\right\} = \operatorname{Im} z$

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