Convergence of Generalized Sum of Complex Numbers/Corollary

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

Suppose that $\ds \sum_{j \mathop \in I} z_j$ converges to $z \in \C$.

Then $\ds \sum_{j \mathop \in I} \overline {z_j}$ 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:


 * $\ds \sum_{j \mathop \in I} \map \Re {z_j} = \map \Re z$
 * $\ds \sum_{j \mathop \in I} \map \Im {z_j} = \map \Im z$

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