Convergence of Series of Complex Numbers by Real and Imaginary Part

Theorem
Let $\left\langle{z_n}\right\rangle$ be a sequence of complex numbers.

Then the series $\displaystyle \sum_{n \mathop = 1}^\infty z_n$ converges to $z \in \C$ the series:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \Re \left({z_n}\right)$

and:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \Im \left({z_n}\right)$

converge to $\Re \left({z}\right)$ and $\Im \left({z}\right)$ respectively.