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:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \Re \paren {z_n}$
 * and:
 * $\displaystyle \sum_{n \mathop = 1}^\infty \Im \paren {z_n}$
 * converge to $\Re \paren z$ and $\Im \paren z$ respectively.
 * converge to $\Re \paren z$ and $\Im \paren z$ respectively.