Convergence of Series of Complex Numbers by Real and Imaginary Part

Theorem
Let $\sequence {z_n}$ be a sequence of complex numbers.

Then:
 * the series $\ds \sum_{n \mathop = 1}^\infty z_n$ converges to $Z \in \C$


 * the series:
 * $\ds \sum_{n \mathop = 1}^\infty \map \Re {z_n}$
 * and:
 * $\ds \sum_{n \mathop = 1}^\infty \map \Im {z_n}$
 * converge to $\map \Re Z$ and $\map \Im Z$ respectively.
 * converge to $\map \Re Z$ and $\map \Im Z$ respectively.

Proof
Let:
 * the $n$th partial sum of $\sequence {z_n}$ be denoted $Z_n$
 * the $n$th partial sum of $\sequence {\map \Re {z_n} }$ be denoted $U_n$
 * the $n$th partial sum of $\sequence {\map \Im {z_n} }$ be denoted $V_n$

Then:
 * $Z_n = U_n + i V_n$

Let:
 * $\lim_{n \mathop \to \infty} U_n = U$
 * $\lim_{n \mathop \to \infty} V_n = V$

By definition of convergent complex sequence:


 * $\lim_{n \mathop \to \infty} Z_n = \lim_{n \mathop \to \infty} U_n + i \lim_{n \mathop \to \infty} V_n$

and so:
 * $\lim_{n \mathop \to \infty} Z_n = U + i V$

and the result follows.