Cauchy's Convergence Criterion/Complex Numbers

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

Then $\left \langle {z_n} \right \rangle$ is a Cauchy sequence it is convergent.

Let $\left \langle {x_n} \right \rangle$ be a real sequence where:
 * $x_n = \Re \left( {z_n} \right)$ for every $n$
 * $\Re \left({z_n}\right)$ is the real part of $z_n$

Let $\left \langle {y_n} \right \rangle$ be a real sequence where:
 * $y_n = \Im \left( {z_n} \right)$ for every $n$
 * $\Im \left( { z_n} \right)$ is the imaginary part of $z_n$

Also see

 * Real Sequence is Cauchy iff Convergent