Cauchy's Convergence Criterion/Complex Numbers/Proof 2

Lemma
Let $\sequence {x_n}$ be a real sequence where:
 * $x_n = \Re \paren {z_n}$ for every $n$
 * $\Re \paren {z_n}$ is the real part of $z_n$

Let $\sequence {y_n}$ be a real sequence where
 * $y_n = \Im \paren {z_n}$ for every $n$
 * $\Im \paren {z_n}$ is the imaginary part of $z_n$

We find:


 * $\sequence {z_n}$ is a Cauchy sequence


 * $\iff \sequence {x_n}$ and $\sequence {y_n}$ are Cauchy sequences (by Lemma)


 * $\iff \sequence {x_n}$ and $\sequence {y_n}$ are convergent (by Real Sequence is Cauchy iff Convergent)


 * $\iff \sequence {z_n}$ is convergent by definition of convergent complex sequence