Cauchy's Convergence Criterion/Complex Numbers/Proof 2

Proof
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 $z_n$

Lemma 2
We have:


 * $\left \langle {z_n} \right \rangle$ is a Cauchy sequence


 * $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ are Cauchy sequences (by Lemma 1)
 * $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ are Cauchy sequences (by Lemma 1)


 * $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ are convergent (by Real Sequence is Cauchy iff Convergent)
 * $\left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ are convergent (by Real Sequence is Cauchy iff Convergent)


 * $\left \langle {z_n} \right \rangle$ is convergent (by Lemma 2)
 * $\left \langle {z_n} \right \rangle$ is convergent (by Lemma 2)