Cauchy's Convergence Criterion/Complex Numbers/Proof 1

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 $z_n$

Necessary Condition
Let $\left \langle {z_n} \right \rangle$ be a Cauchy sequence.

We aim to prove that $\left \langle {z_n} \right \rangle$ is convergent.

We find:


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


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


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


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

Sufficient Condition
Let $\left \langle {z_n} \right \rangle$ be convergent.

We aim to prove that $\left \langle {z_n} \right \rangle$ is a Cauchy sequence.

We find:


 * $\left \langle {z_n} \right \rangle$ is convergent


 * $\implies \left \langle {x_n} \right \rangle$ and $\left \langle {y_n} \right \rangle$ are convergent by Lemma 2


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


 * $\implies \left \langle {z_n} \right \rangle$ is a Cauchy sequence by Lemma 1