Cauchy's Convergence Criterion/Complex Numbers/Proof 1

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 $z_n$

Necessary Condition
Let $\sequence {z_n}$ be a Cauchy sequence.

We aim to prove that $\sequence {z_n}$ is convergent.

We find:


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


 * $\implies \sequence {x_n}$ and $\sequence {y_n}$ are Cauchy sequences by Lemma


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


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

Sufficient Condition
Let $\sequence {z_n}$ be convergent.

We aim to prove that $\sequence {z_n}$ is a Cauchy sequence.

We find:


 * $\sequence {z_n}$ is convergent


 * $\implies \sequence {x_n}$ and $\sequence {y_n} $ are convergent by definition of convergent complex sequence


 * $\implies \sequence {x_n}$ and $\sequence {y_n}$ are Cauchy sequences by Real Sequence is Cauchy iff Convergent


 * $\implies \sequence {z_n}$ is a Cauchy sequence by Lemma