Cauchy's Convergence Criterion/Complex Numbers

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Theorem

Let $\sequence {z_n}$ be a complex sequence.

Then $\sequence {z_n}$ is a Cauchy sequence if and only if it is convergent.

Proof 1

Lemma

Let $\sequence {z_n}$ be a complex sequence.

Let $\NN$ be the domain of $\sequence {z_n}$.

Let $x_n = \map \Re {z_n}$ for every $n \in \NN$.

Let $y_n = \map \Im {z_n}$ for every $n \in \NN$.

Then $\sequence {z_n}$ is a (complex) Cauchy sequence if and only if $\sequence {x_n}$ and $\sequence {y_n}$ are (real) Cauchy sequences.

$\Box$

Let $\sequence {x_n}$ be a real sequence where:

$x_n = \map \Re {z_n}$ for every $n$

$\map \Re {z_n}$ is the real part of $z_n$

Let $\sequence {y_n}$ be a real sequence where:

$y_n = \map \Im {z_n}$ for every $n$

$\map \Im {z_n}$ 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

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

$\leadsto \sequence {x_n}$ and $\sequence {y_n}$ are convergent by Cauchy's Convergence Criterion on Real Numbers

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

$\Box$

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

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

$\leadsto \sequence {x_n}$ and $\sequence {y_n}$ are Cauchy sequences by Cauchy's Convergence Criterion on Real Numbers

$\leadsto \sequence {z_n}$ is a Cauchy sequence by Lemma

$\blacksquare$

Proof 2

Lemma

Let $\sequence {z_n}$ be a complex sequence.

Let $\NN$ be the domain of $\sequence {z_n}$.

Let $x_n = \map \Re {z_n}$ for every $n \in \NN$.

Let $y_n = \map \Im {z_n}$ for every $n \in \NN$.

Then $\sequence {z_n}$ is a (complex) Cauchy sequence if and only if $\sequence {x_n}$ and $\sequence {y_n}$ are (real) Cauchy sequences.

$\Box$

Let $\sequence {x_n}$ be a real sequence where:

$x_n = \map \Re {z_n}$ for every $n$

$\map \Re {z_n}$ is the real part of $z_n$

Let $\sequence {y_n}$ be a real sequence where

$y_n = \map \Im {z_n}$ for every $n$

$\map \Im {z_n}$ is the imaginary part of $z_n$

We find:

$\sequence {z_n}$ is a Cauchy sequence

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

$\leadstoandfrom \sequence {x_n}$ and $\sequence {y_n}$ are convergent by Cauchy's Convergence Criterion on Real Numbers

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

$\blacksquare$

Also see

• Cauchy's Convergence Criterion on Real Numbers