# Definition:Cauchy Sequence/Complex Numbers

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## Definition

Let $\left \langle {z_n} \right \rangle$ be a sequence in $\C$.

Then $\left \langle {z_n} \right \rangle$ is a **Cauchy sequence** if and only if:

- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \left|{z_n - z_m}\right| < \epsilon$

where $\left|{z_n - z_m}\right|$ denotes the complex modulus of $z_n - z_m$.

Considering the complex plane as a metric space, it is clear that this is a special case of the definition for a metric space.

## Also see

- Definition:Complete Metric Space: a metric space in which the converse holds, i.e. a Cauchy sequence is convergent.

Thus in $\C$ a **Cauchy sequence** and a convergent sequence are equivalent concepts.

## Source of Name

This entry was named for Augustin Louis Cauchy.