Definition:Cauchy Sequence/Complex Numbers

Definition
Let $\sequence {z_n}$ be a sequence in $\C$.

Then $\sequence {z_n}$ is a Cauchy sequence :
 * $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \size {z_n - z_m} < \epsilon$

where $\size {z_n - z_m}$ 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

 * Convergent Sequence in Metric Space is Cauchy Sequence


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


 * Complex Plane is Complete Metric Space

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