Definition:Cauchy Sequence/Complex Numbers

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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

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

Source of Name

This entry was named for Augustin Louis Cauchy.