# Definition:Cauchy Sequence/Normed Division Ring

## Definition

Let $\struct {R, \norm {\,\cdot\,} }$ be a normed division ring

Let $\sequence {x_n}$ be a sequence in $R$.

Then $\sequence {x_n}$ is a Cauchy sequence in the norm $\norm {\, \cdot \,}$ if and only if:

$\sequence {x_n}$ is a cauchy sequence in the metric induced by the norm $\norm {\, \cdot \,}$

That is:

$\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$

## Source of Name

This entry was named for Augustin Louis Cauchy.