\begin{definition}[Definition:Cauchy Sequence/Normed Division Ring]
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 $\norm {\, \cdot \,}$''' {{iff}}:
:$\sequence {x_n}$ is a cauchy sequence in the metric induced by the norm $\norm {\, \cdot \,}$
That is:
: $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$
\end{definition}