\begin{definition}[Definition:Cauchy Sequence/Normed Vector Space]
Let $\struct {V, \norm {\,\cdot\,} }$ be a normed vector space.
Let $\sequence {x_n}$ be a sequence in $V$.
Then $\sequence {x_n}$ is a '''Cauchy sequence''' {{iff}}:
: $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$
\end{definition}