Definition:Cauchy Sequence/Normed Vector Space

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Let $\left({V, \left\Vert{\,\cdot\,}\right\Vert}\right)$ be a normed vector space over a normed division ring $\left({R, \left\Vert{\,\cdot\,}\right\Vert_R}\right)$.

Let $\left \langle {x_n} \right \rangle$ be a sequence in $V$.

Then $\left \langle {x_n} \right \rangle$ is a Cauchy sequence if and only if:

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

Also see

Source of Name

This entry was named for Augustin Louis Cauchy.