Definition:Cauchy Sequence/Normed Division Ring
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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:
- $\forall \epsilon \in \R_{>0}: \exists N \in \N: \forall m, n \in \N: m, n \ge N: \norm {x_n - x_m} < \epsilon$
Also see
Source of Name
This entry was named for Augustin Louis Cauchy.
Sources
- 1997: Fernando Q. Gouvea: p-adic Numbers: An Introduction ... (previous) ... (next): $\S 3.2$: Completions
- 2007: Svetlana Katok: p-adic Analysis Compared with Real ... (previous) ... (next): $\S 1.2$: Normed Fields: Definition $1.7$