Difference between revisions of "Cauchy sequence/Short definition"
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| − | Given a [[Metric space|metric space]] {{M|(X,d)}} and a [[Sequence|sequence]] {{M|1=(x_n)_{n=1}^\infty\subseteq X}} is said to be a ''Cauchy sequence''<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref>{{rAPIKM}} if: | + | <onlyinclude>Given a [[Metric space|metric space]] {{M|(X,d)}} and a [[Sequence|sequence]] {{M|1=(x_n)_{n=1}^\infty\subseteq X}} is said to be a ''Cauchy sequence''<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref>{{rAPIKM}} if: |
| − | * {{M|\forall\epsilon > 0\exists N\in\mathbb{N}\forall n,m\in\mathbb{N}[n\ge m> N\implies d(x_m,x_n)<\epsilon]}} | + | * {{M|\forall\epsilon > 0\exists N\in\mathbb{N}\forall n,m\in\mathbb{N}[n\ge m> N\implies d(x_m,x_n)<\epsilon]}}</onlyinclude> |
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{{Definition|Real Analysis|Functional Analysis|Topology|Metric Space}} | {{Definition|Real Analysis|Functional Analysis|Topology|Metric Space}} | ||
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Latest revision as of 13:55, 5 December 2015
Given a metric space (X,d) and a sequence (xn)∞n=1⊆X is said to be a Cauchy sequence[1][2] if:
- ∀ϵ>0∃N∈N∀n,m∈N[n≥m>N⟹d(xm,xn)<ϵ]
Notes
References
- Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici
- Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin
