Difference between revisions of "Cauchy sequence"
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==Definition== | ==Definition== | ||
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> if: | 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> if: | ||
| − | * {{M|\ | + | * {{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]}} |
| − | * For any arbitrary distance apart, there exists a point such that any two points in the sequence after that point are within that arbitrary distance apart. | + | In words it is simply: |
| − | + | * For any arbitrary distance apart, there exists a point such that any two points in the sequence after that point are within that arbitrary distance apart. | |
==References== | ==References== | ||
<references/> | <references/> | ||
{{Definition|Functional Analysis|Metric Space|Real Analysis}} | {{Definition|Functional Analysis|Metric Space|Real Analysis}} | ||
Revision as of 16:29, 23 August 2015
Definition
Given a metric space (X,d) and a sequence (xn)∞n=1⊆X is said to be a Cauchy sequence[1] if:
- ∀ϵ>0∃N∈N∀n,m∈N[n≥m>N⟹d(xm,xn)<ϵ]
In words it is simply:
- For any arbitrary distance apart, there exists a point such that any two points in the sequence after that point are within that arbitrary distance apart.
References
- Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici
