Difference between revisions of "Cauchy sequence"
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Revision as of 14:03, 9 July 2015
Definition
Given a metric space [ilmath](X,d)[/ilmath] and a sequence [ilmath](x_n)_{n=1}^\infty\subseteq X[/ilmath] is said to be a Cauchy sequence[1] if:
- [ilmath]\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][/ilmath]
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
- ↑ Functional Analysis - George Bachman and Lawrence Narici
