Difference between revisions of "Equivalence of Cauchy sequences/Proof"

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Revision as of 20:21, 29 February 2016

Statement

Given two Cauchy sequences, [ilmath](a_n)_{n=1}^\infty[/ilmath] and [ilmath](b_n)_{n=1}^\infty[/ilmath] in a metric space [ilmath](X,d)[/ilmath] we define them as equivalent if[1]:

  • [math]\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(a_n,b_n)<\epsilon][/math]

And that this indeed actually defines an equivalence relation

Proof

Reflexivity

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Transitivity

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Symmetry

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References

  1. ↑ Analysis - Part 1: Elements - Krzysztof Maurin
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