Cauchy sequence/Definition

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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][2] 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][Note 1][Note 2]

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.


  1. Note that in Krzysztof Maurin's notation this is written as [math]\bigwedge_{\epsilon>0}\bigvee_{N\in\mathbb{N} }\bigwedge_{m,n>\mathbb{N} }d(x_n,x_m)<\epsilon[/math] - which is rather elegant
  2. It doesn't matter if we use [ilmath]n\ge m>N[/ilmath] or [ilmath]n,m\ge N[/ilmath] because if [ilmath]n=m[/ilmath] then [ilmath]d(x_n,x_m)=0[/ilmath], it doesn't matter which way we consider them (as [ilmath]n>m[/ilmath] or [ilmath]m>n[/ilmath]) for [ilmath]d(x,y)=d(y,x)[/ilmath] - I use the ordering to give the impression that as [ilmath]n[/ilmath] goes out ahead it never ventures far (as in [ilmath]\epsilon[/ilmath]-distance}}) from [ilmath]x_m[/ilmath]. This has served me well


  1. Functional Analysis - George Bachman and Lawrence Narici
  2. Analysis - Part 1: Elements - Krzysztof Maurin