Cauchy sequence/Short 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]

Notes

References

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