Cauchy sequence

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Cauchy sequence (plural Cauchy sequences)

  1. (analysis) A sequence in a normed vector space such that the difference between any two entries can be made arbitrarily small by stipulating that the two entries be sufficiently far out in the sequence.
  2. (analysis) A sequence in a metric space with metric d such that for every there exists a natural number N so that for every  : .
    • 2000, George Bachman, Lawrence Narici, Functional Analysis, page 52,
      In the case of the real line, every Cauchy sequence converges; that is, being a Cauchy sequence is sufficient to guarantee the existence of a limit. In the general case, however, this is not so. If a metric space does have the property that every Cauchy sequence converges, the space is called a complete metric space.