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.
    \lim_{n,m\to \infty} \|x_n-x_m\|=0
  2. (analysis) A sequence  x_n in a metric space with metric d such that for every  \epsilon > 0 there exists a natural number N so that for every  k, m \ge N  :  d(x_k, x_m) < \epsilon .
    • 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.