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(mathematical analysis) Any sequence x_n in a metric space with metric d such that for every 𝜀>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<𝜀.

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Cauchy sequence

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Cauchy sequence

noun

(mathematical analysis) Any sequence x_n in a metric space with metric d such that for every 𝜀>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<𝜀.

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(mathematical analysis) Any sequence x_n in a metric space with metric d such that for every 𝜀>0 there exists a natural number N such that for all k,m>N, d(x_k,x_m)<𝜀.

However, it is possible to derive topological results from statements about Cauchy sequences; for example, a subset A of the space of real numbers is closed if and only if each Cauchy sequence in A converges to some point of A.

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Cauchy sequence

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