# A Convergent Sequence Satisfies the Cauchy Criterion

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This Demonstration shows that a convergent sequence satisfies the Cauchy criterion. Suppose . For each , there exists , such that for all , . If , then .

Contributed by: Izidor Hafner (March 2011)

Open content licensed under CC BY-NC-SA

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A sequence satisfies the Cauchy criterion if for any , there is a natural number such that for any , |. A metric space is said to be complete if every Cauchy sequence converges. As this Demonstration illustrates, the real numbers are a complete metric space.

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