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A Convergent Sequence Satisfies the Cauchy Criterion


	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
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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.

Cauchy Criterion (Wolfram MathWorld)

Limit of a Sequence (The Wolfram Demonstrations Project)

"A Convergent Sequence Satisfies the Cauchy Criterion" from The Wolfram Demonstrations Project
http://demonstrations.wolfram.com/AConvergentSequenceSatisfiesTheCauchyCriterion/

Contributed by: Izidor Hafner

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