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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