A Convergent Sequence Satisfies the Cauchy Criterion
Initializing live version
Requires a Wolfram Notebook System
Interact on desktop, mobile and cloud with the free Wolfram Player or other Wolfram Language products.
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
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.