Difference between revisions of "Cauchy criterion for convergence"

Revision as of 13:41, 9 July 2015

Iffy page

The Cauchy criterion for convergence requires the space be complete. I encountered it with sequences on $\mathbb{R}$ - there are of course other spaces! As such this page is being refactored.

See Cauchy sequence for a definition

Page resumes

If a sequence converges, it is the same as saying it matches the Cauchy criterion for convergence.

Cauchy Sequence

A sequence $$(a_n)^\infty_{n=1}$$ is Cauchy if:

$$\forall\epsilon>0\exists N\in\mathbb{N}:n> m> N\implies d(a_m,a_n)<\epsilon$$

Theorem

A sequence converges if and only if it is Cauchy

TODO: proof, easy stuff