Cauchy criterion for convergence

Iffy page

The purpose of this page is to show that on a complete space a sequence converges [ilmath]\iff[/ilmath] it is a Cauchy sequence

The Cauchy criterion for convergence requires the space be complete. I encountered it with sequences on [ilmath]\mathbb{R} [/ilmath] - 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