Difference between revisions of "Cauchy criterion for convergence"
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| + | :: '''The purpose of this page is to show that on a complete space a [[Limit (sequence)|sequence converges]] {{M|\iff}} it is a [[Cauchy sequence]]''' | ||
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The Cauchy criterion for convergence requires the space be complete. I encountered it with sequences on {{M|\mathbb{R} }} - there are of course other spaces! As such this page is being refactored. | The Cauchy criterion for convergence requires the space be complete. I encountered it with sequences on {{M|\mathbb{R} }} - there are of course other spaces! As such this page is being refactored. | ||
Latest revision as of 15:26, 24 November 2015
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 [math](a_n)^\infty_{n=1}[/math] is Cauchy if:
[math]\forall\epsilon>0\exists N\in\mathbb{N}:n> m> N\implies d(a_m,a_n)<\epsilon[/math]
Theorem
A sequence converges if and only if it is Cauchy
TODO: proof, easy stuff
