Difference between revisions of "Cauchy criterion for convergence"
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| + | ==Iffy page== | ||
| + | :: '''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. | ||
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| + | '''See [[Cauchy sequence]] for a definition''' | ||
| + | ==Page resumes== | ||
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If a [[Sequence|sequence]] converges, it is the same as saying it matches the Cauchy criterion for convergence. | If a [[Sequence|sequence]] converges, it is the same as saying it matches the Cauchy criterion for convergence. | ||
Latest revision as of 15:26, 24 November 2015
Contents
[hide]Iffy page
- The purpose of this page is to show that on a complete space a sequence converges ⟺ it is a Cauchy sequence
The Cauchy criterion for convergence requires the space be complete. I encountered it with sequences on 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 (an)∞n=1 is Cauchy if:
∀ϵ>0∃N∈N:n>m>N⟹d(am,an)<ϵ
Theorem
A sequence converges if and only if it is Cauchy
TODO: proof, easy stuff
