Difference between revisions of "Cauchy criterion for convergence"

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(Created page with "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>...")
 
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==Iffy page==
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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.
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'''See [[Cauchy sequence]] for a definition'''
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==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.
  
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{{Todo|proof, easy stuff}}
 
{{Todo|proof, easy stuff}}
  
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==Interesting examples==
 
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===<math>f_n(t)=t^n\rightarrow 0</math> in <math>\|\cdot\|_{L^1}</math>===
 
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Using the <math>\|\cdot\|_{L^1}</math> [[Norm|norm]] stated here for convenience: <math>\|f\|_{L^p}=\left(\int^1_0|f(x)|^pdx\right)^\frac{1}{p}</math> so <math>\|f\|_{L^1}=\int^1_0|f(x)|dx</math>
 
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We see that <math>\|f_n\|_{L^1}=\int^1_0x^ndx=\left[\frac{1}{n+1}x^{n+1}\right]^1_0=\frac{1}{n+1}</math>
 
  
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This clearly <math>\rightarrow 0</math> - this is <math>0:[0,1]\rightarrow\mathbb{R}</math> which of course has [[Norm|norm]] {{M|0}}, we think of this from the sequence <math>(\|f_n-0\|_{L^1})^\infty_{n=1}\rightarrow 0\iff f_n\rightarrow 0</math>
 
  
 
{{Definition|Real Analysis|Functional Analysis}}
 
{{Definition|Real Analysis|Functional Analysis}}
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{{Theorem|Real Analysis|Functional Analysis}}
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{{Theorem Of|Real Analysis|Functional Analysis}}

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


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