Difference between revisions of "Cauchy sequence/Definition"

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(Created page with "Given a metric space {{M|(X,d)}} and a sequence {{M|1=(x_n)_{n=1}^\infty\subseteq X}} is said to be a ''Cauchy sequence''<ref name="FA">Functiona...")
 
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Given a [[Metric space|metric space]] {{M|(X,d)}} and a [[Sequence|sequence]] {{M|1=(x_n)_{n=1}^\infty\subseteq X}} is said to be a ''Cauchy sequence''<ref name="FA">Functional Analysis - George Bachman and Lawrence Narici</ref>{{rAPIKM}} if:
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{{:Cauchy sequence/Short definition}}<ref group="Note">Note that in [[Krzysztof Maurin's notation]] this is written as {{MM|1=\bigwedge_{\epsilon>0}\bigvee_{N\in\mathbb{N} }\bigwedge_{m,n>\mathbb{N} }d(x_n,x_m)<\epsilon}} - which is rather elegant</ref><ref group="Note">It doesn't matter if we use {{M|n\ge m>N}} or {{M|n,m\ge N}} because if {{M|1=n=m}} then {{M|1=d(x_n,x_m)=0}}, it doesn't matter which way we consider them (as {{M|n>m}} or {{M|m>n}}) for {{M|1=d(x,y)=d(y,x)}} - I use the ordering to give the impression that as {{M|n}} goes out ahead it never ventures far (as in {{M|\epsilon}}-distance}}) from {{M|x_m}}. This has served me well</ref>
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* {{M|\forall\epsilon > 0\exists N\in\mathbb{N}\forall n,m\in\mathbb{N}[n\ge m> N\implies d(x_m,x_n)<\epsilon]}}<ref group="Note">Note that in [[Krzysztof Maurin's notation]] this is written as {{MM|1=\bigwedge_{\epsilon>0}\bigvee_{N\in\mathbb{N} }\bigwedge_{m,n>\mathbb{N} }d(x_n,x_m)<\epsilon}} - which is rather elegant</ref><ref group="Note">It doesn't matter if we use {{M|n\ge m>N}} or {{M|n,m\ge N}} because if {{M|1=n=m}} then {{M|1=d(x_n,x_m)=0}}, it doesn't matter which way we consider them (as {{M|n>m}} or {{M|m>n}}) for {{M|1=d(x,y)=d(y,x)}} - I use the ordering to give the impression that as {{M|n}} goes out ahead it never ventures far (as in {{M|\epsilon}}-distance}}) from {{M|x_m}}. This has served me well</ref>
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In words it is simply:
 
In words it is simply:
 
* For any arbitrary distance apart, there exists a point such that any two points in the sequence after that point are within that arbitrary distance apart.  
 
* For any arbitrary distance apart, there exists a point such that any two points in the sequence after that point are within that arbitrary distance apart.  

Latest revision as of 13:55, 5 December 2015

Given a metric space (X,d) and a sequence (xn)∞n=1⊆X is said to be a Cauchy sequence[1][2] if:

  • ∀ϵ>0∃N∈N∀n,m∈N[n≥m>N⟹d(xm,xn)<ϵ][Note 1][Note 2]

In words it is simply:

  • For any arbitrary distance apart, there exists a point such that any two points in the sequence after that point are within that arbitrary distance apart.

Notes

  1. Jump up ↑ Note that in Krzysztof Maurin's notation this is written as ⋀ϵ>0⋁N∈N⋀m,n>Nd(xn,xm)<ϵ - which is rather elegant
  2. Jump up ↑ It doesn't matter if we use n≥m>N or n,m≥N because if n=m then d(xn,xm)=0, it doesn't matter which way we consider them (as n>m or m>n) for d(x,y)=d(y,x) - I use the ordering to give the impression that as n goes out ahead it never ventures far (as in ϵ-distance}}) from xm. This has served me well

References

  1. Jump up ↑ Functional Analysis - George Bachman and Lawrence Narici
  2. Jump up ↑ Analysis - Part 1: Elements - Krzysztof Maurin
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