View source for Limit (sequence)

:: '''Note: ''' see [[Limit]] page for other kinds of limits

__TOC__
==Definition==
Given a [[sequence]] {{M|1=(x_n)_{n=1}^\infty\subseteq X}}, a [[metric space]] {{M|(X,d)}} (that is [[complete metric space|complete]]) and a point {{M|x\in X}}, the sequence {{M|1=(x_n)}} is said to<ref name="KMAPI">Krzysztof Maurin - Analysis - Part 1: Elements</ref><ref group="Note">Actually Maurin gives:
* {{MM|\forall\epsilon>0\exists N\in\mathbb{N}\forall n[n\ge N\implies d(x_n,x)<\epsilon]}} (the change is the {{M|\ge}} sign between the {{M|n}} and {{M|N}}) but as we shall see this doesn't matter</ref>:
* have ''limit'' {{M|x}} or ''converge to'' {{M|x}}
When:
* {{MM|1=\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(x,x_n)<\epsilon]}}<ref group="Note">In [[Krzysztof Maurin's notation]] this can be written as:
* {{MM|\bigwedge_{\epsilon>0}\bigvee_{N\in\mathbb{N} }\bigwedge_{n>N}d(x_n,x)<\epsilon}}</ref>
*:: (note that {{M|\epsilon\in\mathbb{R} }}, obviously - as the [[co-domain]] of {{M|d}} is {{M|\mathbb{R} }})
* Read this as:
*: ''for all {{M|\epsilon}} greater than zero, there exists an {{M|N}} in the [[natural numbers]] such that for all {{M|n}} that are also natural we have that:''
*:: ''whenever {{M|n}} is beyond {{M|N}} that {{M|x_n}} is within {{M|\epsilon}} of {{M|x}}''
===Equivalent definitions===
:: ''Note: '' where it is not obvious changes have a {{M|\{ }} underneath them
{{Begin Inline Theorem}}
{{MM|1=\lim_{n\rightarrow\infty}(x_n)=x\iff\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}\left[\underbrace{n\ge N}\implies d(x_n,x)<\epsilon\right]}}
{{Begin Inline Proof}}
Here we have two definitions
# {{MM|1=\lim_{n\rightarrow\infty}(x_n)=x\iff\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}\left[n> N\implies d(x_n,x)<\epsilon\right]}} (given at the top of the page)
# {{MM|1=\lim_{n\rightarrow\infty}(x_n)=x\iff\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}\left[n\ge N\implies d(x_n,x)<\epsilon\right]}}
'''Proof: ''' {{M|1\implies 2}}
: Let {{M|\epsilon >0}} be given.
:: We know {{M|\exists N'\in\mathbb{N} }} such that {{M|n>N'\implies d(x_n,x)<\epsilon}} by assuming {{M|1}} is true
:: Choose {{M|1=N=N'+1}} 
::: Now {{M|1=n\ge N\implies[n>N\vee n=N]}} by definition of {{M|\ge}}, substituting {{M|1=N=N'+1}} we get {{M|1=n\ge N'+1\implies[n>N'+1\vee n=N'+1]}}
:::* '''(Case: {{M|n>N'+1}})''' Note that {{M|n>N'+1>N'}} so by transitivity of {{M|>}} we see {{M|n>N'}}
:::*: We know from {{M|1}} that {{M|n>N'\implies d(x_n,x)<\epsilon}}
:::* '''(Case: {{M|1=n=N'+1}})''', trivially {{M|N'+1>N'}} so we have {{M|n=N'+1>N'}} so {{M|n>N'}}
:::*: We know from {{M|1}} that {{M|n>N'\implies d(x_n,x)<\epsilon}}
::: So in either case we have {{M|d(x_n,x)<\epsilon}}
::* We have shown that if {{M|n\ge N}} we have {{M|d(x_n,x)\epsilon}}
:* Thus choosing {{M|1=N=N'+1}} is "an {{M|N}} that exists" for the given {{M|\epsilon}}
: This completes the first part of the proof
'''Proof: ''' {{M|2\implies 1}}
: Let {{M|\epsilon>0}} be given.
:: We know {{M|\exists N'\in\mathbb{N} }} such that {{M|n\ge N'\implies d(x_n,x)<\epsilon}} by assuming {{M|2}} is true
:: Choose {{M|1=N=N'-1}}
::: Now {{M|1=n> N\implies n>N'-1}}
:::* '''(Case: {{M|1=n=N'}})''' if this is the case we know that {{M|N'>N'-1}} so {{M|n>N}} is satisfied, but also so is {{M|n\ge N'}} (we have equality)
:::*: We know from {{M|2}} that this {{M|\implies d(x_n,x)<\epsilon}}
:::* '''(Case: {{M|1=n>N'}})''' well {{M|n\ge N'}} means "if {{M|n>N'}} or {{M|1=n=N'}}" so {{M|n>N'\implies n\ge N'}}, thus {{M|n\ge N'}}
:::*: We know from {{M|2}} that this {{M|\implies d(x_n,x)<\epsilon}}
::Thus for {{M|n>N}} we see that {{M|d(x_n,x)<\epsilon}}
: This completes the proof

'''(End of proof)'''
[[Category:First-year friendly]]
{{End Proof}}{{End Theorem}}
==Discussion==
===Requiring {{M|x\in X}}===
If {{M|x\notin X}} then {{M|d(x_n,x)}} is undefined, as {{M|d:X\times X\rightarrow\mathbb{R}_{\ge_0} }}, that is the [[metric space|distance metric]] is only defined for things in {{M|X}}.

To sidestep this limitation and talk about sequences that would converge if only their limit was in the space we consider [[Cauchy sequence|Cauchy sequences]]. It is easy to see that all convergent sequences are Cauchy:
====[[Cauchy sequence]]====
Recall a [[Cauchy sequence]] is defined as:<br/>
{{:Cauchy sequence/Short definition}}
{{Begin Theorem}}
[[Every convergent sequence is Cauchy]]
{{Begin Proof}}
{{:Every convergent sequence is Cauchy/Proof}}
{{End Proof}}{{End Theorem}}

===Process===
{{Begin Theorem}}
[[Limit (sequence)/Discussion of definition|Discussion of why the definition is what it is.]]
{{Begin Proof}}
{{:Limit (sequence)/Discussion of definition}}

{{End Proof}}{{End Theorem}}

==See also==
* [[Cauchy sequence]]
* [[Limit]]
==Notes==
<references group="Note"/>
==References==
<references/>
{{Definition|Real Analysis|Metric Space|Topology}}
[[Category:First-year friendly]]