Difference between revisions of "Limit (sequence)"

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(Created page with ":: '''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)}}...")
 
m (→‎Cauchy sequence)
 
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==Discussion==
 
==Discussion==
 
===Requiring {{M|x\in X}}===
 
===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}}
+
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}}.
−
===Process===
+
−
The idea is that defining "tends towards {{M|x}}" is rather difficult, to sidestep this we just say "we can get as close as we like to" instead. This is the purpose of {{M|\epsilon}}.
+
  
−
We say that "if you give me an {{M|\epsilon>0}} - as small as you like - I can find you a point of the sequence ({{M|N}}) where '''''all''''' points after are ''within'' {{M|\epsilon}} of {{M|x}} (where {{M|d(\cdot,\cdot)}} is our notion of distance)
+
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:
−
* That is after {{M|N}} in the sequence, so that's {{M|x_{n+1},x_{n+1},\ldots}} the ''distance'' between {{M|x_{N+i} }} and {{M|x}} is {{M|<\epsilon}}
+
====[[Cauchy sequence]]====
−
*: This is exactly what {{M|n>N\implies d(x_n,x)<\epsilon}} says, it says that:
+
Recall a [[Cauchy sequence]] is defined as:<br/>
−
*:* whenever {{M|n>N}} we must have {{M|d(x_n,x)<\epsilon}}
+
{{:Cauchy sequence/Short definition}}
−
As per the nature of [[implies]] we ''may'' have {{M|d(x_n,x)<\epsilon}} without {{M|n>N}}, it is only important that WHENEVER we are beyond {{M|N}} in the sequence that {{M|d(x_n,x)<\epsilon}}
+
{{Begin Theorem}}
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{| class="wikitable" border="1"
+
[[Every convergent sequence is Cauchy]]
−
|-
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{{Begin Proof}}
−
! colspan="2" | Example
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{{:Every convergent sequence is Cauchy/Proof}}
−
|-
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{{End Proof}}{{End Theorem}}
−
| [[File:Sequencelimit.gif]]
+
 
−
| Here:
+
===Process===
−
* {{M|x}}-axis scale is from {{M|0}} to {{M|12.6}}, marks are shown every unit.
+
{{Begin Theorem}}
−
* {{M|y}}-axis scale starts from {{M|0}} and is marked every {{M|0.25}} units.
+
[[Limit (sequence)/Discussion of definition|Discussion of why the definition is what it is.]]
−
* The sequence is any sequence of points on the wavy function shown.
+
{{Begin Proof}}
−
** The limit of this is clearly {{M|1}}
+
{{:Limit (sequence)/Discussion of definition}}
−
* The two horizontal lines show {{M|1-\epsilon}} and {{M|1+\epsilon}}
+
−
* The vertical line shows one possible value where every point after it is within {{M|\epsilon}} of {{M|1}}
+
−
* '''due to technical limitations the function {{M|1=f(x)=1+\frac{\sin(\pi x)}{\frac{1}{4}x^2} }} is shown'''
+
−
* The curves are bounds on the function.
+
−
|}
+
−
Notice that at {{M|1=x=1}} that {{M|f(1)=1}}, in fact the curve is ''within'' {{M|\pm\epsilon}} several times before we reach the vertical line, this is the significance of the [[implies]] sign, when we write {{M|A\implies B}} we require that ''whenever'' {{M|A}} is true, {{M|B}} must be true, but {{M|B}} may be true regardless of what {{M|A}} is.
+
  
−
Note that ''after'' the vertical line the function is ''always'' within the bounds.
+
{{End Proof}}{{End Theorem}}
  
−
Because of this any {{M|N'>N}} may be used too, as if {{M|n>N'}} and {{M|N'>N}} then {{M|n>N'>N}} so {{M|n>N}} - this proves that if {{M|N}} works then any larger {{M|N'}} will too. There is no requirement to find the smallest {{M|N}} that'll work, just ''an'' {{M|N}} such that {{M|n>N\implies d(x_n,x)<\epsilon}}
 
 
==See also==
 
==See also==
 
* [[Cauchy sequence]]
 
* [[Cauchy sequence]]

Latest revision as of 13:56, 5 December 2015

Note: see Limit page for other kinds of limits

Definition

Given a sequence [ilmath](x_n)_{n=1}^\infty\subseteq X[/ilmath], a metric space [ilmath](X,d)[/ilmath] (that is complete) and a point [ilmath]x\in X[/ilmath], the sequence [ilmath](x_n)[/ilmath] is said to[1][Note 1]:

  • have limit [ilmath]x[/ilmath] or converge to [ilmath]x[/ilmath]

When:

  • [math]\forall\epsilon>0\exists N\in\mathbb{N}\forall n\in\mathbb{N}[n>N\implies d(x,x_n)<\epsilon][/math][Note 2]
    (note that [ilmath]\epsilon\in\mathbb{R} [/ilmath], obviously - as the co-domain of [ilmath]d[/ilmath] is [ilmath]\mathbb{R} [/ilmath])
  • Read this as:
    for all [ilmath]\epsilon[/ilmath] greater than zero, there exists an [ilmath]N[/ilmath] in the natural numbers such that for all [ilmath]n[/ilmath] that are also natural we have that:
    whenever [ilmath]n[/ilmath] is beyond [ilmath]N[/ilmath] that [ilmath]x_n[/ilmath] is within [ilmath]\epsilon[/ilmath] of [ilmath]x[/ilmath]

Equivalent definitions

Note: where it is not obvious changes have a [ilmath]\{ [/ilmath] underneath them

[math]\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][/math]

Here we have two definitions

  1. [math]\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][/math] (given at the top of the page)
  2. [math]\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][/math]

Proof: [ilmath]1\implies 2[/ilmath]

Let [ilmath]\epsilon >0[/ilmath] be given.
We know [ilmath]\exists N'\in\mathbb{N} [/ilmath] such that [ilmath]n>N'\implies d(x_n,x)<\epsilon[/ilmath] by assuming [ilmath]1[/ilmath] is true
Choose [ilmath]N=N'+1[/ilmath]
Now [ilmath]n\ge N\implies[n>N\vee n=N][/ilmath] by definition of [ilmath]\ge[/ilmath], substituting [ilmath]N=N'+1[/ilmath] we get [ilmath]n\ge N'+1\implies[n>N'+1\vee n=N'+1][/ilmath]
  • (Case: [ilmath]n>N'+1[/ilmath]) Note that [ilmath]n>N'+1>N'[/ilmath] so by transitivity of [ilmath]>[/ilmath] we see [ilmath]n>N'[/ilmath]
    We know from [ilmath]1[/ilmath] that [ilmath]n>N'\implies d(x_n,x)<\epsilon[/ilmath]
  • (Case: [ilmath]n=N'+1[/ilmath]), trivially [ilmath]N'+1>N'[/ilmath] so we have so [ilmath]n>N'[/ilmath]
    We know from [ilmath]1[/ilmath] that [ilmath]n>N'\implies d(x_n,x)<\epsilon[/ilmath]
So in either case we have [ilmath]d(x_n,x)<\epsilon[/ilmath]
  • We have shown that if [ilmath]n\ge N[/ilmath] we have [ilmath]d(x_n,x)\epsilon[/ilmath]
  • Thus choosing [ilmath]N=N'+1[/ilmath] is "an [ilmath]N[/ilmath] that exists" for the given [ilmath]\epsilon[/ilmath]
This completes the first part of the proof

Proof: [ilmath]2\implies 1[/ilmath]

Let [ilmath]\epsilon>0[/ilmath] be given.
We know [ilmath]\exists N'\in\mathbb{N} [/ilmath] such that [ilmath]n\ge N'\implies d(x_n,x)<\epsilon[/ilmath] by assuming [ilmath]2[/ilmath] is true
Choose [ilmath]N=N'-1[/ilmath]
Now [ilmath]n> N\implies n>N'-1[/ilmath]
  • (Case: [ilmath]n=N'[/ilmath]) if this is the case we know that [ilmath]N'>N'-1[/ilmath] so [ilmath]n>N[/ilmath] is satisfied, but also so is [ilmath]n\ge N'[/ilmath] (we have equality)
    We know from [ilmath]2[/ilmath] that this [ilmath]\implies d(x_n,x)<\epsilon[/ilmath]
  • (Case: [ilmath]n>N'[/ilmath]) well [ilmath]n\ge N'[/ilmath] means "if [ilmath]n>N'[/ilmath] or [ilmath]n=N'[/ilmath]" so [ilmath]n>N'\implies n\ge N'[/ilmath], thus [ilmath]n\ge N'[/ilmath]
    We know from [ilmath]2[/ilmath] that this [ilmath]\implies d(x_n,x)<\epsilon[/ilmath]
Thus for [ilmath]n>N[/ilmath] we see that [ilmath]d(x_n,x)<\epsilon[/ilmath]
This completes the proof

(End of proof)

Discussion

Requiring [ilmath]x\in X[/ilmath]

If [ilmath]x\notin X[/ilmath] then [ilmath]d(x_n,x)[/ilmath] is undefined, as [ilmath]d:X\times X\rightarrow\mathbb{R}_{\ge_0} [/ilmath], that is the distance metric is only defined for things in [ilmath]X[/ilmath].

To sidestep this limitation and talk about sequences that would converge if only their limit was in the space we consider Cauchy sequences. It is easy to see that all convergent sequences are Cauchy:

Cauchy sequence

Recall a Cauchy sequence is defined as:
Given a metric space [ilmath](X,d)[/ilmath] and a sequence [ilmath](x_n)_{n=1}^\infty\subseteq X[/ilmath] is said to be a Cauchy sequence[2][3] if:

  • [ilmath]\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][/ilmath]

Every convergent sequence is Cauchy

Every convergent sequence is Cauchy/Proof


Process

Discussion of why the definition is what it is.

The idea is that defining "tends towards [ilmath]x[/ilmath]" is rather difficult, to sidestep this we just say "we can get as close as we like to" instead. This is the purpose of [ilmath]\epsilon[/ilmath].

We say that "if you give me an [ilmath]\epsilon>0[/ilmath] - as small as you like - I can find you a point of the sequence ([ilmath]N[/ilmath]) where all points after are within [ilmath]\epsilon[/ilmath] of [ilmath]x[/ilmath] (where [ilmath]d(\cdot,\cdot)[/ilmath] is our notion of distance)

  • That is after [ilmath]N[/ilmath] in the sequence, so that's [ilmath]x_{n+1},x_{n+1},\ldots[/ilmath] the distance between [ilmath]x_{N+i} [/ilmath] and [ilmath]x[/ilmath] is [ilmath]<\epsilon[/ilmath]
    This is exactly what [ilmath]n>N\implies d(x_n,x)<\epsilon[/ilmath] says, it says that:
    • whenever [ilmath]n>N[/ilmath] we must have [ilmath]d(x_n,x)<\epsilon[/ilmath]

As per the nature of implies we may have [ilmath]d(x_n,x)<\epsilon[/ilmath] without [ilmath]n>N[/ilmath], it is only important that WHENEVER we are beyond [ilmath]N[/ilmath] in the sequence that [ilmath]d(x_n,x)<\epsilon[/ilmath]

Example
Sequencelimit.gif Here:
  • [ilmath]x[/ilmath]-axis scale is from [ilmath]0[/ilmath] to [ilmath]12.6[/ilmath], marks are shown every unit.
  • [ilmath]y[/ilmath]-axis scale starts from [ilmath]0[/ilmath] and is marked every [ilmath]0.25[/ilmath] units.
  • The sequence is any sequence of points on the wavy function shown.
    • The limit of this is clearly [ilmath]1[/ilmath]
  • The two horizontal lines show [ilmath]1-\epsilon[/ilmath] and [ilmath]1+\epsilon[/ilmath]
  • The vertical line shows one possible value where every point after it is within [ilmath]\epsilon[/ilmath] of [ilmath]1[/ilmath]
  • due to technical limitations the function [ilmath]f(x)=1+\frac{\sin(\pi x)}{\frac{1}{4}x^2}[/ilmath] is shown
  • The curves are bounds on the function.

Notice that at [ilmath]x=1[/ilmath] that , in fact the curve is within [ilmath]\pm\epsilon[/ilmath] several times before we reach the vertical line, this is the significance of the implies sign, when we write [ilmath]A\implies B[/ilmath] we require that whenever [ilmath]A[/ilmath] is true, [ilmath]B[/ilmath] must be true, but [ilmath]B[/ilmath] may be true regardless of what [ilmath]A[/ilmath] is.

Note that after the vertical line the function is always within the bounds.

Because of this any [ilmath]N'>N[/ilmath] may be used too, as if [ilmath]n>N'[/ilmath] and [ilmath]N'>N[/ilmath] then [ilmath]n>N'>N[/ilmath] so [ilmath]n>N[/ilmath] - this proves that if [ilmath]N[/ilmath] works then any larger [ilmath]N'[/ilmath] will too. There is no requirement to find the smallest [ilmath]N[/ilmath] that'll work, just an [ilmath]N[/ilmath] such that [ilmath]n>N\implies d(x_n,x)<\epsilon[/ilmath]


See also

Notes

  1. ↑ Actually Maurin gives:
    • [math]\forall\epsilon>0\exists N\in\mathbb{N}\forall n[n\ge N\implies d(x_n,x)<\epsilon][/math] (the change is the [ilmath]\ge[/ilmath] sign between the [ilmath]n[/ilmath] and [ilmath]N[/ilmath]) but as we shall see this doesn't matter
  2. ↑ In Krzysztof Maurin's notation this can be written as:
    • [math]\bigwedge_{\epsilon>0}\bigvee_{N\in\mathbb{N} }\bigwedge_{n>N}d(x_n,x)<\epsilon[/math]

References

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