Difference between revisions of "Sequence"
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A sequence is one of the earliest and easiest definitions encountered, but I will restate it. | A sequence is one of the earliest and easiest definitions encountered, but I will restate it. | ||
| − | I was taught to denote the sequence <math>\{a_1,a_2,...\}</math> by <math>\{a_n\}_{n=1}^\infty</math> however I don't like this, as it looks like a set. I have seen the notation <math>(a_n)_{n=1}^\infty</math> and I must say I prefer it. | + | I was taught to denote the sequence <math>\{a_1,a_2,...\}</math> by <math>\{a_n\}_{n=1}^\infty</math> however I don't like this, as it looks like a set. I have seen the notation <math>(a_n)_{n=1}^\infty</math> and I must say I prefer it. This notation is inline with that of a [[Tuple|tuple]] which is a generalisation of [[Ordered pair|an ordered pair]]. |
==Definition== | ==Definition== | ||
| − | Formally a sequence is a function<ref>p46 - Introduction To Set Theory, third edition, Jech and Hrbacek</ref>, <math>f:\mathbb{N}\rightarrow S</math> where {{M|S}} is some set. For a finite sequence it is simply <math>f:\{1,...,n\}\rightarrow S</math> | + | Formally a sequence {{M|1=(A_i)_{i=1}^\infty}} is a function<ref>p46 - Introduction To Set Theory, third edition, Jech and Hrbacek</ref><ref name="Analysis">p11 - Analysis - Part 1: Elements - Krzysztof Maurin</ref>, <math>f:\mathbb{N}\rightarrow S</math> where {{M|S}} is some set. For a finite sequence it is simply <math>f:\{1,...,n\}\rightarrow S</math>. Now we can write: |
| + | * {{M|1=f(i):=A_i}} | ||
| + | This naturally then generalises to [[Indexing set|indexing sets]] | ||
| − | + | ==Subsequence== | |
| + | Given a sequence {{M|1=(x_n)_{n=1}^\infty}} we define a ''subsequence of {{M|1=(x_n)^\infty_{n=1} }}''<ref name="Analysis"/> as a sequence: | ||
| + | * {{M|k:\mathbb{N}\rightarrow\mathbb{N} }} which operates on an {{M|n\in\mathbb{N} }} with {{M|1=n\mapsto k_n:=k(n)}} where: | ||
| + | ** {{M|k_n}} is increasing, that means {{M|k_n\le k_{n+1} }} | ||
| − | == | + | We denote this: |
| − | + | * {{M|1=(x_{k_n})_{n=1}^\infty}} | |
==See also== | ==See also== | ||
| + | * [[Monotonic sequence]] | ||
| + | * [[Bolzano-Weierstrass theorem]] | ||
* [[Cauchy criterion for convergence]] | * [[Cauchy criterion for convergence]] | ||
* [[Convergence of a sequence]] | * [[Convergence of a sequence]] | ||
==References== | ==References== | ||
| − | + | <references/> | |
{{Definition|Set Theory|Real Analysis|Functional Analysis}} | {{Definition|Set Theory|Real Analysis|Functional Analysis}} | ||
Revision as of 23:47, 21 June 2015
A sequence is one of the earliest and easiest definitions encountered, but I will restate it.
I was taught to denote the sequence [math]\{a_1,a_2,...\}[/math] by [math]\{a_n\}_{n=1}^\infty[/math] however I don't like this, as it looks like a set. I have seen the notation [math](a_n)_{n=1}^\infty[/math] and I must say I prefer it. This notation is inline with that of a tuple which is a generalisation of an ordered pair.
Contents
Definition
Formally a sequence [ilmath](A_i)_{i=1}^\infty[/ilmath] is a function[1][2], [math]f:\mathbb{N}\rightarrow S[/math] where [ilmath]S[/ilmath] is some set. For a finite sequence it is simply [math]f:\{1,...,n\}\rightarrow S[/math]. Now we can write:
- [ilmath]f(i):=A_i[/ilmath]
This naturally then generalises to indexing sets
Subsequence
Given a sequence [ilmath](x_n)_{n=1}^\infty[/ilmath] we define a subsequence of [ilmath](x_n)^\infty_{n=1}[/ilmath][2] as a sequence:
- [ilmath]k:\mathbb{N}\rightarrow\mathbb{N} [/ilmath] which operates on an [ilmath]n\in\mathbb{N} [/ilmath] with [ilmath]n\mapsto k_n:=k(n)[/ilmath] where:
- [ilmath]k_n[/ilmath] is increasing, that means [ilmath]k_n\le k_{n+1} [/ilmath]
We denote this:
- [ilmath](x_{k_n})_{n=1}^\infty[/ilmath]
See also
- Monotonic sequence
- Bolzano-Weierstrass theorem
- Cauchy criterion for convergence
- Convergence of a sequence
