Difference between revisions of "Sequence"
(Created page with " ==Introduction== 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,...\}</ma...") |
m |
||
| Line 11: | Line 11: | ||
==Convergence of a sequence== | ==Convergence of a sequence== | ||
| + | ===Topological form=== | ||
| + | A sequence <math>(a_n)_{n=1}^\infty</math> in a [[Topological space|topological space]] {{M|X}} converges if <math>\forall U</math> that are open neighbourhoods of {{M|x}} <math>\exists N\in\mathbb{N}: n> N\implies x_n\in U</math> | ||
| + | ===Metric space form=== | ||
A sequence <math>(a_n)_{n=1}^\infty</math> in a [[Metric space|metric space]] {{M|V}} (Keep in mind it is easy to get a metric given a [[Norm|normed]] [[Vector space|vector space]]) is said to converge to a limit <math>a\in V</math> if: | A sequence <math>(a_n)_{n=1}^\infty</math> in a [[Metric space|metric space]] {{M|V}} (Keep in mind it is easy to get a metric given a [[Norm|normed]] [[Vector space|vector space]]) is said to converge to a limit <math>a\in V</math> if: | ||
| Line 16: | Line 19: | ||
In this case we may write: <math>\lim_{n\rightarrow\infty}(a_n)=a</math> | In this case we may write: <math>\lim_{n\rightarrow\infty}(a_n)=a</math> | ||
| − | + | ===Basic form=== | |
| − | === | + | |
Usually <math>\forall\epsilon>0\exists N\in\mathbb{N}: n > N\implies |a_n-a|<\epsilon</math> is first seen, or even just a [[Null sequence]] then defining converging to {{M|a}} by subtraction, like with [[Continuous map]] you move on to a metric space. | Usually <math>\forall\epsilon>0\exists N\in\mathbb{N}: n > N\implies |a_n-a|<\epsilon</math> is first seen, or even just a [[Null sequence]] then defining converging to {{M|a}} by subtraction, like with [[Continuous map]] you move on to a metric space. | ||
| − | + | ===Normed form=== | |
In a [[Norm|normed]] [[Vector space|vector space]] as you'd expect it's defined as follows: | In a [[Norm|normed]] [[Vector space|vector space]] as you'd expect it's defined as follows: | ||
Revision as of 04:52, 8 March 2015
Contents
Introduction
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.
Definition
Formally a sequence is a function[1], [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]
There is little more to say.
Convergence of a sequence
Topological form
A sequence [math](a_n)_{n=1}^\infty[/math] in a topological space [ilmath]X[/ilmath] converges if [math]\forall U[/math] that are open neighbourhoods of [ilmath]x[/ilmath] [math]\exists N\in\mathbb{N}: n> N\implies x_n\in U[/math]
Metric space form
A sequence [math](a_n)_{n=1}^\infty[/math] in a metric space [ilmath]V[/ilmath] (Keep in mind it is easy to get a metric given a normed vector space) is said to converge to a limit [math]a\in V[/math] if:
[math]\forall\epsilon>0\exists N\in\mathbb{N}:n > N\implies d(a_n,a)<\epsilon[/math] - note the implicit [math]\forall n[/math]
In this case we may write: [math]\lim_{n\rightarrow\infty}(a_n)=a[/math]
Basic form
Usually [math]\forall\epsilon>0\exists N\in\mathbb{N}: n > N\implies |a_n-a|<\epsilon[/math] is first seen, or even just a Null sequence then defining converging to [ilmath]a[/ilmath] by subtraction, like with Continuous map you move on to a metric space.
Normed form
In a normed vector space as you'd expect it's defined as follows:
[math]\forall\epsilon>0\exists N\in\mathbb{N}:n > N\implies\|a_n-a\|<\epsilon[/math], note this it the definition of the sequence [math](\|a_n-a\|)_{n=1}^\infty[/math] tending towards 0
See also
References
- ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
