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| ==Convergence of a sequence== | | ==Convergence of a sequence== |
− | ===Topological form===
| + | * See [[Convergence of a sequence]] |
− | 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>
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− | ===Metric space form===
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− | 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:
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− | | + | |
− | <math>\forall\epsilon>0\exists N\in\mathbb{N}:n > N\implies d(a_n,a)<\epsilon</math> - note the [[Implicit qualifier|implicit <math>\forall n</math>]]
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− | | + | |
− | In this case we may write: <math>\lim_{n\rightarrow\infty}(a_n)=a</math>
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− | ===Basic form===
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− | 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.
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− | ===Normed form===
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− | | + | |
− | In a [[Norm|normed]] [[Vector space|vector space]] as you'd expect it's defined as follows:
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− | | + | |
− | <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
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| ==See also== | | ==See also== |
| * [[Cauchy criterion for convergence]] | | * [[Cauchy criterion for convergence]] |
| + | * [[Convergence of a sequence]] |
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| ==References== | | ==References== |
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| {{Definition|Set Theory|Real Analysis|Functional Analysis}} | | {{Definition|Set Theory|Real Analysis|Functional Analysis}} |
Revision as of 17:13, 8 March 2015
Introduction
A sequence is one of the earliest and easiest definitions encountered, but I will restate it.
I was taught to denote the sequence {a1,a2,...} by {an}∞n=1 however I don't like this, as it looks like a set. I have seen the notation (an)∞n=1 and I must say I prefer it.
Definition
Formally a sequence is a function[1], f:N→S where S is some set. For a finite sequence it is simply f:{1,...,n}→S
There is little more to say.
Convergence of a sequence
See also
References
- Jump up ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek