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...") |
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==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: | ||
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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
[hide]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
Topological form
A sequence (an)∞n=1 in a topological space X converges if ∀U that are open neighbourhoods of x ∃N∈N:n>N⟹xn∈U
Metric space form
A sequence (an)∞n=1 in a metric space V (Keep in mind it is easy to get a metric given a normed vector space) is said to converge to a limit a∈V if:
∀ϵ>0∃N∈N:n>N⟹d(an,a)<ϵ - note the implicit ∀n
In this case we may write: lim
Basic form
Usually \forall\epsilon>0\exists N\in\mathbb{N}: n > N\implies |a_n-a|<\epsilon is first seen, or even just a Null sequence then defining converging to a 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:
\forall\epsilon>0\exists N\in\mathbb{N}:n > N\implies\|a_n-a\|<\epsilon, note this it the definition of the sequence (\|a_n-a\|)_{n=1}^\infty tending towards 0
See also
References
- Jump up ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek