Sequence
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
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
Notes
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.
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