Sequence
From Maths
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 \{a_n\}_{n=1}^\infty however I don't like this, as it looks like a set. I have seen the notation (a_n)_{n=1}^\infty 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
[hide]Definition
Formally a sequence (A_i)_{i=1}^\infty is a function[1][2], f:\mathbb{N}\rightarrow S where S is some set. For a finite sequence it is simply f:\{1,...,n\}\rightarrow S. Now we can write:
- f(i):=A_i
This naturally then generalises to indexing sets
Subsequence
Given a sequence (x_n)_{n=1}^\infty we define a subsequence of (x_n)^\infty_{n=1}[2] as a sequence:
- k:\mathbb{N}\rightarrow\mathbb{N} which operates on an n\in\mathbb{N} with n\mapsto k_n:=k(n) where:
- k_n is increasing, that means k_n\le k_{n+1}
We denote this:
- (x_{k_n})_{n=1}^\infty
See also
- Monotonic sequence
- Bolzano-Weierstrass theorem
- Cauchy criterion for convergence
- Convergence of a sequence
References
- Jump up ↑ p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
- ↑ Jump up to: 2.0 2.1 p11 - Analysis - Part 1: Elements - Krzysztof Maurin