Sequence

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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.

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

References

  1. Jump up p46 - Introduction To Set Theory, third edition, Jech and Hrbacek
  2. Jump up to: 2.0 2.1 p11 - Analysis - Part 1: Elements - Krzysztof Maurin