Given a sequence {xn}n, any infinite subset of the sequence forms a subsequence. We formalize this as follows:


If X is a set and {an}nN is a sequence in X, then a subsequence of {an} is a sequence of the form {anr}rN where {nr}rN is a strictly increasing sequence of natural numbersMathworldPlanetmath.

Equivalently, {yn}n is a subsequence of {xn}n if

  1. 1.

    {yn}n is a sequence of elements of X, and

  2. 2.

    there is a strictly increasing function a: such that

    yn=xa(n) for all n.

Let X= and let {xn} be the sequence


Then, the sequence


is a subsequence of {xn}. The subsequence of natural numbers mentioned in the definition is {n2}n and the functionMathworldPlanetmath a: mentioned above is a(n)=n2.

Title subsequence
Canonical name Subsequence
Date of creation 2013-03-22 12:56:34
Last modified on 2013-03-22 12:56:34
Owner alozano (2414)
Last modified by alozano (2414)
Numerical id 6
Author alozano (2414)
Entry type Definition
Classification msc 00A05