limit inferior

Let S be a set of real numbers. Recall that a limit pointPlanetmathPlanetmath of S is a real number x such that for all ϵ>0 there exist infinitely many yS such that


We define lim infS, pronounced the limit inferior of S, to be the infimumMathworldPlanetmath of all the limit points of S. If there are no limit points, we define the limit inferior to be +.

The two most common notations for the limit inferior are

lim infS



An alternative, but equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath, definition is available in the case of an infiniteMathworldPlanetmath sequenceMathworldPlanetmath of real numbers x0,x1,x2,,. For each k, let yk be the infimum of the kth tail,


This construction produces a non-decreasing sequence


which either converges to its supremum, or diverges to +. We define the limit inferior of the original sequence to be this limit;

lim infkxk=limkyk.
Title limit inferior
Canonical name LimitInferior
Date of creation 2013-03-22 12:22:01
Last modified on 2013-03-22 12:22:01
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 10
Author rmilson (146)
Entry type Definition
Classification msc 26A03
Synonym liminf
Synonym infimum limit
Related topic LimitSuperior