limit superior

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


We define lim supS=lim¯, pronounced the limit superior of S, to be the supremumMathworldPlanetmathPlanetmath of all the limit points of S. If there are no limit points, we define the limit superior to be -.

We can generalize the above definition to the case of a mapping f:X. Now, we define a limit point of f to be an x such that for all ϵ>0 there exist infinitely many yX such that


We then define lim supf, to be the supremum of all the limit points of f, or - if there are no limit points. We recover the previous definition as a special case by considering the limit superior of the inclusion mapping ι:S.

Since a sequence of real numbers x0,x1,x2,, is just a mapping from to , we may adapt the above definition to arrive at the notion of the limit superior of a sequence. However for the case of sequences, an alternative, but equivalentMathworldPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath definition is available. For each k, let yk be the supremum of the kth tail,


This construction produces a non-increasing sequence


which either converges to its infimumMathworldPlanetmath, or diverges to -. We define the limit superior of the original sequence to be this limit;

lim supkxk=limkyk.
Title limit superior
Canonical name LimitSuperior
Date of creation 2013-03-22 12:21:58
Last modified on 2013-03-22 12:21:58
Owner rmilson (146)
Last modified by rmilson (146)
Numerical id 12
Author rmilson (146)
Entry type Definition
Classification msc 26A03
Synonym limsup
Synonym supremum limit
Related topic LimitInferior