every bounded sequence has limit along an ultrafilter

Theorem 1.

Let F be an ultrafilterMathworldPlanetmath on N and (xn) be a real boundedPlanetmathPlanetmathPlanetmath sequence. Then F-limxn exists.


Let (xn) be a bounded sequence. Choose a0 and b0 such that a0xnb0. Put c0:=a0+b02. Then precisely one of the sets {n;xna0,c0}, {n;xnc0,b0} belongs to the filter . (Their union is and the filter is an ultrafilter.) We choose a1,b1 as that subinterval from a0,c0 and c0,b0 for which C:={n;xna1,b1} belongs to .

Now we again bisect the interval a1,b1 by putting c1=a1+b12. Denote A:={n;xna1,c1}, B:={n;xnc1,b1}. It holds BA(C)=. By the alternative characterization of ultrafilters we get that one of these sets is in . The set C doesn’t belong to , therefore it must be one of the sets A and B. We choose the corresponding interval for a2,b2.

By inductionMathworldPlanetmath we obtain the monotonous sequences (an), (bn) with the same limit limnan=limnbn:=L such that for any n it holds {n;xna1,b1}.

We claim that -limxn=L. Indeed, for any ε>0 there is n such that an,bn(L-ε,L+ε), thus {n;xnan,bn}A(ε). The set {n;xna1,b1} belongs to , hence A(ε) as well. ∎

Note that, if we modify the definition of -limit in a such way that we admit the values ±, then every sequence has -limit along an ultrafilter . (The limit is + if for each neighborhoodMathworldPlanetmathPlanetmath V of infinityMathworldPlanetmathPlanetmath, the set {n;xnV} belongs to . Similarly for -.)


  • 1 M. A. Alekseev, L. Yu. Glebsky, and E. I. Gordon, On approximations of groups, group actions and Hopf algebras, Journal of Mathematical Sciences 107 (2001), no. 5, 4305–4332.
  • 2 B. Balcar and P. Štěpánek, Teorie množin, Academia, Praha, 1986 (Czech).
  • 3 K. Hrbacek and T. Jech, Introduction to set theoryMathworldPlanetmath, Marcel Dekker, New York, 1999.
Title every bounded sequence has limit along an ultrafilter
Canonical name EveryBoundedSequenceHasLimitAlongAnUltrafilter
Date of creation 2013-03-22 15:32:26
Last modified on 2013-03-22 15:32:26
Owner kompik (10588)
Last modified by kompik (10588)
Numerical id 4
Author kompik (10588)
Entry type Theorem
Classification msc 40A05
Classification msc 03E99
Related topic Ultrafilter