Limiter Spaces: A Universal Extension for Limits of Real Sequences

Abstract

We introduce the Limiter, a universal extension of the real numbers and of the limit functional that assigns a canonical limit in an enlarged space to every real sequence. Motivated by generalized summation methods such as Borel summation and Ramanujan's assignments to divergent series, we require our extension to respect classical limits and assign limits in a way that depends only on the cluster points of a sequence and varies continuously when the cluster set is slightly modified.