Limiter Spaces: A Universal Extension for Limits of Real Sequences

TL;DR

The paper introduces the Limiter, a universal extension of real numbers and limits that assigns canonical limits to all sequences, inspired by generalized summation methods, respecting classical limits and cluster points.

Contribution

It presents the Limiter as a new universal extension that unifies classical and generalized limits for sequences, with continuity and cluster point dependence.

Findings

The Limiter assigns limits to all sequences, including divergent ones.

It respects classical limits and depends only on cluster points.

The extension varies continuously with the cluster set.

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.