A more intuitive definition of limit

TL;DR

This paper proposes a new, more intuitive axiomatic framework for defining limits, aiming to simplify the derivation of convergence results through additional axioms or an abstract order-based approach.

Contribution

It introduces a novel axiomatic approach to defining limits, including extensions with additional axioms and an abstract framework based solely on order relations.

Findings

Derived basic convergence results from the axioms

Proposed two methods to extend the axiomatic definition

Outlined a future abstract framework for limits

Abstract

Limit can be defined by two axioms: 1. Strict inequality between limits implies, ultimately, strict inequality between functions. 2. For constant functions limit is trivial. How can basic results on convergence be derived from these axioms? In this paper we propose two answers: a) at the most elementary level- add two more axioms, b) at somewhat higher level, do it in three steps, and, in our forthcoming paper "Axiomatic definition of limit", a third answer- c) do it neater - in an abstract framework, where only order relations are present.