An elementary proof of the existence and uniqueness of solutions to an initial value problem

TL;DR

This paper provides an elementary proof of the local existence and uniqueness of solutions to initial value problems under Lipschitz conditions, avoiding integration and using basic analysis tools.

Contribution

It introduces a simple, elementary approach to proving existence and uniqueness without involving integration, relying solely on basic analysis techniques.

Findings

Cauchy iterates converge on a dense subset of the interval

The limit function extends to a solution on the entire interval

The proof avoids the use of integration, simplifying the classical approach

Abstract

In this note, we show a classical result on the local existence and uniqueness of a solution to an initial value problem subject to a Lipschitz condition. We use only elementary tools from mathematical analysis, without involving any integration. We proceed by showing that the Cauchy iterates converge on a dense subset of the interval and subsequently proving that the extension of this limit function to the whole interval is a solution to the Cauchy problem.