A note on Cauchy integrability

TL;DR

The paper provides an elementary proof that Cauchy's and Riemann's definitions of integrability are equivalent by showing that for any bounded function, Riemann sums can approximate the upper Darboux sum arbitrarily closely.

Contribution

It introduces a simple proof demonstrating the equivalence of Cauchy's and Riemann's integrability concepts using partition-based approximations.

Findings

Riemann sums can approximate upper Darboux sums within any epsilon

Elementary proof of the equivalence of Cauchy's and Riemann's integrability

Partition constructions for approximation of integrals

Abstract

We show that for any bounded function $f:[a,b]\rightarrow{\mathbb R}$ and $\epsilon>0$ there is a partition $P$ of $[a,b]$ with respect to which the Riemann sum of $f$ using right endpoints is within $\epsilon$ of the upper Darboux sum of $f$. This leads to an elementary proof of the theorem of Gillespie \cite{G} showing that Cauchy's and Riemann's definitions of integrability coincide.