A note on Cauchy integrability

Abstract

We show that for any bounded function $f:[a,b]\rightarrow{\mathbb R}$ and $ε>0$ there is a partition $P$ of $[a,b]$ with respect to which the Riemann sum of $f$ using right endpoints is within $ε$ 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.