A note on Cauchy integrability
  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. 
Scott Schneider
math.HO math.CA