Cheng, Lu Pien

The Riemann-integral of a function defined on [qb] is well known. It is also well known though not easily found in literature that the Riemann integral can be defined as the limit of Riemann sums using divisions as "directed sets". Many properties of the Riemann integral will then have straightforward proofs. We shall develop equi-Riemann convergence and D-onvergence using such an approach called the Moore-Smith limits or Generalised limits.