Alternative proofs of some theorems on Riemann integration

In introductory calculus, on the assumption that a real‐valued function is Riemann integrable on a compact interval and prior to having studied the Fundamental Theorem of Calculus, the student typically practices computing the integral by evaluating limits of sequences of Riemann sums over equipartitions. When moving on to the first proof course in analysis, this approach, when applied to upper and lower integrals (whose existence is assured for bounded functions), can be helpful not only computationally, but also in providing proofs (alternative to those appearing in the texts) of some of the fundamental theorems on Riemann integration.