ABSTRACT
In this chapter, the author develops the theory of the Riemann Integral for a bounded function on the interval. He does this carefully in (Peterson (8) 2020) also, but now that the author is going to handle integration more abstractly, it is good to have a review of the proofs before the author moves on to the extension to functions of bounded variation. The author shows how the author proves that the set of Riemann integrable functions is quite rich and varied. The author now proves a series of properties of the Riemann integral. Let’s start with a lemma about inflmums and supremums. The author establishes the familiar summation property of the Riemann integral over an interval. Using the Fundamental Theorem of Calculus, the author derives many useful tools. This discussion is a precursor for more complicated discussions of such things using integrals defined by what are called measures and integrals extending Riemann integration to what is called Riemann - Stieltjes integration.
