Chapter 7
The Riemann Integral
Sections 7.2-7.6 develop the essential core material on the
Riemann integral. The Darboux sum approach is used be-
cause it seems the most natural at this level. Rigor is not
compromised at any point, although the chapter is organized
so that more esoteric matters can be skipped without sacri-
ficing essential understanding. The elementary transcenden-
tal functions are defined rigorously in (optional) Section 7.7,
and Lebesgue's criterion for Riemann integrability is proved
in (optional) Section 7.9.7.1 Refresher on Suprema, lnfima, and the
Forcing Principle
In defining the Riemann integral J: f and establishing its properties we will
make frequent use of the concepts of suprema and infima of sets of real numbers,
and use them in new ways. It is a good idea to review Section 1.6 at this time.
In particular, we shall need the following definitions and facts.
D e finition 7.1.1 If A <;;;; IR, and x E IR, then
x+A={x+a:aEA};
xA = {xa: a EA};
-A = {-a : a E A}.3 5 7