ABSTRACT
The real number system could be treated in two ways, either as a number system already known to possess certain familiar properties, or as something to be constructed rigorously from the rational number system. This chapter treats the real number system as a number system already known to possess certain familiar properties and provides an in-depth real analysis of the rigorous construction of the real numbers as equivalence classes of Cauchy sequences of rational numbers. Both the rational and real number systems form ordered fields. The property of the real number system making it the proper place to do calculus is completeness. The field axioms capture all the important properties of the algebraic structure of the real number system. The chapter analyzes the field axioms by using a theorem. It also includes exercise problems related to the real number system.
