Chapter 1
The Real Number System
Sections 1.1-1.4 are optional, containing background on or-
dered fields and the rational numbers. Students "know" these
"facts" but may not have proved them unless they have had
a course in abstract algebra. They can be outlined in one day
or covered completely in four. Sections 1.5 and 1.6 on the
Archimedean and completeness properties, respectively, are
essential. Section 1. 7 is an optional section, outlining a proof
of the uniqueness of the complete ordered field.We begin our study of analysis with an investigation of the real number system
because, ultimately, the entire subject of real analysis rests upon this system.
Every result presented in this course is derived from the properties of the real
number system. This may seem like an exaggerated claim to you. If so, I ask
for your patience. Before the end of the course you will understand that this is
no exaggeration.
There is no universal agreement on what constitutes the best approach to
the study of the real number system. There are at least three popular view-
points. The "constructive" view insists that any proper description of the real
number system must start with the most primitive number system of all , the
natural number system, and construct the real number system by a strictly
rigorous (and tedious) process. A second, "descriptive," view suggests that it is
better to begin directly with the real number system itself, listing its fundamen-
tal axioms and deriving all of its properties from them. A third, "pragmatic,"
view downplays the importance of a rigorous development of the real number
system, holding that it is enough to cover only those aspects of the number
system that seem especially interesting or important in light of their usefulness
in analysis. In this book we take the second, descriptive approach. It is both
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