xvi Preface
AN OVERVIEW OF THE BOOKEvery student should read the "To the Student" section that follows. It provides
some motivating rationale and some helpful words of advice. Instructors are
advised to read "To the Instructor" for pedagogical rationale and suggestions
for coverage.
As already suggested, this book develops elementary real analysis as a
deductive system founded upon the axioms of the real number system. Thus,
familiarity with the principles of logic and proof techniques will be helpful.
Many universities provide a "transitional" course for this purpose. Appendix A
provides a summary of the essential rules and notation of logic, and may be
consulted as needed. Appendix B reviews the mathematics of sets and functions.
Chapter 1 is included here because without this (or equivalent) material it
would be impossible to rigorously develop the subject of analysis. Nevertheless,
much of it is not analysis; it is a prelude to analysis. Sections 1.1- 1.4 and 1. 7
can be skimmed lightly by those wanting to avoid getting bogged down in these
preliminary issues. Sections 1.5 and 1.6 are the only sections of Chapter 1 that
must be covered thoroughly. They discuss the Archimedean and completeness
properties, which are the first topics with the characteristic flavor of analysis.
Chapter 2, on sequences, serves as the entrance into real analysis. It is of
crucial importance in our development of the subject. The key concept of limit
is introduced early and used extensively. The student learns the methodology
of "epsilonics" by working through many examples, from concrete to abstract.
I believe that the concept of limit is best learned in the context of sequences. In
fact, throughout the remainder of the book the reader will find "sequential crite-
ria" for various other concepts: cluster points, closed sets, compactness, dense-
ness, limits of functions, continuity, uniform continuity, and even for integrals.
Thus, sequences will serve as a unifying theme running through the course.
Chapter 2 contains several powerful theorems, such as the monotone con-
vergence theorem, Cantor's nested intervals theorem, the Bolzano-Weierstrass
theorem for sequences, and Cauchy's convergence criterion. Countable and un-
countable sets are also discussed in this chapter, since a countable set is simply
the range of a sequence. Chapter 2 ends with an optional section on upper and
lower limits.
Chapter 3 contains just enough topology of the real number system to allow
us to convey the results of elementary real analysis with the force and clarity of
contemporary topological language. Excessive generality and unnecessary vo-
cabulary are avoided. The Cantor set and compactness are each given thorough
coverage, but in optional sections. To get through the remaining chapters of
this text the only definition of a compact set that one needs^1 is that it is closed
and bounded.
- Except in proving that Lebesgue's criterion is a sufficient condition for Riemann integra-
bility (Theorem 7.9.7).