Preface
Elementary real analysis has earned its place as a core subject in the undergrad-
uate mathematics curriculum. It deserves this status for several reasons. First,
it develops the key concepts of calculus from a mature perspective. By laying
a rigorous foundation for the theory of calculus and establishing its important
results by logical deduction from a reasonably small set of assumptions, we
organize the subject into a bona fide deductive system. In fact, we shall show
that the entire subject follows from the properties of the real number system.
Students generally have not considered this possibility; many find it surprising,
even exciting, to be immersed in making this possibility a reality.
Second, because this co urse traces the powerful techniques of calculus back
to their logical origins in the real number system, it is an essential part of
the preparation of every mathematics teacher, particularly one who intends
to teach calculus. Finally, a course in elementary real analysis opens the door
to further study in real analysis, which is one of the cornerstone subjects in
contemporary pure and applied mathematics. The concepts and techniques
learned in this course will be explored further as part of any contemporary
graduate-level program in mathematics. Indeed, the concepts of elementary real
analysis belong in the repertoire of every mathematical scientist or teacher.
The motivation for writing yet another textbook in elementary real anal-
ysis comes from my many years of experience teaching the subject. My goal
has been to make the subject accessible to as wide an a udience as possible,
without sacrificing content or rigor. At my university all students majoring in
mathematics (including those preparing to be secondary school teachers) take
one semester of elementary real analysis, usually in their junior or senior year.
A second semester course is available as an elective. Students often report that
the first real analysis course is one of the most challenging courses in the un-
dergraduate mathematics curriculum. This is due partly to the nature of the
material itself, and partly to the methodology that must be learned for success
in the subject. To overcome this ch allenge the instructor and the textbook must
meet students at their level. The elegant brevity of professional mathematical
writing, so satisfying to t he mathematician, is inappropriate for many students
at this level. Unfortunately, the writing in most elementary real analysis books
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