xiv Preface
tends toward this style. Many of my students find such textbooks indigestible,
despite t heir excellence, so I decided to attempt to write one that my students
would find more reader-friendly.
The resulting book is a straightforward, comprehensive presentation of the
concepts and methodology of elementary real analysis, written at a level and
in a style that can be understood by the typical undergraduate majoring in
one of the mathematical sciences at a mainstream college or university. Its
prerequisites are the usual core of courses in calculus and linear algebra, and
at least one "transitional" course in which logic and proof techniques are taken
seriously. The book is elementary in the sense that it stops short of the theory
of Lebesgue measure and integral. It is also elementary in presentation. I have
tried to keep the focus of the book on the central core of analysis. Material that
is beyond this core is confined to "projects" or clearly labeled with an asterisk,
"·"
This book is suitable for courses ranging in length from one term to one
year. My colleagues and I have had considerable experience using preliminary
versions of the text for a one-semester course covering the most essential topics
(material not labeled with "")in Chapters 1-7. Additional material has been
added in these and later chapters to allow flexibility for instructors with differ-
ing priorities. For an outline of a specific one-semester course, see the "To the
Instructor" section that follows. To cover the entire book at a reasonable pace
a full year should be allowed.
FEATURES OF THIS TEXT- Written at the undergraduate student's level; designed to be read. Exposi-
tion is often conversational, explaining both the details and the underlying
motivation. - Stresses the underlying ideas and unity of the subject; connects analysis
with previously learned mathematics, and prepares students for graduate
level analysis. - Respects the deductive organization of mathematics. Analysis is devel-
oped as a deductive system based upon the axioms of the real number
system. - Proofs are written in a style appropriate for undergraduates to emulate
in their homework rather than in the elegant style of the professional
mathematician. - Selected logical symbols (especially==?,\:/, and 3) are used frequently and
consistently. They improve clarity of thought by calling attention to the
presence of formal patterns of thinking that students might not otherwise
recognize. A more complete rationale for the use of logical symbolism can
be found in the "To the Instructor" section that follows.