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xviii Preface

tests are discussed in detail, as well as Raabe's test, Dirichlet's test, and Abel's

test. A little more attention is given to the consequences of absolute convergence

than is customary. The chapter concludes with a project that develops the

elementary functions from infinite series.

Chapter 9 focuses on the notion of convergence of sequences and series

of functions. Uniform convergence is contrasted with pointwise convergence,

and the implications of uniform convergence in calculus are explored. By dis-

cussing the notion of convergence of functions in the setting of normed vector

spaces we prepare the student for further courses in modern analysis. The chap-

ter culminates in proofs of two famous results of Weierstrass: the existence of

an everywhere continuous, nowhere differentiable function, and his celebrated

"polynomial approximation" theorem. An historical appreciation of Weierstrass

himself is included. A concluding section points students toward interesting top-

ics for further study.

THE ROLE OF RIGOR

A few words need to be said about the role of rigor in this course. While

intuition is a powerful motivator of progress in mathematics, it is not reliable

as a guarantor of truth. Indeed, many of the results of analysis run counter

to a beginning student's intuition. Some notable examples are the countability

of the rational numbers versus the uncountability of the irrational numbers;

the existence of functions that are continuous on the irrational numbers and

discontinuous on the rational numbers, versus the impossibility of functions that

are continuous on the rational numbers and discontinuous on the irrational

numbers; and the existence of functions that are continuous everywhere and

differentiable nowhere. Contrary to the suspicions of many students, rigor plays

a very practical, even indispensable, role in analysis.

ACKNOWLEDGMENTS

I am very grateful to Millersville University for providing support of various
kinds during the gestation period of this book, but especially for a sabbatical
leave during which the bulk of Chapters 1- 7 were completed. I owe a special
debt to my colleagues in the mathematics department at Millersville for their
encouragement and moral support. Dr. Robert Smith, then Chairman and now
Dean, could not have been more supportive. Special thanks go to Drs. Ximena
Catepillan and Antonia Cardwell, who class-tested the manuscript as it evolved,
and provided valuable critique.
My most heartfelt appreciation goes to the hundreds of students who stud-
ied real analysis in my classes over the years. Their enthusiasm for learning real
analysis and their willingness to work hard toward that end have been a source
of deep satisfaction. Unfortunately, I shall have to leave them unnamed, as to
name a few of them would be an injustice to those not named. I have benefited