To the Student xx iiistatements and proofs by calling the reader's attention to the presence of for-
mal logical patterns. In particular, the symbols =} ("implies" ) , 'v' ("for all"), 3
( "there exists"), and ::i ("such that") are used frequently for this purpose. You
are advised to get used to these symbols as soon as possible.
You can review the basic principles of logic in the first two sections of
Appendix A. If you are not familiar with this material, time spent learning it
will repay you r ichly. The third section of Appendix A outlines some common
strategies of proof.
Nearly all significant results of analysis are expressed in terms of sets and/or
functions. The facts you need to know about them are reviewed in Appendix
B , which you can consult as needed.
WORDS OF ADVICE FROM THE AUTHOR:
SEVEN RULES FOR SUCC E S S IN THIS COURSEE lementary real analysis is not an easy subject. In fact, it is one of the most
challenging courses in the undergraduate curriculum. While calculus is one of
the most appli cable areas of mathematics, analysis is highly theoretical in spirit
and makes uncompromising demands for rigor.
This book is student-oriented. It is designed to be readable, and therefore to
be read. It represents my best attempt to make the subject as understandable
as possible without compromising rigor. I offer these words of advice to those
who really want to succeed:
l. Read the book, word-by-word, page-by-page, except where your instruc-
tor may chart an alternative path for you. Do not skip over the reading
and head straight for the exercises, as you might have done in your cal-
culus courses! If you do, you will miss much of the course.- Some of the material is marked with an asterisk, "*·" Let your instructor
decide how much of that to cover. - Study the proofs. Tear them apart and examine them critically until you
are sure that you understand them completely. Ask for help where you do
not understand. No one can claim to understand analysis who does not
understand its theorems and proofs. They serve as models of the kind of
thinking required to develop new results in analysis. Your instructor may
require that you learn some of the proofs well enough to explain them to
your classma tes or to do them on examinations. - Make sure you understand the definitions. Learn (even memorize) them!
This is a far more serious issue than most students realize. Definitions are
the place to start when proving results about a new concept. - Learning mathematics is not a spectator sport! You learri mathematics
by doing mathema tics. You cannot expect to learn analysis by reading