Table Of ContentThe Elements
of Real Analysis
Robert G. Bartle
DepartmentofMothematics,Universityof Illinois
John Wiley & Sons, Inc., New York • London • Sydney
Copyright ® 1964byJohnWiley&Sons,Inc.
Allrightsreserved.
Thisbookorany partthereof
mustnot bereproducedinanyform
withoutthe written permission ofthe publisher.
FIRST CORRECTED PRINTING, AUGUST, 1967
Libraryof CongressCatalogCard Number: 64-20061
Printedinthe UnitedStatesofAmerica
To my parents
Preface
There was a time when an undergraduate student of mathematics
was expected to develop technique in solving problems that involved
considerable computation; however, he was not expected to master theo
retical subtleties such as uniform convergence or uniform continuity.
The student was expected to be able to use the Implicit Function Theo
rem, but was not expected to know its hypotheses. The situation has
changed. Now it is generally agreed that it is important for all stu
dents- whether future mathematicians, physicists, engineers, or econ
omists- to grasp the basic theoretical nature of the subject. For, having
done so, they will understand both the power and the limitation of the
general theory and they will be better equipped to devise specific tech
niques to attack particular problems as they arise.
This text has developed from my experience in teaching courses in
elementary real analysis at the University of Illinois since 1955. My
audience has ranged from well-prepared freshman students to graduate
students; the majority in these classes are usually not mathematics
majors. Generally they have taken at least the equivalent of three
semesters of non-rigorous calculus, including multiple integrals, vector
calculus, line integrals, infinite series, and the like.
It would be desirable to have the students take a semester either in
linear or modern algebra before this analysis course, for such a back
ground facilitates the study of rigorous analysis. However, since the
students I encounter do not all have this background, I purposely delay
the study of analysis and first explore the notion of an ordered field
to provide practice in giving proofs. Thus the first six sections of this
text are mostly preparatory in nature; they can be covered in about
three weeks in a normal class and more rapidly in a well-prepared one.
vii
vm PREFACE
It has been my experience that it is possible to discuss Sections 1-13,
15-17.1, 19, and most of 22 in one semester. The entire text provides
about the right amount of material for two semesters at this level.
Most of the topics generally associated with courses in "advanced cal
culus" are treated here in a reasonably sophisticated way. The main
exception is line and surface integrals and the various formulations of
Stokes's Theorem. This topic is not discussed, since an intuitive and
informal development is properly a part of calculus, and a rigorous
treatment is advanced in nature.
The arrangement of this text is indicated by the adjoining diagram.
A solid line in this diagram indicates the dependence on the preceding
section; a dotted line indicates that there is a dependence on one or
two results in the preceding section. All definitions, theorems, corollaries,
PREFACE
and lemmas are numbered consecutively and according to the section
number. Similarly, the exercises and projects are labeled according to
section number. I have given names to some of the more important
theorems whenever I could think of one I felt was reasonably descrip
tive. The proofs, in order to be set off from the rest of the text, start
with the heading PROOF and end with Q.E.D. This deviates somewhat
from the classical tradition, since the conclusion is not always stated
explicitly at the end of the proof.
In writing this text I have drawn from my classroom experience and
have beeninfluenced by many sources which I cannot enumerate. I have
benefited by many conversations with students and colleagues and I am
indebtedto all of them. In particular, I am deeply grateful to Professors
K.W. Anderson, W. G. Bade, and A. L. Peressini for their critical read
ingofthis textinits various manuscript forms; I valued their comments
even when I chose to disregard them. I have enjoyed unusually good
secretarial help and am very grateful to Jelane Huber and Barbara
Beadles, who typed the earlier version, and to Mari Hollos and Julie
Sutter, who typed the later versions of the manuscript. Finally, I am
appreciative of the staff at John Wiley and Sons for their assistance
andcooperation.
ROBERT G. BARTLE
Urbana, Illinois
May 5, 1964
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Chapter Summaries
Introduction: A Glimpse at Set Theory 1
1. The Algebra of Sets, 1
Inclusion, intersection, union, complement, Cartesian product
2. Functions, 11
Transformations, composition of functions, inverse functions,
direct and inverse images
3. Finite and Infinite Sets, 23
Finite, countable, and infinite sets
I. The Real Numbers 27
4. Fields, 28
Definition, elementary property of fields
5. Ordered Fields, 34
Definition, properties of ordered fields, absolute value, intervals,
Archimedean ordered fields, nested intervals
6. The Real Number System, 45
Complete Archimedean fields, suprema and infima, the Su
premum Principle, Dedekind cuts, the Ca.ntor set
II. The Topology of Cartesian Spaces 58
7. Cartesian Spaces, 59
Definition, algebra of vectors, inner product, norm, basic in
equalities
8. Elementary Topological Concepts, 69
Open sets, closed sets, neighborhoods, intervals, The Nested
Intervals Theorem, Bolzano-Weierstrass Theorem, connected
sets
xi
Description:elementary real analysis at the University of Illinois since 1955. My audience has the study of analysis and first explore the notion of an ordered field.
