Table Of ContentUndergraduate Texts in Mathematics
Editors
S. Axler
F. W. Gehring
K.A. Ribet
Springer Science+Business Media, LLC
Undergraduate Texts in Mathematics
Anglin: Mathematics: A Concise History Devlin: The Joy of Sets: Fundamentals
and Philosophy. of Contemporary Set Theory.
Readings in Mathematics. Second edition.
Anglin/Lambek: The Heritage of Dixmier: General Topology.
Thales. Driver: Why Math?
Readings in Mathematics. Ebbinghaus/Flumffhomas:
Apostol: Introduction to Analytic Mathematical Logic. Second edition.
Number Theory. Second edition. Edgar: Measure, Topology, and Fractal
Armstrong: Basic Topology. Geometry.
Armstrong: Groups and Symmetry. Elaydi: Introduction to Difference
Axler: Linear Algebra Done Right. Equations.
Second edition. Exner: An Accompaniment to Higher
Beardon: Limits: A New Approach to Mathematics.
Real Analysis. FineJRosenberger: The Fundamental
BakINewman: Complex Analysis. Theory of Algebra.
Second edition. Fischer: Intermediate Real Analysis.
BanchoffIWermer: Linear Algebra Flanigan/Kazdan: Calculus Two: Linear
Through Geometry. Second edition. and Nonlinear Functions. Second
Berberian: A First Course in Real edition.
Analysis. Fleming: Functions of Several Variables.
Bremaud: An Introduction to Second edition.
Probabilistic Modeling. Foulds: Combinatorial Optimization for
Bressoud: Factorization and Primality Undergraduates.
Testing. Foulds: Optimization Techniques: An
Bressoud: Second Year Calculus. Introduction.
Readings in Mathematics. Franklin: Methods of Mathematical
Brickman: Mathematical Introduction Economics.
to Linear Programming and Game Gordon: Discrete Probability.
Theory. HairerlWanner: Analysis by Its History.
Browder: Mathematical Analysis: Readings in Mathematics.
An Introduction. Halmos: Finite-Dimensional Vector
Buskeslvan Rooij: Topological Spaces: Spaces. Second edition.
From Distance to Neighborhood. Halmos: Naive Set Theory.
Cederberg: A Course in Modem HiimmerlinIHoffmann: Numerical
Geometries. Mathematics.
Childs: A Concrete Introduction to Readings in Mathematics.
Higher Algebra. Second edition. Hijab: Introduction to Calculus and
Chung: Elementary Probability Theory Classical Analysis.
with Stochastic Processes. Third HiitonIHoltonlPedersen: Mathematical
edition. Reflections: In a Room with Many
CoxILittleJO'Shea: Ideals, Varieties, Mirrors.
and Algorithms. Second edition. IoosslJoseph: Elementary Stability
Croom: Basic Concepts of Algebraic and Bifurcation Theory. Second
Topology. edition.
Curtis: Linear Algebra: An Introductory Isaac: The Pleasures of Probability.
Approach. Fourth edition. Readings in Mathematics.
Sterling K. Berberian
A First Course
in Real Analysis
With 19 Illustrations
Springer
Sterling K. Berberian
Department of Mathematics
The University of Texas at Austin
Austin, TX 78712
USA
Editorial Board
S. Axler EW. Gehring K.A. Ribet
Department of Department of Department of
Mathematics Mathematics Mathematics
San Francisco State University University of Michigan University of California
San Francisco, CA 94132 Ann Arbor, MI 48109 at Berkeley
USA USA Berkeley, CA 94720-3840
USA
On the cover: Formula for the fundamental theorem of calculus.
Mathematics Subject Classification (1991): 26-01
Library of Congress Cataloging-in-Publication Data
Berberian, Sterling K., 1926-
A first course in real analysis/Sterling K. Berberian
p. cm. - (Undergraduate texts in mathematics)
lncIudes bibliographical references (p. - ) and index.
ISBN 978-1-4612-6433-0 ISBN 978-1-4419-8548-4 (eBook)
DOI 10.1007/978-1-4419-8548-4
1. Mathematical analysis. 2. Numbers, Real. I. Title.
II. Series.
QA300.B457 1994
S15-dc20 93-46020
Printed on acid-free paper.
© 1994 Springer Science+Business Media New York
Origina11y published by Springer-Verlag New York, lne. in 1994
Softcover reprint of the hardcover Ist edition 1994
Ali rights reserved. This work may not be translated or copied in whole or in part without the
written permission of the publisher (Springer Science+Business Media, LLC except for brief
excerpts in connection with reviews or scholarly analysis. Use in connection with any form
of information storage and retrieval, electronic adaptation, computer software, or by similar
or dissimilar methodology now known or hereafter deve10ped is forbidden.
The use of general descriptive names, trade names, trademarks, etc., in this publication even
if the former are not especially identified, is not to be taken as a sign that such names, as
understood by the Trade Marks and Merchandise Marks Act, may accordingly be used freely
byanyone.
Production managed by Francine McNei11; manufacturing supervised by Genieve Shaw.
Photocomposed copy prepared using the author's TeX files.
9 8 7 6 5 4 3 2 (Corrected seeond printing, 1998)
ISBN 978-1-4612-6433-0 SPIN 10662888
For Carol
Preface
Mathematics is the music of science, and real analysis is the Bach of
mathematics. There are many other foolish things I could say about the
subject of this book, but the foregoing will give the reader an idea of where
my heart lies.
The present book was written to support a first course in real analysis,
normally taken after a year of elementary calculus. Real analysis is, roughly
speaking, the modern setting for Calculus, "real" alluding to the field of
real numbers that underlies it all. At center stage are functions, defined
and taking values in sets of real numbers or in sets (the plane, 3-space,
etc.) readily derived from the real numbers; a first course in real analysis
traditionally places the emphasis on real-valued functions defined on sets
of real numbers.
The agenda for the course: (1) start with the axioms for the field ofreal
numbers, (2) build, in one semester and with appropriate rigor, the foun
dations of calculus (including the "Fundamental Theorem"), and, along
the way, (3) develop those skills and attitudes that enable us to continue
learning mathematics on our own. Three decades of experience with the
exercise have not diminished my astonishment that it can be done.
This paragraph is about prerequisites. The first three sections of the
Appendix summarize the notations and terminology of sets and functions,
and a few principles of elementary logic, that are the language in which
the subject of the book is expressed. Ideally, the student will have met
such matters in a preliminary course (on almost any subject) that provides
some experience in the art of theorem-proving. Nonetheless, proofs are
written out in sufficient detail that the book can also be used for a "first
post-calculus, theorem-proving course"; a couple of days of orientation on
material in the Appendix will be needed, and the going will be slower, at
least at the beginning, but it can be done. Some topics will have to be
sacrificed to accommodate the slower pace; I recommend sacrificing all of
Chapters 10 (Infinite series) and 11 (Beyond the Riemann integral), the
last four sections of Chapter 9, and Theorem 8.5.5 (used only in one of the
sections of Chapter 9 just nominated for omission).
Chapter 11 is an attempt to bring some significant part of the theory of
the Lebesgue integral-a powerful generalization of the integral of elemen
tary calculus-within reach of an undergraduate course in analysis, while
vii
viii Preface
avoiding the sea of technical details that usually accompanies it. It will
not be easy to squeeze this chapter into a one-semester course- perhaps
it should not be tried (I never did in my classes)- but it is good to have
a glimpse of what lies over the horizon. Probably the best use for this
chapter is a self-study project for the student headed for a graduate course
in real analysis; just reading the chapter introduction should be of some
help in navigating the above-mentioned sea of details.
Each section is followed by an exercise set. The division of labor between
text and exercises is straightforward: no proof in the text depends on the
solution of an exercise and, throughout Chapters 1-10, the exercises are
meant to be solvable using concepts and theorems already covered in the
text (the very few exceptions to this rule are clearly signaled). On the
other hand, many of the 'exercises' in Chapter 11 are included primarily
for the reader's information and do require results not covered in the text;
where this is the case, references to other sources are provided.
Some exercises contain extensions and generalizations (for example, the
Riemann-Stieltjes integral and metric space topology) of material in the
text proper; an exceptionally well-prepared class can easily integrate such
topics into the mainstream. Other exercises offer alternative proofs of the
orems proved in the text. Typically, exercises are for 'hands-on' practice,
to strengthen the reader's grasp of the concepts introduced in that section;
the easiest are good candidates for inclusion on a test, which is how many
of them came into being.
Experience persuades me that sequential convergence is the most natural
way to introduce limits. This form of the limit concept has much intu
itive support (decimal expansions, the half-the-distance-to-the-wall story,
etc.), but 1 acknowledge that the decision is subjective and that 1 have not
always taught the course that way. The equivalent epsilon-delta and 'neigh
borhood' formulations of limits (superior to sequences in some situations)
are also discussed fully.
My observation is that the introductory course in analysis, more than
any other in the undergraduate mathematics curriculum, is a course that
makes a difference. The right course at the right time, distilling millennia
of human reflection into a brief semester's time, it is a wonderful learning
machine whose output is, no less, Mathematicians.
Austin, Texas Sterling Berberian
Summer, 1993
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . vii
CHAPTER 1
Axioms for the Field IR of Real Numbers 1
§ 1.1. The field axioms . . . . . 1
§1.2. The order axioms . . . . . . . . . . . 4
§1.3. Bounded sets, LUB and GLB . . . . . . 8
§1.4. The completeness axiom (existence of LUB's) .11
CHAPTER 2
First Properties of IR . . . . . . . . . . . . . . . . . 15
§2.1. Dual of the completeness axiom (existence of GLB's) 15
§2.2. Archimedean property . · 16
§2.3. Bracket function · 19
§2.4. Density of the rationals · 20
§2.5. Monotone sequences . . · 21
§2.6. Theorem on nested intervals · 23
§2.7. Dedekind cut property . 26
§2.8. Square roots . · 28
§2.9. Absolute value . . . .30
CHAPTER 3
Sequences of Real Numbers, Convergence .. 33
§3.1. Bounded sequences · 33
§3.2. Ultimately, frequently · 35
§3.3. Null sequences .36
§3.4. Convergent sequences · 39
§3.5. Subsequences, Weierstrass-Bolzano theorem · 43
§3.6. Cauchy's criterion for convergence. . . .48
§3.7. limsup and liminf of a bounded sequence . . · 50
ix
x Contents
CHAPTER 4
Special Subsets of lR. ... 56
§4.1. Intervals . . . .56
§4.2. Closed sets . . .61
§4.3. Open sets, neighborhoods .65
§4.4. Finite and infinite sets . . .71
§4.5. Heine-Borel covering theorem . 75
CHAPTER 5
Continuity . . . . . . . . . . . . . . .. 80
§5.1. Functions, direct images, inverse images .80
§5.2. Continuity at a point .84
§5.3. Algebra of continuity .88
§5.4. Continuous functions .90
§5.5. One-sided continuity. . 92
§5.6. Composition . . . . .96
CHAPTER 6
Continuous Functions on an Interval .. 98
§6.1. Intermediate value theorem . . . .98
§6.2. n'th roots 100
§6.3. Continuous functions on a closed interval . 101
§6.4. Monotonic continuous functions 103
§6.5. Inverse function theorem 105
§6.6. Uniform continuity . . . . . 106
CHAPTER 7
Limits of Functions 110
§7.1. Deleted neighborhoods 110
§7.2. Limits . . . . . . . 112
§7.3. Limits and continuity 115
§7.4. E,8 characterization of limits 117
§7.5. Algebra of limits . . . . . 118
CHAPTER 8
Derivatives . 121
§8.1. Differentiability . . . 121
§8.2. Algebra of derivatives 124
§8.3. Composition (Chain Rule) 126
§8.4. Local max and min 128
§8.5. Mean value theorem . . . 129
