introduction to real analysis third edition robert g bartle and donald r sherbert
Introduction to Real Analysis
INTRODUCTION TO REAL ANALYSIS
Third Edition
Robert G. Bartle
Donald R. Sherbert
Easten Michigan University, Ypsilanti
University of Illinois, Urbana-Champaign
John Wiley & Sons, Inc.
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Libray of Congress Caaloging in ublication Daa:
Bartle, Robert Gardner, 1927Introduction to real analysis / Robert G . Bartle, Donald R., Sherbert. - 3rd ed.
cm.
p.
Includes bibliographical references and index.
ISBN 0-47 1-32148-6 (a1k. paper)
1. Mathematical analysis. 2. Functions of real variables.
1. Sherbert, Donald R., 1935- . II. Title.
QA300.B294
515-dc21
2000
99-13829
CIP
A. M. S. Classiication 26-01
Printed in the United States of America
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II
PREFACE
The study of real analysis is indispensible for a prospective graduate student of pure or
applied mathematics. It also has great value for any undergraduate student who wishes
to go beyond the routine manipulations of formulas to solve standard problems, because
it develops the ability to think deductively, analyze mathematical situations, and extend
ideas to a new context. n recent years, mathematics has become valuable in many areas,
including economics and management science as well as the physical sciences, engineering,
and computer science. Our goal is to provide an accessible, reasonably paced textbook in
the undamental concepts and techniques of real analysis for students in these areas. This
book is designed for students who have studied calculus as it is raditionally presented in
the United States. While students ind this book challenging, our experience is that serious
students at this level are fully capable of mastering the material presented here.
The irst two editions of this book were very well received, and we have taken pains
to maintain the same spirit and user-friendly approach. In preparing this edition, we have
examined every section and set of exercises, sreamlined some arguments, provided a few
new examples, moved certain topics to new locations, and made revisions. Except for the
new Chapter 10, which deals with the generalized Riemann integral, we have not added
much new material. While there is more materil than can be covered in one semester,
instructors may wish to use certain topics as honors projects or extra credit assignments.
It is desirable that the student have had some exposure to proofs, but we do not assume
that to be the case. To provide some help for students in analyzing proofs of theorems,
we include an appendix on "Logic and Proofs" that discusses topics such as implications,
quantifiers, negations, contrapositives, and diferent types of proofs. We have kept the
discussion informal to avoid becoming mired in the technical details of formal logic. We
feel that it is a more useful experience to len how to construct proofs by irst watching
and then doing than by reading about techniques of proof.
We have adopted a medium level of generality consistently throughout the book: we
present results that are general enough to cover cases that actually arise, but we do not strive
for maximum generality. In the main, we proceed from the particular to the general. Thus
we consider continuous functions on open and closed intervals in detail, but we are careul
to present proofs that can readily be adapted to a more general situation. (In Chapter 1 1
we take particular advantage of the approach.) We believe that it is important to provide
students with many examples to aid them in their understanding, and we have compiled
rather extensive lists of exercises to challenge them. While we do leave routine proofs as
exercises, we do not try to attain brevity by relegating diicult proofs to the exercises.
owever, in some of the later sections, we do break down a moderately difficult exercise
into a sequence of steps.
In Chapter 1 we present a brief summary of the notions and notations for sets and
functions that we use. A discussion of Mathematical Induction is also given, since inductive
proofs arise frequently. We also include a short section on finite, countable and infinite sets.
We recommend that this chapter be covered quickly, or used as background material,
retuning later as necessary.
v
vi
PREFACE
Chapter 2 presents the properties of the real number system . The irst two sections
deal with the Algebraic and Order Properties and provide some practice in writing proofs
of elementry results. The crucial Completeness Property is given in Section 2.3 as the
Supremum Property, and its ramifications are discussed throughout the remainder of this
chapter.
In Chapter 3 we give a thorough treatment of sequences in R and the associated limit
concepts. The material is of the greatest importance; fortunately, students find it rather
natural although it takes some time for them to become fully accustomed to the use of €.
In the new Section 3.7, we give a brief introduction to infinite series, so that this important
topic will not be omitted due to a shortage of time.
Chapter 4 on limits of functions and Chapter 5 on continuous functions constitute
the heart of the book. Our discussion of limits and continuity relies heavily on the use of
sequences, and the closely parallel approach of these chapters reinforces the understanding
of these essential topics. The fundamental properties of continuous functions (on intervals)
are discussed in Section 5.3 and 5.4. The notion of a "gauge" is introduced in Section 5.5
and used to give altenative proofs of these properties. Monotone functions are discussed
in Section 5.6.
The basic theory of the derivative is given in the first part of Chapter 6. This important
material is standard, except that we have used a result of Caratheodory to give simpler
proofs of the Chain Rule and the Inversion Theorem. The remainder of this chapter consists
of applications of the Mean Value Theorem and may be explored as time permits.
Chapter 7, dealing with the Riemann integral, has been completely revised in this
edition. Rather than introducing upper and lower integrals (as we did in the previous
editions), we here define the integral as a limit of Riemann sums. This has the advantage that
it is consistent with the students ' irst exposure to the integral in calculus and in applications;
since it is not dependent on order properties, it permits immediate generalization to complex
and vector-valued functions that students may encounter in later courses. Contrary to
popular opinion, this limit approach is no more dificult than the order approach. It also is
consistent with the generalized Riemann integral that is discussed in detail in Chapter 10.
Section 7.4 gives a brief discussion of the familiar numerical methods of calculating the
integral of continuous functions.
Sequences of functions and uniform convergence are discussed in the irst two sec
tions of Chapter 8, and the basic transcendental functions are put on a firm foundation in
Section 8.3 and 8.4 by using uniform convergence. Chapter 9 completes our discussion of
ininite series. Chapters 8 and 9 are intrinsically important, and they also show how the
material in the earlier chapters can be applied.
Chapter 10 is completely new; it is a presentation of the generalized Riemann integral
(sometimes called the "Henstock-Kurzweil" or the "gauge" integral). It will be new to many
readers, and we think they will be amazed that such an apparently minor modification of
the deinition of the Riemann integral can lead to an integral that is more general than the
Lebesgue integral. We believe that this relatively new approach to integration theory is both
accessible and exciting to anyone who has studied the basic Riemann integral.
The final Chapter 1 1 deals with topological concepts. Earlier proofs given for intervals
are extended to a more abstract setting. For example, the concept of compactness is given
proper emphasis and metric spaces are introduced. This chapter will be very useful for
students continuing to graduate courses in mathematics.
Throughout the book we have paid more attention to topics from numerical analysis
and approximation theory than is usual. We have done so because of the importance of
these areas, and to show that real analysis is not merely an exercise in abstract thought.
PREFACE
vii
We have provided rather lengthy lists of exercises, some easy and some challenging.
We have provided "hints" for many of these exercises, to help students get started toward a
solution or to check their "answer". More complete solutions of almost every exercise are
given in a separate Instructor' s Manual, which is available to teachers upon request to the
publisher.
It is a satisfying experience to see how the mathematical maturity of the students
increases and how the students gradually len to work comfortably with concepts that
initially seemed so mysterious. But there is no doubt that a lot of hard work is required on
the part of both the students and the teachers.
In order to enrich the historical perspective of he book, we include brief biographical
sketches of some famous mathematicians who contributed to this area. We are particularly
indebted to Dr. Patrick Muldowney for providing us with his photograph of Professors
Henstock and Kurzweil. We also thank John Wiley & Sons for obtaining photographs of
the other mathematicians.
We have received many helpful comments from colleagues at a wide variety of in
stitutions who have taught from earlier editions and liked the book enough to express
their opinions about how to improve it. We appreciate their remarks and suggestions, even
though we did not always follow their advice. We thank them for communicating with us
and wish them well in their endeavors to impart the challenge and excitement of lening
real analysis and "real" mathematics. It is our hope that they will ind this new edition even
more helpful than the earlier ones.
Februay 24, 1999
Ypsilanti and Urbana
Robert G. Bartle
Donald R. Sherbert
THE GREEK ALPHABET
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To our wives, Carolyn and Janice,
with our appreciation for their
patience, support, and love.
CONTENTS
CHAPTER 1
PRELIMINARIES 1
CHAPTER 2
THE REAL NUMBERS 22
CHAPTER 3
CHAPTER 4
C.HAPTER 5
1 . 1 Sets and Functions· 1
1.2 Mathematical Induction 12
1 .3 Finite and Ininite Sets 16
2.1
2.2
2.3
2.4
2.5
The Algebraic and Order Properties of R
Absolute Value and Real Line 31
The Completeness Property of R 34
Applications of the Supremum Property
Intervals 44
22
38
SEQUENCES AND SERIES 52
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Sequences and Their Limits 53
Limit Theorems 60
Monotone Sequences 68
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion 80
Properly Divergent Sequences 86
Introduction to Ininite Series 89
75
LIMITS 96
4. 1 Limits of Functions 97
4.2 Limit Theorems 105
4.3 Some Extensions of the Limit Concept
111
CONTINUOUS FUNCTIONS 119
5.1
5.2
5.3
5.4
5.5
5.6
Continuous Functions 120
Combinations of Continuous Functions
Continuous Functions on Intervals 129
Uniform Continuity 136
Continuity and Gauges 145
Monotone and Inverse Functions 149
125
ix
x
CONTENTS
CHAPTER 6
CHAPTER 7
CHAPTER 8
CHAPTER 9
DIFFERENTIATION 157
6.1
6.2
6.3
6.4
The Derivative 158
The Mean Value Theorem
L'Hospital's Rules 176
Taylor' s Theorem 183
168
TE RIEMANN INTEGRAL 193
7.1
7.2
7.3
7.4
The Riemann Integral 194
Riemann Integrable Functions 202
The Fundamental Theorem 210
Approximate Integration 219
SEQUENCES OF FUNCTIONS 227
8.1
8.2
8.3
8.4
Pointwise and Uniform Convergence 227
Interchange of Limits 233
The Exponential and Logarithmic Functions
The Trigonometric Functions 246
239
INFINITE SERIES 253
9.1
9.2
9.3
9.4
Absolute Convergence 253
Tests for Absolute Convergence 257
Tests for Nonabsolute Convergence 263
Series of Functions 266
CHAPTER 10
THE GENERALIZED RIEMANN INTEGRAL 274
CHAPTER 11
A GLIMPSE INTO TOPOLOGY 312
10.1
10.2
10.3
10.4
1 1. 1
1 1.2
1 1.3
1 1.4
Definition and Main Properties 275
Improper and Lebesgue Integrals 287
Infinite Intervals 294
Convergence Theorems 301
Open and Closed Sets in R
Compact Sets 319
Continuous Functions 323
Metric Spaces 327
312
APPENDIX A
LOGIC AND PROOFS 334
APPENDIX B
FINITE AND COUNTABLE SETS 343
APPENDIX C
THE RIEMANN AND LEBESGUE CRITERIA 347
CONTENTS
APPENDIX D
APPROXIMATE INTEGRATION 351
APPENDIX E
TWO EXAMPLES 354
REFERENCES 357
PHOTO CREDITS 358
HINTS FOR SELECTED EXERCISES 359
INDEX 381
xi
CHAPTER 1
PRELIMINARIES
In this initial chapter we will present the background needed for the study of real analysis.
Section 1.1 consists of a brief survey of set operations and functions, two vital tools for all
of mathematics. In it we establish the notation and state the basic deinitions and properties
that will be used throughout the book. We will regard the word "set" as synonymous with
the words "class", "collection", and "family", and we will not deine these terms or give a
list of axioms for set theory. This approach, often referred to as "naive" set theory, is quite
adequate for working with sets in the context of real analysis.
Section 1 .2 is concened with a special method of proof called Mathematical Induction.
t is related to the fundamental properties of the natural number system and, though it is
restricted to proving particular types of statements, it is important and used frequently. An
informal discussion of the diferent types of proofs that are used in mathematics, such as
contrapositives and proofs by contradiction, can be found in Appendix A.
In Section 1 .3 we apply some of the tools presented in the irst two sections of this
chapter to a discussion of what it means for a set to be inite or ininite. Creful deinitions
are given and some basic consequences of these deinitions are derived. The important
result that the set of rational numbers is countably ininite is established.
In addition to introducing basic concepts and establishing terminology and notation,
this chapter also provides the reader with some initial experience in working with precise
deinitions and writing proofs. The careul study of real analysis unavoidably entails the
reading and writing of proofs, and like any skill, it is necessary to practice. This chapter is
a starting point.
Section 1.1
Sets and Functions
In this section we give a brief review of the terminology and notation that
will be used in this text. We suggest that you look through quickly and come back later
when you need to recall the meaning of a term or a symbol.
If an element is in a set A, we write
To the reader:
x
and say that is a member of A, or that
x
X EA
x
x
belongs to A . If x is
¢ A.
not in A, we write
If�very element of a set A also belongs to a set B , we say that A is a subset of B and write
or
We say that a set A is a proper subset of a set B if A � B, but there is at least one element
of B that is not in A. In this case we sometimes write
A C B.
1
2
CHAPTER 1
PRELINARlES
1.1.1 Deinition wo sets A and B are said to be equal. and we write A = B. if they
contain the same elements.
Thus. to prove hat the sets A and B are equal. we must show that
A � B and B � A.
A set is normally deined by either listing its elements explicitly. or by specifying a
property that determines he elements of the set. If P denotes a property that is meaningful
and unambiguous for elements of a set S. then we write
{x E S P(x)}
:
for the set of all elements x in S for which the property P is true. If the set S is understood
rom the context. then it is oten omitted in this notation.
Several special sets are used throughout this book. and they are denoted by standard
symbols. le will use the symbol : = to mean that the symbol on the left is being deined
by the symbol on the right.)
The set of natural numbers N := {I. 2. 3 }.
The set of integers Z : = to. 1. -1.2, -2, · · . },
The set ofrational numbers Q : = {min : m, n E Z and n - OJ.
The set of real numbers R
The set R of real numbers is of fundamental importance for us and will be discussed
at length in Chapter 2.
•
•
. . .
•
•
•
1.1.2 Examples (a)
The set
{x E N : x2 - 3x + 2 = O}
consists of those natural numbers satisfying the stated equation. Since the only solutions of
this quadratic equation re x = 1 and x = 2, we can denote this set more simply by {I, 2}.
(b) A natural number n is even if it has the form n = 2k for some k E N. The set of even
natural numbers can be written
{2k : k E N},
which is less cumbersome than {n E N : n = 2k, k E N}. Similarly, the set of odd natural
numbers can be written
{2k - 1 k E N}.
:
o
Set Operations
We now deine the methods of obtaining new sets from given ones. Note that these set
operations are based on the meaning of the words "or", "and", and "not". For the union,
it is important to be aware of the fact that the word "or" is used in the inclusive sense,
allowing the possibility that x may belong to both sets. In legal terminology, this inclusive
sense is sometimes indicated by "andlor".
1.1.3 Deinition
(a)
The union of sets A and B is the set
AU B := {x : x E A or x E B} .
1.1 SETS AND FUNCTIONS
3
(b) The intersection of the sets A and B is the set
AnB
:=
{x : x E A and x E B} .
(c) The complement of B relative to A is the set
A\B
A U B ID
:=
{x : x E A and x
(a) A U B
Fiure 1.1.1
t
(b) A n B
B} .
(c) A\B
A\B �
The set that has no elements is called the empty set and is denoted by the symbol 0.
Two sets A and B are said to be disjoint if they have no elements in common; this can be
expressed by writing A n B = 0.
To illustrate the method of proving set equalities, we will next establish one of the
for three sets. The proof of the other one is let as an exercise.
DeMorgan laws
1.1.4 Theorem
f A, B, C
re sets, then
(a) A\(B U C) = (A\B) n (A\C),
(b) A\(B n C) = (A\B) U (A\C) .
To prove (a), we will show that every element in A\ (B U C) is contained in both
(A\B) and (A \C), and conversely.
If x is in A\(B U C), then x is in A, but x is not in B U C . Hence x is in A, but x
is neither in B nor in C . Therefore, x is in A but not B, and x is in A but not C . Thus,
x E A\B and x E A\C, which shows that x E (A \B) n (A\C).
Conversely, if x E (A\B) n (A\C), then x E (A\B) and x E (A\C). Hence x E A
and both x t B and x t C . Therefore, x E A and x t (B U C), so that x E A\ (B U C) .
Since the sets (A\B) n (A\ C) and A\ (B U C) contain the same elements, they are
Proo.
equal by Deinition
1.1.1.
Q.E.D.
There are times when it is desirable to form unions and intersections of more than two
sets. For a finite collection of sets {A I ' A 2 , .. . , An }, their union is the set A consisting of
all elements that belong to
of the sets Ak , and their intersection consists of all
el�ments that belong to of the sets Ak •
This is extended to an ininite collection of sets {A I' A2 , ••• , An ' . . .} as follows. Their
union is the set of elements that belong to
of the sets An ' In this case we
write
all
at least one
at least one
0
U An
n =1
:=
{x : x E A n for some n
E N} .
CHTER 1
4
PRELIMNARIES
Similarly, their intersection is the set of elements that belong to all of these sets An' In this
case we write
n An := {x : x E An for all n E N} .
0
n=l
Cartesian Products
In order to discuss unctions, we deine the Cartesian product of two sets.
1.1.5 Deinition If A and B are nonempty sets, then the Cartesian product A x B of A
and B is the set of all ordered pairs (a, b) with a E A and b E B. That is,
A
x
B
:=
{(a, b) : a E A, bE B} .
Thus if A = {l, 2, 3} and B = {I, 5}, then the set A x B is the set whose elements are
the ordered pairs
(1, 1), (1, 5), (2, 1), (2, 5), (3, 1), (3, 5).
We may visualize the set A x B as the set of six points in the plane with the coordinates
that we have just listed.
We oten draw a diagram (such as Figure 1.1.2) to indicate the Cartesian product of
two sets A and B . However, it should be realized that this diagram may be a simplification.
For example, if A := {x E R : 1 ::: x ::: 2} and B : = {y E R : 0 :::y :::1 or 2 :::y :::3},
then instead of a rectangle, we should have a drawing such as Figure 1.1 . 3 .
We will now discuss the undamental notion of afunction or a mapping.
To the mathematician of the early nineteenth century, the word "function" meant a
deinite formula, such as f (x) := x2 + 3x - 5, which associates to each real number x
another number f(x) . (Here, f(O) = -5, f(1) = - 1, f(5) = 35 .) This understanding
excluded the case of diferent formulas on diferent intervals, so that unctions could not
be deined "in pieces".
AxB
B
b
------
3
2
AxB
1 (a , b)
I
I
I
I
I
a
A
Figure 1.1.2
2
Figure 1.1.3
1.1 SETS AD UNCTIONS
5
As mathematics developed, it became clear that a more general definition of "function"
would be useful. It also became evident that it is important to make a clear distinction
between the function itself and the values of the function. A revised definition might be:
A function
f from a set A into a set B is a rule of correspondence that assigns to
x in A a uniquely determined element f (x) in B .
each element
But however suggestive this revised deinition might be, there is the difficulty o f interpreting
the phrase "rule of correspondence". In order to clarify this, we will express the definition
entirely in terms of sets; in efect, we will define a function to be its
graph. W hile this has
the disadvantage of being somewhat artificial, it has the advantage of being unambiguous
and clearer.
1.1.6 Deinition
pairs in
A
x
B
other words, if
A and B be sets. Then a function from A to B is a set f of ordered
E A there exists a unique E B with
E f. (In
E f and
E f, then
Let
a
(a, b')
such that for each
(a, b)
b
b = b'.)
(a, b)
A of first elements of a function f is called the domain of f and is often
D(f). The set of all second elements in f is called the range of f and is
denoted by R(f). Note that, although D(f)
A, we only have R(f) � B. (See
The set
denoted by
often
=
Figure 1.1.4.)
The essential condition that:
(a, b') E f implies that b= b'
is sometimes called the vertical line test. In geometrical terms it says every vertical line
x = a with a E A intersects the graph of f exactly once.
(a, b) E f
and
The notation
f : A;B
f i s a function from A
mapping of A into B, or that f maps A into B . If
is often used to indicate that
into
B.
W e will also say that
f is a
(a, b) is an element in f, it is customary
to write
b = f(a)
a b.
o r sometimes
+
;
r�IE
-�
_____
Figure 1.1.4
A=DV)
�.1
______
a
A unction as a graph
6
If
CHAPTER 1
PRELIMINARIES
b = f(a), we often refer to b as the value of f at a, or as the image of a under f.
ransformations and Machines
tansformation
(a, b) f,
1 . 1.5,
Aside from using graphs, we can visualize a function as a
of the set DU) =
A into the set RU) � B. In this phraseology, when
E
we think of as taking
the element from A and "ransforming" or "mapping" it into an element =
in
RU) � B . We often draw a diagram, such as Figure
even when the sets A and B are
not subsets of the plane.
a
b =!(a)
Figue 1.1.5
f
b f(a)
Rf)
A unction as a transfonnation
machine
There is another way of visualizing a function: namely, as a
that accepts
elements of DU) = A as
and produces corresponding elements of RU) � B as
f we take an element x E D U) and put it into then out comes the corresponding
value
If we put a diferent element y E DU) into then out comes ey ) which may
or may not difer rom
If we ry to insert something that does not belong to DU)
into we find that it is not accepted, for can operate only on elements from DU). (See
Figure
This last visualization makes clear the distinction between and
the first is the
machine itself, and the second is the output of the machine when is the input. Whereas
no one is likely to confuse a meat grinder with ground meat, enough people have confused
functions with their values that it is worth distinguishing between them notationally.
outputs.
f(x).
f,
1.1.6.)
inputs
f(x).
f,
f,
f
f
f
x
t
!
!(x )
Figue 1.1.6
A function as a machine
f
x
f (x):
1.1 SETS AND UNCTIONS
7
Direct and Inverse Images
Let
f:A
�
DC!) = A and range RC!) � B.
If E is a subset of A, then the direct image of E under f is the subset
B be a function with domain
1.1.7 Deinition
f(E) of B given by
f(E) := {f(x) : x E E } .
,
If H is a subset of B, then the inverse image of H under f is the subset f - (H) of A
given by
f-' (H) := {x E A : f(x) E H } .
f-' (H)
Remark The notation
used in this connection has its disadvantages. However,
we will use it since it is the standard notation.
E
y,
f - ' (H)
f(E)
2 f(x2 )
Thus, if we are given a set � A, then a point E B is in the direct image
Similarly, given
if and only if there exists at least one point E such that Y, =
a set � B, then a point is in the inverse image
if and only if Y : =
belongs to
(See Figure 1 . 1 .7.)
H
x2
H.
x, E
f(x ,).
x2 • Then the direct image
of the set E := {x : 0 � x � 2} is the set f(E) = {y : 0 � Y � 4}.
If G := {y : 0 � Y � 4}, then the inverse image of G is the set f -' (G) = {x : -2 �
x � 2}. Thus, in this case, we see that f - ' C!(E» = E.
On the other hand, we have f (J - '(G ») = G. But if H : = {y : -1 � Y � I}, then
we have f (J - ' (H») = {y : 0 � y � I } = H.
A sketch of the graph of f may help to visualize these sets.
(b) Let f : A
B, and let G, H be subsets of B . We will show that
f- ' (G n H) � f - ' (G) n f-' (H).
For, if x E f - ' (G n H), then f(x) E G n H, so that f(x) E G and f(x) E H. But this
implies that x E f - ' (G) and x E f - ' (H), whence x E f -' (G) n f - ' (H). Thus the stated
1.1.8 Examples (a) Let f : R � R be deined by f(x)
:=
�
implication is proved. [The opposite inclusion is also true, so that we actually have set
0
equality between these sets; see Exercise 13.]
Further facts about direct and inverse images are given in the exercises.
E
�
f
H
Figue 1.1.7
Drct nd inverse images
8
CHAPTER 1
PRELIMINARIES
Special ypes of Functions
The following deinitions identify some very important types of functions.
1.1.9 Deinition Let I : A + B be a unction from A to B .
(a) The function I i s said to be ijective (or to b e one-one) if whenever x , = x2 , then
I (x,) = l (x2 ) · f I is an injective function, we also say that I is an ijection.
(b) The function I is said to be surjective (or to map A onto B) if I (A) = B; that is, if
the range
= B. If I is a surjective function, we also say that I is a surjection.
(c) If I is both injective and sujective, then I is said to be bijective. If I is bijective, we
also say that I is a bijection.
R(f)
•
In order to prove that a function I is injective, we must establish that:
for all x" x2 in A , if I (x,) = l (x2 ) , then x, = x2•
To do this we assume that I (x,) = l (x2) and show that x, = x2 .
[In other words, the graph of I satisies the
line y = with E B intersects the graph I in
one point.]
To prove that a function I is sujective, we must show that for any
least one x E A such that I (x) =
[In other words, the graph of I satisies the
line y = with E B intersects the graph I in
one point.]
b
•
irst horizontal line test: Every horizontal
at most
b E B there exists at
second horizontal line test: Every horizontal
at least
b
b.
b
b
1.1.10 Example Let A := {x E R : x = l } and deine / (x) := 2x/(x -l) for all x A.
To show that I is injective, we take x, and x2 in A and assume that I (x,) = l (x2). Thus
we have
2x2
2x,
- = -,
x, x2 -
E
1
1
which implies that x, (x2 - 1) = x2 (x, - 1), and hence x, = x2 . Therefore I is injective.
To determine the range of I, we solve the equation y = 2x/(x - for x in tenus of
y. We obtain x = y / (y - 2), which is meaningful for y = 2. Thus the range of I is the set
0
B : = { y E R : y = 2}. Thus, I is a bijection of A onto B.
1)
Inverse Functions
If I is a function from A into B, then I is a special subset of A x B (namely, one passing
the
The set of ordered pairs in B x A obtained by interchanging the
members of ordered pairs in I is not generally a function. (That is, the set I may not pass
of the
However, if I is a bijection, then this interchange does
lead to a function, called the "inverse function" of I.
vertical line test.)
both
horizontal line tests.)
1.1.11 Deinition If I : A + B is a bijection of A onto B, then
g
:=
feb, a) E B
x
A:
(a, b) E f}
is a function on B into A. This function is called the inverse function of I, and is denoted
by 1-'. The function 1-' is also called the inverse of I .
1 . 1 SETS AND UNCTIONS
9
I and its inverse I - I by noting that
I
R(f) D(f - )
b = I(a) if and only if a = I - I (b).
For example, we saw in Example 1.1.10 that the function
I(x) : = x 2x- I
is a bijection of A := {x E R : x = I} onto the set B := {y E R : y = 2}. The function
inverse to I is given by
I - I (y) := y--y 2 for y E B.
I
Remark We introduced the notation I - (H) in Deinition 1.1.7. It makes sense even if
I does not have an inverse unction. However, if the inverse function I - I does exist, then
I- I (H) is the direct image of the set H � B under I- I .
We can also express the connection between
=
and
=
and that
D(f) R(f- I )
Composition of Functions
I, g
I (x)
g (f (x));
I (x)
g.
all I(x),
I
g.
1.1.8.)
1.1.12 Deinition If I A
B and g : B
C, and if R(f) � D(g) = B, then the
composite function g o I (note the orderl) is the function from A into C deined by
(g 0 f)(x) := g(f(x)) for ll x E A .
1.1.13 Examples (a) The order of the composition must be careully noted. For, let I
and g be the functions whose values at x E R are given by
I(x) := 2x and g(x) := 3x 2 1.
Since D(g) = R and R(f) � R = D(g), then the domain D(g 0 f) is also equal to R , and
the composite function g 0 I is given by
(g 0 f)(x) = 3 (2x) 2 1 = 12x 2 1.
It often happens that we want to "compose" two functions
by first inding
and
then applying to get
however, this is possible only when
belongs to the
domain of In order to be able to do this for
we must assume that the range of
is contained in the domain of (See Figure
g
:
�
�
-
-
A
-
B
I
�
gol
Fiue 1.1.8
he composition of f and g
c
10
CHAPTER 1
PRELIMINARIES
l o g is also R, but
(f g)(x) = 2(3x 2 - 1) = 6x 2 2.
Thus, in this case, we have g I = l o g.
(b) In considering g I, some care must be exercised to be sure that the range of 1 is
contained in the domain of g. For example, if
I(x) := l - x 2 and g(x ) : = X,
then, since D(g) = {x : x � O}, the composite unction g o I is given by the formula
(g f)(x) = 7
only for x E D(f) that satisfy I(x) � 0; that is, for x satisfying - 1 x 1 .
We note that if we reverse the order, then the composition l o g is given by the fomula
(f o g)(x) = l - x,
but only for those x in the domain D(g) = {x : x � O}.
o
On the other hand, the domain of the composite function
0
-
0
0
0
�
�
We now give the relationship between composite functions and inverse images. The
proof is left as an instructive exercise.
1.1.14 Theorem
C. Then we have
Note the
Let
I:A
+
B nd
g : B C be functions nd let H be a subset of
+
reversal in the order of the functions.
Restrictions of Functions
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
: A + B is a function and if A l A, we can define a function II : A l + B by
II (x) := I(x) for x E A I •
The function II is called the restriction of I to A I . Sometimes it is denoted by II = I I A I'
It may seem strange to the reader that one would ever choose to throw away a pt of a
function, but there are some good reasons for doing so. For example, if I : R + R is the
C
If I
squaring function:
for
I
x E R,
I
then is not injective, so it cannot have an inverse function. However, if we restrict to
is a bijection of onto
O}, then the restriction
the set
Therefore,
this restriction has an inverse function, which is the positive square root function. (Sketch
a graph.)
Similarly, the trigonomeric functions
sin and
cos are not injective
on all of R However, by making suitable restrictions of these unctions, one can obtain
the inverse sine and the inverse cosine functions that the reader has undoubtedly already
encountered.
A l := {x : x �
IIA I
S(x) := x
Al
C(x) :=
A I'
x
1.1 SETS AD FUNCTIONS
11
Exercises for Section 1.1
1. If A and B are sets, show that A � B if and only if An B = A.
2. Prove the second De Morgan Law [Theorem 1.1.4(b)].
3. Prove the Distributive Laws:
(a) A n (B U C) = (An B) U (An C),
(b) AU (Bn C) = (A U B) n (A U C).
4. The symmetric diference of two sets A and B is the set D of all elements hat belong to either
A or B but not both. Represent D with a diagram.
(a) Show that D = (A\B) U (B\A) .
(b) Show that D is also given by D = (A U B)\(An B).
5. For each n E N, let An = {en + I)k : k E N}.
(a) What is A n Az?
(b) Determinel the sets UlAn : n E N} and n{An : n E N}.
6. Draw diagrams in the plane of the Cartesian products A x B for the given sets A and B.
(a) A = {x E R : 1::x::2 or 3::x::4}, B = {x E R : x = 1 or x = 2}.
b) A = { I , 2, 3}, B = {x E R : 1:: x::3}.
7. Let A := B := {x E R : -1::x:: I} and consider the subset C := {(x, y) : xZ + l = I } of
A x B. Is this set a unction? Explain.
8. Let f(x) := I/xz, x . 0, X E R.
(a) Determine the direct image fee) where E := {x E R : 1::x::2}.
(b) Determine the inverse image f-I(G) where G := {x E R : 1::x::4}.
9. Let g(x) := XZ and f(x) := x + 2 for x E R, and let h be the composite unction h := g 0 f.
(a) Find the direct image h(E) of E := {x E R : 0::x:: I }.
b) Find the inverse image h-I (G) of G := {x E R : 0::x :: 4}.
10. Let f(x) := XZ for x E R, and let E : = {x E R : -1::x::O} and F := {x E R : 0::x:: l}.
Show that En F = {O} and feEn F) = {O}, while fee) = f(F) = { y E R : 0::y:: I }.
Hence feEn F) is a proper subset of fee) n f(F). What happens if 0 is deleted from the sets
E and F?
11. Let f and E. F be as in Exercise 10. Find the sets E\F and f(E)\f(F) and show that it is not
true that f(E\F) � f(E)\f(F).
12. Show that if f : A � B and E, F are subsets of A, then feE U F) = fee) U f(F) and
feE n F) � fee) n f(F).
13. Show that if f : A � B and G, H are subsets of B, hen rl (G U H) = f-I(G) U f-I(H)
and rl(Gn H) = rl (G) n rl (H).
14. Show that the unction f deined by f(x) := x/VxZ + 1, x E R, is a bijection of R onto
{y : -1 < y < I}.
15. For a, b E R with a < b, ind an explicit bijection of A := {x : a < x < b} onto B := {y : 0 <
y < I}.
16. Give an example of two functions f, g on R to R such that f . g, but such that fog = g 0 f.
17. (a) Show that if f : A � B is injective and E � A, then f-I(f(E» = E. Give an example
to show that equality need not hold if f is not injective.
(b) Show that if f : A � B is sujective and H � B, then f(rl (H» = H. Give an example
to show that equality need not hold if f is not surjective.
18. (a) Suppose that f is an injection . Show that f-I 0 f(x) = x for all x E D(f) and that
fori (y) = y for all y E R(f) .
(b) If f is a bijection of A onto B, show that f-I is a bijection of B onto A.
CHAPTER 1
12
PRELMINARIES
19. Prove that if I : A � B is bijective and g : B � C is bijective, then the composite g 0 I is a
bijective map of A onto C.
20. Let I : A � B and g : B � C be functions.
(a) Show that if g 0 I is injective, then I is injective.
(b) Show that if g o I is surjective, then g is sujective.
21. Prove Theorem 1.1.14.
22. Let I, g be functions such that (g 0 f)(x) = x for all x E D(f) and (f 0 g)(y) = y for all
y E D(g). Prove that g = 1-'.
Section 1.2
Mathematical Induction
Mathematical Induction is a powerful method of proof that is requently used to establish
the validity of statements that are given in terms of the natural numbers. Although its utility
is restricted to this rather special context, Mathematical Induction is an indispensable tool
in all branches of mathematics. Since many induction proofs follow the same formal lines
of argument, we will often state only that a result follows from Mathematical Induction
and leave it to the reader to provide the necessary details. In this section, we will state the
principle and give several examples to illustrate how inductive proofs proceed.
We shall assume familiarity with the set of natural numbers:
N
:=
{l, 2, 3, . . . },
with the usual arithmetic operations of addition and multiplication, and with the meaning
of a natural number being less than another one. We will also assume the following
fundamental property of N.
1.2.1 Well-Ordering Property of N
Evey nonempy subset of N has a least element.
A more detailed statement of this property is as follows: If S is a subset of N and if
S . 0, then there exists E S such that ::: k for all k E S.
On the basis of the Well-Ordering Property, we shall derive a version of the Principle
of Mathematical Induction that is expressed in terms of subsets of N.
m
m
1.2.2 Principle of Mathematical Induction
Let S be a subset of N that possesses he
two properties:
(1) The number 1 E S.
(2) For every k E N, if k E S, then k + 1 E S.
Then we have S = N.
Suppose to the contrary that S . N. Then the set N\S is not empty, so by the
Well-Ordering Principle it has a least element Since 1 E S by hypothesis (1), we know
that > 1 . But this implies that
1 is also a natural number. Since
1 < and
since is the least element in N such that
S, we conclude that
S.
We now apply hypothesis (2) to the element k :=
1 in S, to infer that k + 1 =
1 ) 1 = belongs to S. But this statement contradicts the fact that
Since
was obtained from the assumption that N\S is not empty, we have obtained a contradiction.
Q.E.D.
Therefore we must have S = N.
Proo.
m
m
(m
-
m
+
m
-
m.
mt
m
-
m
m
m 1E
m t s.
m
-
-
1.2 MATEMATICAL INDUCTION
13
The Principle of Mathematical Induction is often set forth in the framework of proper
ties or statements about natural numbers. If P(n) is a meaningul statement about n E N,
then P(n) may be true for some values of n and false for others. For example, if PI (n) is
the statement: "n 2 = n", then PI (1) is true while PI (n) is false for all n > 1 , n E N. On
the other hand, if P2 (n) is the statement: "n 2 >
then P2 (1) is false, while P2 (n) is true
for all n > 1, n E N.
In this context, the Principle of Mathematical Induction can be formulated as follows.
I",
For each n E N, let P (n) be a statement about n. Suppose that:
(I') P(1) is rue.
(2') For evey k E N, if P (k) is rue, then
Then
P(n) is rue for all n E N.
P (k + 1) is true.
The connection with the preceding version of Mathematical Induction, given in 1 .2.2,
is made by letting S : = {n E N : P(n) is true}. Then the conditions (1) and (2) of 1 .2.2
correspond exactly to the conditions (1') and (2'), respectively. The conclusion that S = N
in 1 .2.2 corresponds to the conclusion that P(n) is true for all n E N.
In (2') the assumption "if P(k) is true" is called the induction hypothesis. In estab
lishing (2'), we are not concened with the actual truth or falsity of P(k), but only with
the validity of the implication "if P(k), then P(k + I)". For example, if we consider the
statements P(n): "n = n + 5", then (2') is logically correct, for we can simply add 1 to
both sides of P(k) to obtain P(k + 1). However, since the statement P (1): "1 = is false,
we cannot use Mathematical Induction to conclude that n = n + 5 for all n E N.
It may happen that statements P (n) re false for certain natural numbers but then are
true for all n ::no for some particular no. The Principle of Mathematical Induction can be
modiied to deal with this situation. We will formulate the modiied principle, but leave its
veriication as an exercise. (See Exercise 12.)
6"
1.2.3 Principle of Mathematical Induction (second version)
be a statement for each natural number n ::no. Suppose that:
(1) he statement P (no) is true.
(2) For all k ::no' the ruth of P(k) implies the ruth of P(k +
Let no E N and let P (n)
1).
P (n) is true for all n ::nO"
Sometimes the number n o in (1) is called the base, since it serves as the starting point,
and the implication in (2), which can be written P(k) } P(k + 1), is called the bridge,
since it connects the case k to the case k + 1 .
hen
The following examples illusrate how Mathematical Induction is used to prove asser
tions about natural numbers.
1.2.4 Examples
(a) For each n E N, the sum of the first n natural numbers is given by
1 + 2 + . . . + n = !n(n + 1).
To prove this formula, we let S be the set of all n E N for which the formula i s true.
We must veriy that conditions ( 1) and (2) of 1 .2.2 are satisfied. If n = 1, then we have
1 = ! . 1 . (1 + 1) so that 1 E S, and (1) is satisied. Next, we assume that k E S and wish
to infer from this assumption that k + 1 E S. Indeed, if k E S, then
1 + 2 + . . . + k = !k(k + 1).
14
CER
1
PELIMINAES
+ 1 to both sides of the assumed equality, we obtain
1 + 2 + . . . + k + (k + 1) = �k(k + 1) + (k + 1)
= �(k + 1)(k + 2).
Since this is the stated formula for n = k + 1 , we conclude that k + 1 E S. Therefore,
condition (2) of 1 .2.2 is satisied. Consequently, by the Principle of Mathematical Induction,
we infer that S = N, so the formula holds for all n E N.
(b) For each n E N, the sum of the squares of the first n natural numbers is given by
1 2 + 22 + . . . + n 2 = �n(n + 1)(2n + 1).
To establish this formula, we note that it is true for n = 1, since 1 2 = � . 1 2 3 . If
we assume it is true for k, then adding (k + 1) 2 to both sides of the assumed formula gives
1 2 + 22 + . . . + k2 + (k + 1) 2 = �k(k + 1)(2k + 1) + (k + 1) 2
= �(k + 1)(2k2 + k + 6k + 6)
= �(k + 1)(k + 2)(2k + 3 ).
Consequently, the formula is valid for all n E N .
(c) Given two real numbers a and b, we will prove that a - b is a factor of a n - bn for
all n E N.
First we see that the statement is clearly true for n = 1. If we now assume that a - b
k
is a factor of a - bk , then
aHI bk+ 1 = ak+k1 abk k + abk k bk+ 1
= a(a - b ) + b (a - b).
k k
By the induction hypothesis, a - b is a factor of a(a - b ) and it is plainly a factor of
kb (a - b). Therefore, a - b is a factor of a HI - bk+ 1 , and it follows from Mathematical
Induction that a - b is a factor of a n - bn for all n E N.
A variety of divisibility results can be derived rom this fact. For example, since
1 1 - 7 = 4, we see that 1 1 n - 7n is divisible by4 for all n E N.
(d) The inequality 2n > 2n + 1 is false for n = 1 , 2, but it is true for n = 3 . If we assume
k
that 2 > 2k + 1 , then multiplication by 2 gives, when 2k + 2 > 3, the inequality
2k+ 1 > 2(2k + 1) = 4k + 2 = 2k + (2k + 2) > 2k + 3 = 2(k + 1) + 1 .
Since 2k + 2 > 3 for all k : 1 , the bridge is valid for all k : 1 (even though the statement
is false for k = 1 , 2). Hence, with the base no = 3, we can apply Mathematical Induction
to conclude that the inequality holds for all n : 3.
(e) The inequality 2n ::: (n + 1 ) ! can be established by Mathematical Induction.
first observe that it is true for n = 1, since 2 1 = 2 = 1 + 1 . If we assume that
k2 :::We
(k + 1 ) !, it follows from the fact that 2 :::k + 2 that
2k + l = 2 . 2k :::2(k + I ) ! ::: (k + 2)(k + I ) ! = (k + 2) !.
Thus, if the inequality holds for k, then it also holds for k + 1 . Therefore, Mathematical
Induction implies that the inequality is true for all n E N.
() If r E �, r - 1, and n E N, then
1...rn + 1
1 + + r 2 + . . . + rn =
l -r
If we add k
·
_
_
r
_
·
1 .2 MATEMATICAL INDUCTION
15
This is the fonnu1a for the sum of the tenns in a "geometric progression". It can
be established using Mathematical Induction as follows. First, if
1 , then 1 r
k
2
(1 - r )/(1 - r). If we assume the truth of the fonu1a for
and add the tenn r + 1 to
both sides, we get (after a little algebra)
k 1
k 2
1-r +
1-r +
k
k ...
k
r +1
1 r r
r +1
l -r
l -r
which is the fonnu1a for
1 . Therefore, Mathematical Induction implies the validity
of the fonula for all
[This result can also be proved without using Mathematical Induction. If we let
n
n
2
2
r , then rSn r r
r +l , so that
1 r r
n
n l
( 1 - r) n
n - rSn 1 - r + .
n=k
+ + + + =
+
n=k+
n E N.
s := + + + . . . +
= + +...+
s =s
=
If we divide by 1 r, we obtain the stated fonnula.]
n=
=
+ =
,
-
(g) Careless use of the Principle of Mathematical Induction can lead to obviously absurd
conclusions. The reader is invited to ind the error in the "proof" of the following assertion.
n E N and if the maximum of the natural numbers p and is n, then p =
"Proof." Let S b e the subset of N for which the claim i s true. Evidently, 1 E S since if
E N and their maximum is 1 , then both equal 1 and so p = Now assume that k E S
and that the maximum of p and is k + 1 . Then the maximum of - 1 and - 1 is k. But
since k E S, then p - 1 = - 1 and therefore p = Thus, k + 1 E S, and we conclude
that the assertion is true for all n E N.
(h) There are statements that are true for many natural numbers but that are not true for
all of them.
2
For example, the fonnula p (n) : = n - n + 4 1 gives a prime number for n = 1 , 2, . . .
Claim:
q
If
p, q
q
q
q.
q.
p
q.
q
40. However, p(41) is obviously divisible by 41, so it is not a prime number.
,
0
Another version of the Principle of Mathematical Induction is sometimes quite useful.
It is called the " Principle of Strong Induction", even though it is in fact equivalent to 1 .2.2.
Let be a subset ofN such that
1.2.5 Principle of Strong Induction
S
"
(1 ) 1 S.
(2" )
2, . . .
; S,
E
For evey k E N, f {I,
Then S = N.
, k}
then k + 1 E S.
We will leave it to the reader to establish the equivalence of 1 .2.2 and 1 .2.5.
Exercises for Section 1.2
n/(n + 1) for all n E N.
Prove that 13 + 23 + . + n3 = [ n(n + 1) ] 2 for all n E No
Prove that 3 + 1 1 + . . . + (8n 5) = 4n2 - n for all n E No
Prove that 12 + 32 + . . . + (2n - 1)2 = (4n3 - n)/3 for all n E N.
5. Prove that 12 - 22 + 32 + . . + (- It+1n2 = (_ 1)n+ln(n + 1)/2 for all n E N.
1.
2.
3.
4.
Prove that 1/1 · 2 + 1/2 · 3 + . . . + l /n(n + 1)
. .
-
.
�
=
16
6.
7.
8.
9.
10.
CTER 1
Prove that n3
PRELIMNARIES
+
5n is divisible by 6 for all n E N.
- 1 is divisible by 8 for all n E N.
Prove that
Prove that 5" - 4n - 1 is divisible by 16 for all n E N.
Prove that n3 + (n + 1)3 + (n + 2)3 is divisible by 9 for all n E N.
Conjecture a formula for the sum 1/1 . 3 + 1/3 · 5 + . . . + 1/(2n - 1) (2n + 1), and prove your
52n
conjecture by using Mathematical Induction.
1 1. Conjecture a formula for the sum of the irst n odd natural numbers 1 + 3 + . . . + (2n - 1),
and prove your formula by using Mathematical Induction.
12.
13.
14.
15.
16.
17.
Prove the Principle of Mathematical Induction 1 .2.3 (second version).
Prove that n < 2" for all n E N.
Prove that 2" < n ! for all n � 4, n E N.
2
Prove that 2n - 3 ::2 - for all n � 5, n E N.
n
2
Find all natural numbers n such that n < 2" . Prove your assertion.
Find the largest natural number m such that n 3 - n is divisible by m for all n E No Prove your
assertion.
18. Prove that 1/0 + 1/2 + . . . + I/n > n for all n E N.
19. Let S be a subset of N such that (a) 2k E S for all k E N, and (b) if k E S and k � 2, then
k - 1 E S. Prove that S = N.
20. Let the numbers be deined as follows: X l := 1, := 2, and
:=
+ for ll
n E N. Use the Principle of Srong Induction (1.2.5) to show that 1 :: :: 2 for all n E N.
x2
xn
Section 1.3
x"+2 �(xn+ l xn )
xn
Finite and Ininite Sets
When we count the elements in a set, we say "one, two, three, . . . ", stopping when we
have exhausted the set. From a mathematical perspective, what we are doing is defining a
bijective mapping between the set and a portion of the set of natural numbers. If the set is
such that the counting does not terminate, such as the set of natural numbers itself, then we
describe the set as being ininite.
The notions of "finite" and "infinite" are extremely primitive, and it is very likely
that the reader has never examined these notions very carefully. In this section we will
define these terms precisely and establish a few basic results and state some other importnt
results that seem obvious but whose proofs are a bit tricky. These proofs can be found in
Appendix B and can be read later
INTRODUCTION TO REAL ANALYSIS
Third Edition
Robert G. Bartle
Donald R. Sherbert
Easten Michigan University, Ypsilanti
University of Illinois, Urbana-Champaign
John Wiley & Sons, Inc.
ACQUISITION EDITOR
Barbara Holland
ASSOCIATE EDITOR
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Libray of Congress Caaloging in ublication Daa:
Bartle, Robert Gardner, 1927Introduction to real analysis / Robert G . Bartle, Donald R., Sherbert. - 3rd ed.
cm.
p.
Includes bibliographical references and index.
ISBN 0-47 1-32148-6 (a1k. paper)
1. Mathematical analysis. 2. Functions of real variables.
1. Sherbert, Donald R., 1935- . II. Title.
QA300.B294
515-dc21
2000
99-13829
CIP
A. M. S. Classiication 26-01
Printed in the United States of America
20
19
18
17
16
15
14
13
12
II
PREFACE
The study of real analysis is indispensible for a prospective graduate student of pure or
applied mathematics. It also has great value for any undergraduate student who wishes
to go beyond the routine manipulations of formulas to solve standard problems, because
it develops the ability to think deductively, analyze mathematical situations, and extend
ideas to a new context. n recent years, mathematics has become valuable in many areas,
including economics and management science as well as the physical sciences, engineering,
and computer science. Our goal is to provide an accessible, reasonably paced textbook in
the undamental concepts and techniques of real analysis for students in these areas. This
book is designed for students who have studied calculus as it is raditionally presented in
the United States. While students ind this book challenging, our experience is that serious
students at this level are fully capable of mastering the material presented here.
The irst two editions of this book were very well received, and we have taken pains
to maintain the same spirit and user-friendly approach. In preparing this edition, we have
examined every section and set of exercises, sreamlined some arguments, provided a few
new examples, moved certain topics to new locations, and made revisions. Except for the
new Chapter 10, which deals with the generalized Riemann integral, we have not added
much new material. While there is more materil than can be covered in one semester,
instructors may wish to use certain topics as honors projects or extra credit assignments.
It is desirable that the student have had some exposure to proofs, but we do not assume
that to be the case. To provide some help for students in analyzing proofs of theorems,
we include an appendix on "Logic and Proofs" that discusses topics such as implications,
quantifiers, negations, contrapositives, and diferent types of proofs. We have kept the
discussion informal to avoid becoming mired in the technical details of formal logic. We
feel that it is a more useful experience to len how to construct proofs by irst watching
and then doing than by reading about techniques of proof.
We have adopted a medium level of generality consistently throughout the book: we
present results that are general enough to cover cases that actually arise, but we do not strive
for maximum generality. In the main, we proceed from the particular to the general. Thus
we consider continuous functions on open and closed intervals in detail, but we are careul
to present proofs that can readily be adapted to a more general situation. (In Chapter 1 1
we take particular advantage of the approach.) We believe that it is important to provide
students with many examples to aid them in their understanding, and we have compiled
rather extensive lists of exercises to challenge them. While we do leave routine proofs as
exercises, we do not try to attain brevity by relegating diicult proofs to the exercises.
owever, in some of the later sections, we do break down a moderately difficult exercise
into a sequence of steps.
In Chapter 1 we present a brief summary of the notions and notations for sets and
functions that we use. A discussion of Mathematical Induction is also given, since inductive
proofs arise frequently. We also include a short section on finite, countable and infinite sets.
We recommend that this chapter be covered quickly, or used as background material,
retuning later as necessary.
v
vi
PREFACE
Chapter 2 presents the properties of the real number system . The irst two sections
deal with the Algebraic and Order Properties and provide some practice in writing proofs
of elementry results. The crucial Completeness Property is given in Section 2.3 as the
Supremum Property, and its ramifications are discussed throughout the remainder of this
chapter.
In Chapter 3 we give a thorough treatment of sequences in R and the associated limit
concepts. The material is of the greatest importance; fortunately, students find it rather
natural although it takes some time for them to become fully accustomed to the use of €.
In the new Section 3.7, we give a brief introduction to infinite series, so that this important
topic will not be omitted due to a shortage of time.
Chapter 4 on limits of functions and Chapter 5 on continuous functions constitute
the heart of the book. Our discussion of limits and continuity relies heavily on the use of
sequences, and the closely parallel approach of these chapters reinforces the understanding
of these essential topics. The fundamental properties of continuous functions (on intervals)
are discussed in Section 5.3 and 5.4. The notion of a "gauge" is introduced in Section 5.5
and used to give altenative proofs of these properties. Monotone functions are discussed
in Section 5.6.
The basic theory of the derivative is given in the first part of Chapter 6. This important
material is standard, except that we have used a result of Caratheodory to give simpler
proofs of the Chain Rule and the Inversion Theorem. The remainder of this chapter consists
of applications of the Mean Value Theorem and may be explored as time permits.
Chapter 7, dealing with the Riemann integral, has been completely revised in this
edition. Rather than introducing upper and lower integrals (as we did in the previous
editions), we here define the integral as a limit of Riemann sums. This has the advantage that
it is consistent with the students ' irst exposure to the integral in calculus and in applications;
since it is not dependent on order properties, it permits immediate generalization to complex
and vector-valued functions that students may encounter in later courses. Contrary to
popular opinion, this limit approach is no more dificult than the order approach. It also is
consistent with the generalized Riemann integral that is discussed in detail in Chapter 10.
Section 7.4 gives a brief discussion of the familiar numerical methods of calculating the
integral of continuous functions.
Sequences of functions and uniform convergence are discussed in the irst two sec
tions of Chapter 8, and the basic transcendental functions are put on a firm foundation in
Section 8.3 and 8.4 by using uniform convergence. Chapter 9 completes our discussion of
ininite series. Chapters 8 and 9 are intrinsically important, and they also show how the
material in the earlier chapters can be applied.
Chapter 10 is completely new; it is a presentation of the generalized Riemann integral
(sometimes called the "Henstock-Kurzweil" or the "gauge" integral). It will be new to many
readers, and we think they will be amazed that such an apparently minor modification of
the deinition of the Riemann integral can lead to an integral that is more general than the
Lebesgue integral. We believe that this relatively new approach to integration theory is both
accessible and exciting to anyone who has studied the basic Riemann integral.
The final Chapter 1 1 deals with topological concepts. Earlier proofs given for intervals
are extended to a more abstract setting. For example, the concept of compactness is given
proper emphasis and metric spaces are introduced. This chapter will be very useful for
students continuing to graduate courses in mathematics.
Throughout the book we have paid more attention to topics from numerical analysis
and approximation theory than is usual. We have done so because of the importance of
these areas, and to show that real analysis is not merely an exercise in abstract thought.
PREFACE
vii
We have provided rather lengthy lists of exercises, some easy and some challenging.
We have provided "hints" for many of these exercises, to help students get started toward a
solution or to check their "answer". More complete solutions of almost every exercise are
given in a separate Instructor' s Manual, which is available to teachers upon request to the
publisher.
It is a satisfying experience to see how the mathematical maturity of the students
increases and how the students gradually len to work comfortably with concepts that
initially seemed so mysterious. But there is no doubt that a lot of hard work is required on
the part of both the students and the teachers.
In order to enrich the historical perspective of he book, we include brief biographical
sketches of some famous mathematicians who contributed to this area. We are particularly
indebted to Dr. Patrick Muldowney for providing us with his photograph of Professors
Henstock and Kurzweil. We also thank John Wiley & Sons for obtaining photographs of
the other mathematicians.
We have received many helpful comments from colleagues at a wide variety of in
stitutions who have taught from earlier editions and liked the book enough to express
their opinions about how to improve it. We appreciate their remarks and suggestions, even
though we did not always follow their advice. We thank them for communicating with us
and wish them well in their endeavors to impart the challenge and excitement of lening
real analysis and "real" mathematics. It is our hope that they will ind this new edition even
more helpful than the earlier ones.
Februay 24, 1999
Ypsilanti and Urbana
Robert G. Bartle
Donald R. Sherbert
THE GREEK ALPHABET
A
B
r
x
J
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y
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/
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E
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H
I
K
A
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Alpha
Beta
Gamma
Delta
Epsilon
Zeta
Eta
Theta
Iota
Kappa
Lambda
Mu
N
�
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0
l
P
;
T
1
v
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0
r
p
a
r
v
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J
Q
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X
II
X
1
Nu
i
Omicron
Pi
Rho
Sigma
Tau
Upsilon
Phi
Chi
Psi
Omega
To our wives, Carolyn and Janice,
with our appreciation for their
patience, support, and love.
CONTENTS
CHAPTER 1
PRELIMINARIES 1
CHAPTER 2
THE REAL NUMBERS 22
CHAPTER 3
CHAPTER 4
C.HAPTER 5
1 . 1 Sets and Functions· 1
1.2 Mathematical Induction 12
1 .3 Finite and Ininite Sets 16
2.1
2.2
2.3
2.4
2.5
The Algebraic and Order Properties of R
Absolute Value and Real Line 31
The Completeness Property of R 34
Applications of the Supremum Property
Intervals 44
22
38
SEQUENCES AND SERIES 52
3.1
3.2
3.3
3.4
3.5
3.6
3.7
Sequences and Their Limits 53
Limit Theorems 60
Monotone Sequences 68
Subsequences and the Bolzano-Weierstrass Theorem
The Cauchy Criterion 80
Properly Divergent Sequences 86
Introduction to Ininite Series 89
75
LIMITS 96
4. 1 Limits of Functions 97
4.2 Limit Theorems 105
4.3 Some Extensions of the Limit Concept
111
CONTINUOUS FUNCTIONS 119
5.1
5.2
5.3
5.4
5.5
5.6
Continuous Functions 120
Combinations of Continuous Functions
Continuous Functions on Intervals 129
Uniform Continuity 136
Continuity and Gauges 145
Monotone and Inverse Functions 149
125
ix
x
CONTENTS
CHAPTER 6
CHAPTER 7
CHAPTER 8
CHAPTER 9
DIFFERENTIATION 157
6.1
6.2
6.3
6.4
The Derivative 158
The Mean Value Theorem
L'Hospital's Rules 176
Taylor' s Theorem 183
168
TE RIEMANN INTEGRAL 193
7.1
7.2
7.3
7.4
The Riemann Integral 194
Riemann Integrable Functions 202
The Fundamental Theorem 210
Approximate Integration 219
SEQUENCES OF FUNCTIONS 227
8.1
8.2
8.3
8.4
Pointwise and Uniform Convergence 227
Interchange of Limits 233
The Exponential and Logarithmic Functions
The Trigonometric Functions 246
239
INFINITE SERIES 253
9.1
9.2
9.3
9.4
Absolute Convergence 253
Tests for Absolute Convergence 257
Tests for Nonabsolute Convergence 263
Series of Functions 266
CHAPTER 10
THE GENERALIZED RIEMANN INTEGRAL 274
CHAPTER 11
A GLIMPSE INTO TOPOLOGY 312
10.1
10.2
10.3
10.4
1 1. 1
1 1.2
1 1.3
1 1.4
Definition and Main Properties 275
Improper and Lebesgue Integrals 287
Infinite Intervals 294
Convergence Theorems 301
Open and Closed Sets in R
Compact Sets 319
Continuous Functions 323
Metric Spaces 327
312
APPENDIX A
LOGIC AND PROOFS 334
APPENDIX B
FINITE AND COUNTABLE SETS 343
APPENDIX C
THE RIEMANN AND LEBESGUE CRITERIA 347
CONTENTS
APPENDIX D
APPROXIMATE INTEGRATION 351
APPENDIX E
TWO EXAMPLES 354
REFERENCES 357
PHOTO CREDITS 358
HINTS FOR SELECTED EXERCISES 359
INDEX 381
xi
CHAPTER 1
PRELIMINARIES
In this initial chapter we will present the background needed for the study of real analysis.
Section 1.1 consists of a brief survey of set operations and functions, two vital tools for all
of mathematics. In it we establish the notation and state the basic deinitions and properties
that will be used throughout the book. We will regard the word "set" as synonymous with
the words "class", "collection", and "family", and we will not deine these terms or give a
list of axioms for set theory. This approach, often referred to as "naive" set theory, is quite
adequate for working with sets in the context of real analysis.
Section 1 .2 is concened with a special method of proof called Mathematical Induction.
t is related to the fundamental properties of the natural number system and, though it is
restricted to proving particular types of statements, it is important and used frequently. An
informal discussion of the diferent types of proofs that are used in mathematics, such as
contrapositives and proofs by contradiction, can be found in Appendix A.
In Section 1 .3 we apply some of the tools presented in the irst two sections of this
chapter to a discussion of what it means for a set to be inite or ininite. Creful deinitions
are given and some basic consequences of these deinitions are derived. The important
result that the set of rational numbers is countably ininite is established.
In addition to introducing basic concepts and establishing terminology and notation,
this chapter also provides the reader with some initial experience in working with precise
deinitions and writing proofs. The careul study of real analysis unavoidably entails the
reading and writing of proofs, and like any skill, it is necessary to practice. This chapter is
a starting point.
Section 1.1
Sets and Functions
In this section we give a brief review of the terminology and notation that
will be used in this text. We suggest that you look through quickly and come back later
when you need to recall the meaning of a term or a symbol.
If an element is in a set A, we write
To the reader:
x
and say that is a member of A, or that
x
X EA
x
x
belongs to A . If x is
¢ A.
not in A, we write
If�very element of a set A also belongs to a set B , we say that A is a subset of B and write
or
We say that a set A is a proper subset of a set B if A � B, but there is at least one element
of B that is not in A. In this case we sometimes write
A C B.
1
2
CHAPTER 1
PRELINARlES
1.1.1 Deinition wo sets A and B are said to be equal. and we write A = B. if they
contain the same elements.
Thus. to prove hat the sets A and B are equal. we must show that
A � B and B � A.
A set is normally deined by either listing its elements explicitly. or by specifying a
property that determines he elements of the set. If P denotes a property that is meaningful
and unambiguous for elements of a set S. then we write
{x E S P(x)}
:
for the set of all elements x in S for which the property P is true. If the set S is understood
rom the context. then it is oten omitted in this notation.
Several special sets are used throughout this book. and they are denoted by standard
symbols. le will use the symbol : = to mean that the symbol on the left is being deined
by the symbol on the right.)
The set of natural numbers N := {I. 2. 3 }.
The set of integers Z : = to. 1. -1.2, -2, · · . },
The set ofrational numbers Q : = {min : m, n E Z and n - OJ.
The set of real numbers R
The set R of real numbers is of fundamental importance for us and will be discussed
at length in Chapter 2.
•
•
. . .
•
•
•
1.1.2 Examples (a)
The set
{x E N : x2 - 3x + 2 = O}
consists of those natural numbers satisfying the stated equation. Since the only solutions of
this quadratic equation re x = 1 and x = 2, we can denote this set more simply by {I, 2}.
(b) A natural number n is even if it has the form n = 2k for some k E N. The set of even
natural numbers can be written
{2k : k E N},
which is less cumbersome than {n E N : n = 2k, k E N}. Similarly, the set of odd natural
numbers can be written
{2k - 1 k E N}.
:
o
Set Operations
We now deine the methods of obtaining new sets from given ones. Note that these set
operations are based on the meaning of the words "or", "and", and "not". For the union,
it is important to be aware of the fact that the word "or" is used in the inclusive sense,
allowing the possibility that x may belong to both sets. In legal terminology, this inclusive
sense is sometimes indicated by "andlor".
1.1.3 Deinition
(a)
The union of sets A and B is the set
AU B := {x : x E A or x E B} .
1.1 SETS AND FUNCTIONS
3
(b) The intersection of the sets A and B is the set
AnB
:=
{x : x E A and x E B} .
(c) The complement of B relative to A is the set
A\B
A U B ID
:=
{x : x E A and x
(a) A U B
Fiure 1.1.1
t
(b) A n B
B} .
(c) A\B
A\B �
The set that has no elements is called the empty set and is denoted by the symbol 0.
Two sets A and B are said to be disjoint if they have no elements in common; this can be
expressed by writing A n B = 0.
To illustrate the method of proving set equalities, we will next establish one of the
for three sets. The proof of the other one is let as an exercise.
DeMorgan laws
1.1.4 Theorem
f A, B, C
re sets, then
(a) A\(B U C) = (A\B) n (A\C),
(b) A\(B n C) = (A\B) U (A\C) .
To prove (a), we will show that every element in A\ (B U C) is contained in both
(A\B) and (A \C), and conversely.
If x is in A\(B U C), then x is in A, but x is not in B U C . Hence x is in A, but x
is neither in B nor in C . Therefore, x is in A but not B, and x is in A but not C . Thus,
x E A\B and x E A\C, which shows that x E (A \B) n (A\C).
Conversely, if x E (A\B) n (A\C), then x E (A\B) and x E (A\C). Hence x E A
and both x t B and x t C . Therefore, x E A and x t (B U C), so that x E A\ (B U C) .
Since the sets (A\B) n (A\ C) and A\ (B U C) contain the same elements, they are
Proo.
equal by Deinition
1.1.1.
Q.E.D.
There are times when it is desirable to form unions and intersections of more than two
sets. For a finite collection of sets {A I ' A 2 , .. . , An }, their union is the set A consisting of
all elements that belong to
of the sets Ak , and their intersection consists of all
el�ments that belong to of the sets Ak •
This is extended to an ininite collection of sets {A I' A2 , ••• , An ' . . .} as follows. Their
union is the set of elements that belong to
of the sets An ' In this case we
write
all
at least one
at least one
0
U An
n =1
:=
{x : x E A n for some n
E N} .
CHTER 1
4
PRELIMNARIES
Similarly, their intersection is the set of elements that belong to all of these sets An' In this
case we write
n An := {x : x E An for all n E N} .
0
n=l
Cartesian Products
In order to discuss unctions, we deine the Cartesian product of two sets.
1.1.5 Deinition If A and B are nonempty sets, then the Cartesian product A x B of A
and B is the set of all ordered pairs (a, b) with a E A and b E B. That is,
A
x
B
:=
{(a, b) : a E A, bE B} .
Thus if A = {l, 2, 3} and B = {I, 5}, then the set A x B is the set whose elements are
the ordered pairs
(1, 1), (1, 5), (2, 1), (2, 5), (3, 1), (3, 5).
We may visualize the set A x B as the set of six points in the plane with the coordinates
that we have just listed.
We oten draw a diagram (such as Figure 1.1.2) to indicate the Cartesian product of
two sets A and B . However, it should be realized that this diagram may be a simplification.
For example, if A := {x E R : 1 ::: x ::: 2} and B : = {y E R : 0 :::y :::1 or 2 :::y :::3},
then instead of a rectangle, we should have a drawing such as Figure 1.1 . 3 .
We will now discuss the undamental notion of afunction or a mapping.
To the mathematician of the early nineteenth century, the word "function" meant a
deinite formula, such as f (x) := x2 + 3x - 5, which associates to each real number x
another number f(x) . (Here, f(O) = -5, f(1) = - 1, f(5) = 35 .) This understanding
excluded the case of diferent formulas on diferent intervals, so that unctions could not
be deined "in pieces".
AxB
B
b
------
3
2
AxB
1 (a , b)
I
I
I
I
I
a
A
Figure 1.1.2
2
Figure 1.1.3
1.1 SETS AD UNCTIONS
5
As mathematics developed, it became clear that a more general definition of "function"
would be useful. It also became evident that it is important to make a clear distinction
between the function itself and the values of the function. A revised definition might be:
A function
f from a set A into a set B is a rule of correspondence that assigns to
x in A a uniquely determined element f (x) in B .
each element
But however suggestive this revised deinition might be, there is the difficulty o f interpreting
the phrase "rule of correspondence". In order to clarify this, we will express the definition
entirely in terms of sets; in efect, we will define a function to be its
graph. W hile this has
the disadvantage of being somewhat artificial, it has the advantage of being unambiguous
and clearer.
1.1.6 Deinition
pairs in
A
x
B
other words, if
A and B be sets. Then a function from A to B is a set f of ordered
E A there exists a unique E B with
E f. (In
E f and
E f, then
Let
a
(a, b')
such that for each
(a, b)
b
b = b'.)
(a, b)
A of first elements of a function f is called the domain of f and is often
D(f). The set of all second elements in f is called the range of f and is
denoted by R(f). Note that, although D(f)
A, we only have R(f) � B. (See
The set
denoted by
often
=
Figure 1.1.4.)
The essential condition that:
(a, b') E f implies that b= b'
is sometimes called the vertical line test. In geometrical terms it says every vertical line
x = a with a E A intersects the graph of f exactly once.
(a, b) E f
and
The notation
f : A;B
f i s a function from A
mapping of A into B, or that f maps A into B . If
is often used to indicate that
into
B.
W e will also say that
f is a
(a, b) is an element in f, it is customary
to write
b = f(a)
a b.
o r sometimes
+
;
r�IE
-�
_____
Figure 1.1.4
A=DV)
�.1
______
a
A unction as a graph
6
If
CHAPTER 1
PRELIMINARIES
b = f(a), we often refer to b as the value of f at a, or as the image of a under f.
ransformations and Machines
tansformation
(a, b) f,
1 . 1.5,
Aside from using graphs, we can visualize a function as a
of the set DU) =
A into the set RU) � B. In this phraseology, when
E
we think of as taking
the element from A and "ransforming" or "mapping" it into an element =
in
RU) � B . We often draw a diagram, such as Figure
even when the sets A and B are
not subsets of the plane.
a
b =!(a)
Figue 1.1.5
f
b f(a)
Rf)
A unction as a transfonnation
machine
There is another way of visualizing a function: namely, as a
that accepts
elements of DU) = A as
and produces corresponding elements of RU) � B as
f we take an element x E D U) and put it into then out comes the corresponding
value
If we put a diferent element y E DU) into then out comes ey ) which may
or may not difer rom
If we ry to insert something that does not belong to DU)
into we find that it is not accepted, for can operate only on elements from DU). (See
Figure
This last visualization makes clear the distinction between and
the first is the
machine itself, and the second is the output of the machine when is the input. Whereas
no one is likely to confuse a meat grinder with ground meat, enough people have confused
functions with their values that it is worth distinguishing between them notationally.
outputs.
f(x).
f,
1.1.6.)
inputs
f(x).
f,
f,
f
f
f
x
t
!
!(x )
Figue 1.1.6
A function as a machine
f
x
f (x):
1.1 SETS AND UNCTIONS
7
Direct and Inverse Images
Let
f:A
�
DC!) = A and range RC!) � B.
If E is a subset of A, then the direct image of E under f is the subset
B be a function with domain
1.1.7 Deinition
f(E) of B given by
f(E) := {f(x) : x E E } .
,
If H is a subset of B, then the inverse image of H under f is the subset f - (H) of A
given by
f-' (H) := {x E A : f(x) E H } .
f-' (H)
Remark The notation
used in this connection has its disadvantages. However,
we will use it since it is the standard notation.
E
y,
f - ' (H)
f(E)
2 f(x2 )
Thus, if we are given a set � A, then a point E B is in the direct image
Similarly, given
if and only if there exists at least one point E such that Y, =
a set � B, then a point is in the inverse image
if and only if Y : =
belongs to
(See Figure 1 . 1 .7.)
H
x2
H.
x, E
f(x ,).
x2 • Then the direct image
of the set E := {x : 0 � x � 2} is the set f(E) = {y : 0 � Y � 4}.
If G := {y : 0 � Y � 4}, then the inverse image of G is the set f -' (G) = {x : -2 �
x � 2}. Thus, in this case, we see that f - ' C!(E» = E.
On the other hand, we have f (J - '(G ») = G. But if H : = {y : -1 � Y � I}, then
we have f (J - ' (H») = {y : 0 � y � I } = H.
A sketch of the graph of f may help to visualize these sets.
(b) Let f : A
B, and let G, H be subsets of B . We will show that
f- ' (G n H) � f - ' (G) n f-' (H).
For, if x E f - ' (G n H), then f(x) E G n H, so that f(x) E G and f(x) E H. But this
implies that x E f - ' (G) and x E f - ' (H), whence x E f -' (G) n f - ' (H). Thus the stated
1.1.8 Examples (a) Let f : R � R be deined by f(x)
:=
�
implication is proved. [The opposite inclusion is also true, so that we actually have set
0
equality between these sets; see Exercise 13.]
Further facts about direct and inverse images are given in the exercises.
E
�
f
H
Figue 1.1.7
Drct nd inverse images
8
CHAPTER 1
PRELIMINARIES
Special ypes of Functions
The following deinitions identify some very important types of functions.
1.1.9 Deinition Let I : A + B be a unction from A to B .
(a) The function I i s said to be ijective (or to b e one-one) if whenever x , = x2 , then
I (x,) = l (x2 ) · f I is an injective function, we also say that I is an ijection.
(b) The function I is said to be surjective (or to map A onto B) if I (A) = B; that is, if
the range
= B. If I is a surjective function, we also say that I is a surjection.
(c) If I is both injective and sujective, then I is said to be bijective. If I is bijective, we
also say that I is a bijection.
R(f)
•
In order to prove that a function I is injective, we must establish that:
for all x" x2 in A , if I (x,) = l (x2 ) , then x, = x2•
To do this we assume that I (x,) = l (x2) and show that x, = x2 .
[In other words, the graph of I satisies the
line y = with E B intersects the graph I in
one point.]
To prove that a function I is sujective, we must show that for any
least one x E A such that I (x) =
[In other words, the graph of I satisies the
line y = with E B intersects the graph I in
one point.]
b
•
irst horizontal line test: Every horizontal
at most
b E B there exists at
second horizontal line test: Every horizontal
at least
b
b.
b
b
1.1.10 Example Let A := {x E R : x = l } and deine / (x) := 2x/(x -l) for all x A.
To show that I is injective, we take x, and x2 in A and assume that I (x,) = l (x2). Thus
we have
2x2
2x,
- = -,
x, x2 -
E
1
1
which implies that x, (x2 - 1) = x2 (x, - 1), and hence x, = x2 . Therefore I is injective.
To determine the range of I, we solve the equation y = 2x/(x - for x in tenus of
y. We obtain x = y / (y - 2), which is meaningful for y = 2. Thus the range of I is the set
0
B : = { y E R : y = 2}. Thus, I is a bijection of A onto B.
1)
Inverse Functions
If I is a function from A into B, then I is a special subset of A x B (namely, one passing
the
The set of ordered pairs in B x A obtained by interchanging the
members of ordered pairs in I is not generally a function. (That is, the set I may not pass
of the
However, if I is a bijection, then this interchange does
lead to a function, called the "inverse function" of I.
vertical line test.)
both
horizontal line tests.)
1.1.11 Deinition If I : A + B is a bijection of A onto B, then
g
:=
feb, a) E B
x
A:
(a, b) E f}
is a function on B into A. This function is called the inverse function of I, and is denoted
by 1-'. The function 1-' is also called the inverse of I .
1 . 1 SETS AND UNCTIONS
9
I and its inverse I - I by noting that
I
R(f) D(f - )
b = I(a) if and only if a = I - I (b).
For example, we saw in Example 1.1.10 that the function
I(x) : = x 2x- I
is a bijection of A := {x E R : x = I} onto the set B := {y E R : y = 2}. The function
inverse to I is given by
I - I (y) := y--y 2 for y E B.
I
Remark We introduced the notation I - (H) in Deinition 1.1.7. It makes sense even if
I does not have an inverse unction. However, if the inverse function I - I does exist, then
I- I (H) is the direct image of the set H � B under I- I .
We can also express the connection between
=
and
=
and that
D(f) R(f- I )
Composition of Functions
I, g
I (x)
g (f (x));
I (x)
g.
all I(x),
I
g.
1.1.8.)
1.1.12 Deinition If I A
B and g : B
C, and if R(f) � D(g) = B, then the
composite function g o I (note the orderl) is the function from A into C deined by
(g 0 f)(x) := g(f(x)) for ll x E A .
1.1.13 Examples (a) The order of the composition must be careully noted. For, let I
and g be the functions whose values at x E R are given by
I(x) := 2x and g(x) := 3x 2 1.
Since D(g) = R and R(f) � R = D(g), then the domain D(g 0 f) is also equal to R , and
the composite function g 0 I is given by
(g 0 f)(x) = 3 (2x) 2 1 = 12x 2 1.
It often happens that we want to "compose" two functions
by first inding
and
then applying to get
however, this is possible only when
belongs to the
domain of In order to be able to do this for
we must assume that the range of
is contained in the domain of (See Figure
g
:
�
�
-
-
A
-
B
I
�
gol
Fiue 1.1.8
he composition of f and g
c
10
CHAPTER 1
PRELIMINARIES
l o g is also R, but
(f g)(x) = 2(3x 2 - 1) = 6x 2 2.
Thus, in this case, we have g I = l o g.
(b) In considering g I, some care must be exercised to be sure that the range of 1 is
contained in the domain of g. For example, if
I(x) := l - x 2 and g(x ) : = X,
then, since D(g) = {x : x � O}, the composite unction g o I is given by the formula
(g f)(x) = 7
only for x E D(f) that satisfy I(x) � 0; that is, for x satisfying - 1 x 1 .
We note that if we reverse the order, then the composition l o g is given by the fomula
(f o g)(x) = l - x,
but only for those x in the domain D(g) = {x : x � O}.
o
On the other hand, the domain of the composite function
0
-
0
0
0
�
�
We now give the relationship between composite functions and inverse images. The
proof is left as an instructive exercise.
1.1.14 Theorem
C. Then we have
Note the
Let
I:A
+
B nd
g : B C be functions nd let H be a subset of
+
reversal in the order of the functions.
Restrictions of Functions
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
_
: A + B is a function and if A l A, we can define a function II : A l + B by
II (x) := I(x) for x E A I •
The function II is called the restriction of I to A I . Sometimes it is denoted by II = I I A I'
It may seem strange to the reader that one would ever choose to throw away a pt of a
function, but there are some good reasons for doing so. For example, if I : R + R is the
C
If I
squaring function:
for
I
x E R,
I
then is not injective, so it cannot have an inverse function. However, if we restrict to
is a bijection of onto
O}, then the restriction
the set
Therefore,
this restriction has an inverse function, which is the positive square root function. (Sketch
a graph.)
Similarly, the trigonomeric functions
sin and
cos are not injective
on all of R However, by making suitable restrictions of these unctions, one can obtain
the inverse sine and the inverse cosine functions that the reader has undoubtedly already
encountered.
A l := {x : x �
IIA I
S(x) := x
Al
C(x) :=
A I'
x
1.1 SETS AD FUNCTIONS
11
Exercises for Section 1.1
1. If A and B are sets, show that A � B if and only if An B = A.
2. Prove the second De Morgan Law [Theorem 1.1.4(b)].
3. Prove the Distributive Laws:
(a) A n (B U C) = (An B) U (An C),
(b) AU (Bn C) = (A U B) n (A U C).
4. The symmetric diference of two sets A and B is the set D of all elements hat belong to either
A or B but not both. Represent D with a diagram.
(a) Show that D = (A\B) U (B\A) .
(b) Show that D is also given by D = (A U B)\(An B).
5. For each n E N, let An = {en + I)k : k E N}.
(a) What is A n Az?
(b) Determinel the sets UlAn : n E N} and n{An : n E N}.
6. Draw diagrams in the plane of the Cartesian products A x B for the given sets A and B.
(a) A = {x E R : 1::x::2 or 3::x::4}, B = {x E R : x = 1 or x = 2}.
b) A = { I , 2, 3}, B = {x E R : 1:: x::3}.
7. Let A := B := {x E R : -1::x:: I} and consider the subset C := {(x, y) : xZ + l = I } of
A x B. Is this set a unction? Explain.
8. Let f(x) := I/xz, x . 0, X E R.
(a) Determine the direct image fee) where E := {x E R : 1::x::2}.
(b) Determine the inverse image f-I(G) where G := {x E R : 1::x::4}.
9. Let g(x) := XZ and f(x) := x + 2 for x E R, and let h be the composite unction h := g 0 f.
(a) Find the direct image h(E) of E := {x E R : 0::x:: I }.
b) Find the inverse image h-I (G) of G := {x E R : 0::x :: 4}.
10. Let f(x) := XZ for x E R, and let E : = {x E R : -1::x::O} and F := {x E R : 0::x:: l}.
Show that En F = {O} and feEn F) = {O}, while fee) = f(F) = { y E R : 0::y:: I }.
Hence feEn F) is a proper subset of fee) n f(F). What happens if 0 is deleted from the sets
E and F?
11. Let f and E. F be as in Exercise 10. Find the sets E\F and f(E)\f(F) and show that it is not
true that f(E\F) � f(E)\f(F).
12. Show that if f : A � B and E, F are subsets of A, then feE U F) = fee) U f(F) and
feE n F) � fee) n f(F).
13. Show that if f : A � B and G, H are subsets of B, hen rl (G U H) = f-I(G) U f-I(H)
and rl(Gn H) = rl (G) n rl (H).
14. Show that the unction f deined by f(x) := x/VxZ + 1, x E R, is a bijection of R onto
{y : -1 < y < I}.
15. For a, b E R with a < b, ind an explicit bijection of A := {x : a < x < b} onto B := {y : 0 <
y < I}.
16. Give an example of two functions f, g on R to R such that f . g, but such that fog = g 0 f.
17. (a) Show that if f : A � B is injective and E � A, then f-I(f(E» = E. Give an example
to show that equality need not hold if f is not injective.
(b) Show that if f : A � B is sujective and H � B, then f(rl (H» = H. Give an example
to show that equality need not hold if f is not surjective.
18. (a) Suppose that f is an injection . Show that f-I 0 f(x) = x for all x E D(f) and that
fori (y) = y for all y E R(f) .
(b) If f is a bijection of A onto B, show that f-I is a bijection of B onto A.
CHAPTER 1
12
PRELMINARIES
19. Prove that if I : A � B is bijective and g : B � C is bijective, then the composite g 0 I is a
bijective map of A onto C.
20. Let I : A � B and g : B � C be functions.
(a) Show that if g 0 I is injective, then I is injective.
(b) Show that if g o I is surjective, then g is sujective.
21. Prove Theorem 1.1.14.
22. Let I, g be functions such that (g 0 f)(x) = x for all x E D(f) and (f 0 g)(y) = y for all
y E D(g). Prove that g = 1-'.
Section 1.2
Mathematical Induction
Mathematical Induction is a powerful method of proof that is requently used to establish
the validity of statements that are given in terms of the natural numbers. Although its utility
is restricted to this rather special context, Mathematical Induction is an indispensable tool
in all branches of mathematics. Since many induction proofs follow the same formal lines
of argument, we will often state only that a result follows from Mathematical Induction
and leave it to the reader to provide the necessary details. In this section, we will state the
principle and give several examples to illustrate how inductive proofs proceed.
We shall assume familiarity with the set of natural numbers:
N
:=
{l, 2, 3, . . . },
with the usual arithmetic operations of addition and multiplication, and with the meaning
of a natural number being less than another one. We will also assume the following
fundamental property of N.
1.2.1 Well-Ordering Property of N
Evey nonempy subset of N has a least element.
A more detailed statement of this property is as follows: If S is a subset of N and if
S . 0, then there exists E S such that ::: k for all k E S.
On the basis of the Well-Ordering Property, we shall derive a version of the Principle
of Mathematical Induction that is expressed in terms of subsets of N.
m
m
1.2.2 Principle of Mathematical Induction
Let S be a subset of N that possesses he
two properties:
(1) The number 1 E S.
(2) For every k E N, if k E S, then k + 1 E S.
Then we have S = N.
Suppose to the contrary that S . N. Then the set N\S is not empty, so by the
Well-Ordering Principle it has a least element Since 1 E S by hypothesis (1), we know
that > 1 . But this implies that
1 is also a natural number. Since
1 < and
since is the least element in N such that
S, we conclude that
S.
We now apply hypothesis (2) to the element k :=
1 in S, to infer that k + 1 =
1 ) 1 = belongs to S. But this statement contradicts the fact that
Since
was obtained from the assumption that N\S is not empty, we have obtained a contradiction.
Q.E.D.
Therefore we must have S = N.
Proo.
m
m
(m
-
m
+
m
-
m.
mt
m
-
m
m
m 1E
m t s.
m
-
-
1.2 MATEMATICAL INDUCTION
13
The Principle of Mathematical Induction is often set forth in the framework of proper
ties or statements about natural numbers. If P(n) is a meaningul statement about n E N,
then P(n) may be true for some values of n and false for others. For example, if PI (n) is
the statement: "n 2 = n", then PI (1) is true while PI (n) is false for all n > 1 , n E N. On
the other hand, if P2 (n) is the statement: "n 2 >
then P2 (1) is false, while P2 (n) is true
for all n > 1, n E N.
In this context, the Principle of Mathematical Induction can be formulated as follows.
I",
For each n E N, let P (n) be a statement about n. Suppose that:
(I') P(1) is rue.
(2') For evey k E N, if P (k) is rue, then
Then
P(n) is rue for all n E N.
P (k + 1) is true.
The connection with the preceding version of Mathematical Induction, given in 1 .2.2,
is made by letting S : = {n E N : P(n) is true}. Then the conditions (1) and (2) of 1 .2.2
correspond exactly to the conditions (1') and (2'), respectively. The conclusion that S = N
in 1 .2.2 corresponds to the conclusion that P(n) is true for all n E N.
In (2') the assumption "if P(k) is true" is called the induction hypothesis. In estab
lishing (2'), we are not concened with the actual truth or falsity of P(k), but only with
the validity of the implication "if P(k), then P(k + I)". For example, if we consider the
statements P(n): "n = n + 5", then (2') is logically correct, for we can simply add 1 to
both sides of P(k) to obtain P(k + 1). However, since the statement P (1): "1 = is false,
we cannot use Mathematical Induction to conclude that n = n + 5 for all n E N.
It may happen that statements P (n) re false for certain natural numbers but then are
true for all n ::no for some particular no. The Principle of Mathematical Induction can be
modiied to deal with this situation. We will formulate the modiied principle, but leave its
veriication as an exercise. (See Exercise 12.)
6"
1.2.3 Principle of Mathematical Induction (second version)
be a statement for each natural number n ::no. Suppose that:
(1) he statement P (no) is true.
(2) For all k ::no' the ruth of P(k) implies the ruth of P(k +
Let no E N and let P (n)
1).
P (n) is true for all n ::nO"
Sometimes the number n o in (1) is called the base, since it serves as the starting point,
and the implication in (2), which can be written P(k) } P(k + 1), is called the bridge,
since it connects the case k to the case k + 1 .
hen
The following examples illusrate how Mathematical Induction is used to prove asser
tions about natural numbers.
1.2.4 Examples
(a) For each n E N, the sum of the first n natural numbers is given by
1 + 2 + . . . + n = !n(n + 1).
To prove this formula, we let S be the set of all n E N for which the formula i s true.
We must veriy that conditions ( 1) and (2) of 1 .2.2 are satisfied. If n = 1, then we have
1 = ! . 1 . (1 + 1) so that 1 E S, and (1) is satisied. Next, we assume that k E S and wish
to infer from this assumption that k + 1 E S. Indeed, if k E S, then
1 + 2 + . . . + k = !k(k + 1).
14
CER
1
PELIMINAES
+ 1 to both sides of the assumed equality, we obtain
1 + 2 + . . . + k + (k + 1) = �k(k + 1) + (k + 1)
= �(k + 1)(k + 2).
Since this is the stated formula for n = k + 1 , we conclude that k + 1 E S. Therefore,
condition (2) of 1 .2.2 is satisied. Consequently, by the Principle of Mathematical Induction,
we infer that S = N, so the formula holds for all n E N.
(b) For each n E N, the sum of the squares of the first n natural numbers is given by
1 2 + 22 + . . . + n 2 = �n(n + 1)(2n + 1).
To establish this formula, we note that it is true for n = 1, since 1 2 = � . 1 2 3 . If
we assume it is true for k, then adding (k + 1) 2 to both sides of the assumed formula gives
1 2 + 22 + . . . + k2 + (k + 1) 2 = �k(k + 1)(2k + 1) + (k + 1) 2
= �(k + 1)(2k2 + k + 6k + 6)
= �(k + 1)(k + 2)(2k + 3 ).
Consequently, the formula is valid for all n E N .
(c) Given two real numbers a and b, we will prove that a - b is a factor of a n - bn for
all n E N.
First we see that the statement is clearly true for n = 1. If we now assume that a - b
k
is a factor of a - bk , then
aHI bk+ 1 = ak+k1 abk k + abk k bk+ 1
= a(a - b ) + b (a - b).
k k
By the induction hypothesis, a - b is a factor of a(a - b ) and it is plainly a factor of
kb (a - b). Therefore, a - b is a factor of a HI - bk+ 1 , and it follows from Mathematical
Induction that a - b is a factor of a n - bn for all n E N.
A variety of divisibility results can be derived rom this fact. For example, since
1 1 - 7 = 4, we see that 1 1 n - 7n is divisible by4 for all n E N.
(d) The inequality 2n > 2n + 1 is false for n = 1 , 2, but it is true for n = 3 . If we assume
k
that 2 > 2k + 1 , then multiplication by 2 gives, when 2k + 2 > 3, the inequality
2k+ 1 > 2(2k + 1) = 4k + 2 = 2k + (2k + 2) > 2k + 3 = 2(k + 1) + 1 .
Since 2k + 2 > 3 for all k : 1 , the bridge is valid for all k : 1 (even though the statement
is false for k = 1 , 2). Hence, with the base no = 3, we can apply Mathematical Induction
to conclude that the inequality holds for all n : 3.
(e) The inequality 2n ::: (n + 1 ) ! can be established by Mathematical Induction.
first observe that it is true for n = 1, since 2 1 = 2 = 1 + 1 . If we assume that
k2 :::We
(k + 1 ) !, it follows from the fact that 2 :::k + 2 that
2k + l = 2 . 2k :::2(k + I ) ! ::: (k + 2)(k + I ) ! = (k + 2) !.
Thus, if the inequality holds for k, then it also holds for k + 1 . Therefore, Mathematical
Induction implies that the inequality is true for all n E N.
() If r E �, r - 1, and n E N, then
1...rn + 1
1 + + r 2 + . . . + rn =
l -r
If we add k
·
_
_
r
_
·
1 .2 MATEMATICAL INDUCTION
15
This is the fonnu1a for the sum of the tenns in a "geometric progression". It can
be established using Mathematical Induction as follows. First, if
1 , then 1 r
k
2
(1 - r )/(1 - r). If we assume the truth of the fonu1a for
and add the tenn r + 1 to
both sides, we get (after a little algebra)
k 1
k 2
1-r +
1-r +
k
k ...
k
r +1
1 r r
r +1
l -r
l -r
which is the fonnu1a for
1 . Therefore, Mathematical Induction implies the validity
of the fonula for all
[This result can also be proved without using Mathematical Induction. If we let
n
n
2
2
r , then rSn r r
r +l , so that
1 r r
n
n l
( 1 - r) n
n - rSn 1 - r + .
n=k
+ + + + =
+
n=k+
n E N.
s := + + + . . . +
= + +...+
s =s
=
If we divide by 1 r, we obtain the stated fonnula.]
n=
=
+ =
,
-
(g) Careless use of the Principle of Mathematical Induction can lead to obviously absurd
conclusions. The reader is invited to ind the error in the "proof" of the following assertion.
n E N and if the maximum of the natural numbers p and is n, then p =
"Proof." Let S b e the subset of N for which the claim i s true. Evidently, 1 E S since if
E N and their maximum is 1 , then both equal 1 and so p = Now assume that k E S
and that the maximum of p and is k + 1 . Then the maximum of - 1 and - 1 is k. But
since k E S, then p - 1 = - 1 and therefore p = Thus, k + 1 E S, and we conclude
that the assertion is true for all n E N.
(h) There are statements that are true for many natural numbers but that are not true for
all of them.
2
For example, the fonnula p (n) : = n - n + 4 1 gives a prime number for n = 1 , 2, . . .
Claim:
q
If
p, q
q
q
q.
q.
p
q.
q
40. However, p(41) is obviously divisible by 41, so it is not a prime number.
,
0
Another version of the Principle of Mathematical Induction is sometimes quite useful.
It is called the " Principle of Strong Induction", even though it is in fact equivalent to 1 .2.2.
Let be a subset ofN such that
1.2.5 Principle of Strong Induction
S
"
(1 ) 1 S.
(2" )
2, . . .
; S,
E
For evey k E N, f {I,
Then S = N.
, k}
then k + 1 E S.
We will leave it to the reader to establish the equivalence of 1 .2.2 and 1 .2.5.
Exercises for Section 1.2
n/(n + 1) for all n E N.
Prove that 13 + 23 + . + n3 = [ n(n + 1) ] 2 for all n E No
Prove that 3 + 1 1 + . . . + (8n 5) = 4n2 - n for all n E No
Prove that 12 + 32 + . . . + (2n - 1)2 = (4n3 - n)/3 for all n E N.
5. Prove that 12 - 22 + 32 + . . + (- It+1n2 = (_ 1)n+ln(n + 1)/2 for all n E N.
1.
2.
3.
4.
Prove that 1/1 · 2 + 1/2 · 3 + . . . + l /n(n + 1)
. .
-
.
�
=
16
6.
7.
8.
9.
10.
CTER 1
Prove that n3
PRELIMNARIES
+
5n is divisible by 6 for all n E N.
- 1 is divisible by 8 for all n E N.
Prove that
Prove that 5" - 4n - 1 is divisible by 16 for all n E N.
Prove that n3 + (n + 1)3 + (n + 2)3 is divisible by 9 for all n E N.
Conjecture a formula for the sum 1/1 . 3 + 1/3 · 5 + . . . + 1/(2n - 1) (2n + 1), and prove your
52n
conjecture by using Mathematical Induction.
1 1. Conjecture a formula for the sum of the irst n odd natural numbers 1 + 3 + . . . + (2n - 1),
and prove your formula by using Mathematical Induction.
12.
13.
14.
15.
16.
17.
Prove the Principle of Mathematical Induction 1 .2.3 (second version).
Prove that n < 2" for all n E N.
Prove that 2" < n ! for all n � 4, n E N.
2
Prove that 2n - 3 ::2 - for all n � 5, n E N.
n
2
Find all natural numbers n such that n < 2" . Prove your assertion.
Find the largest natural number m such that n 3 - n is divisible by m for all n E No Prove your
assertion.
18. Prove that 1/0 + 1/2 + . . . + I/n > n for all n E N.
19. Let S be a subset of N such that (a) 2k E S for all k E N, and (b) if k E S and k � 2, then
k - 1 E S. Prove that S = N.
20. Let the numbers be deined as follows: X l := 1, := 2, and
:=
+ for ll
n E N. Use the Principle of Srong Induction (1.2.5) to show that 1 :: :: 2 for all n E N.
x2
xn
Section 1.3
x"+2 �(xn+ l xn )
xn
Finite and Ininite Sets
When we count the elements in a set, we say "one, two, three, . . . ", stopping when we
have exhausted the set. From a mathematical perspective, what we are doing is defining a
bijective mapping between the set and a portion of the set of natural numbers. If the set is
such that the counting does not terminate, such as the set of natural numbers itself, then we
describe the set as being ininite.
The notions of "finite" and "infinite" are extremely primitive, and it is very likely
that the reader has never examined these notions very carefully. In this section we will
define these terms precisely and establish a few basic results and state some other importnt
results that seem obvious but whose proofs are a bit tricky. These proofs can be found in
Appendix B and can be read later