basic elements of real analysis - m. protter

In particular, the system of real numbers satisfies these axioms, and weindicate how the customary laws of elementary algebra concerning addi-tion, subtraction, multiplication, and divisi

Murray H Protter

Springer

Berkeley, CA 94720

USA

Editorial Board

MathematicsDepartment MathematicsDepartment Department of

University University of Michigan University of CaliforniaSan Francisco, CA 94132 Ann Arbor, MI 48109 at Berkeley

USA

Front cover illustration: f n convergesto f , but f1f n doesnot converge to f1f (See p 167 of text for explanation.)

Mathematics Subject Classification (1991): 26-01, 26-06, 26A54

Library of Congress Cataloging-in-Publication Data

Protter, Murray H.

Basic elements of real analysis / Murray H Protter.

p cm.—(Undergraduate textsin mathematics)

Includesbibliographical referencesand index.

ISBN 0-387-98479-8 (hardcover : alk paper)

1 Mathematical analysis I Title II Series.

QA300.P9678 1998

c

 1998 Springer-Verlag New York, Inc.

All rightsreserved Thiswork may not be translated or copied in whole or in part without the written permission of the publisher (Springer-Verlag New York, Inc., 175 Fifth Avenue, New York, NY 10010, USA), except for brief excerptsin connection with reviewsor scholarly analysis Use in connection with any form of information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed isforbidden.

The use of general descriptive names, trade names, trademarks, etc., in this publication, even if the former are not especially identified, is not to be taken as a sign that such names, asunderstood by the Trade Marksand Merchandise MarksAct, may accordingly be used freely by anyone.

ISBN 0-387-98479-8 Springer-Verlag New York Berlin Heidelberg SPIN 10668224

To Barbara and Philip

Some time ago CharlesB Morrey and I wrote A First Course in Real

Analy-sis, a book that provides material sufficient for a comprehensive one-year

course in analysis for those students who have completed a standard mentary course in calculus The book has been through two editions, thesecond of which appeared in 1991; small changes and corrections of mis-printshave been made in the fifth printing of the second edition, whichappeared recently

ele-However, for many students of mathematics and for those studentswho intend to study any of the physical sciences and computer science,the need is for a short one-semester course in real analysis rather than

a lengthy, detailed, comprehensive treatment To fill this need the book

Basic Elements of Real Analysis provides, in a brief and elementary way,

the most important topics in the subject

The first chapter, which deals with the real number system, gives thereader the opportunity to develop facility in proving elementary theo-rems Since most students who take this course have spent their efforts indeveloping manipulative skills, such an introduction presents a welcomechange The last section of this chapter, which establishes the technique

of mathematical induction, is especially helpful for those who have notpreviously been exposed to this important topic

Chapters2 through 5 cover the theory of elementary calculus, ing differentiation and integration Many of the theoremsthat are “statedwithout proof” in elementary calculusare proved here

includ-It isimportant to note that both the Darboux integral and the Riemannintegral are described thoroughly in Chapter 5 of this volume Here we

vii

of functionsin N -dimensional spaces with N > 2 Here derivativesof functionsof N variablesare developed, and the Darboux and Riemann integrals, as described in Chapter 5, are extended in Chapter 7 to N -

dimensional space

Infinite series is the subject of Chapter 8 After a review of the usualtests for convergence and divergence of series, the emphasis shifts touniform convergence The reader must master this concept in order tounderstand the underlying ideas of both power series and Fourier se-ries Although Fourier seriesare not included in thistext, the readershould find it fairly easy reading once he or she masters uniform con-vergence For those interested in studying computer science, not onlyFourier series but also the application of Fourier series to wavelet theory is

recommended (See, e.g., Ten Lectures on Wavelets by Ingrid Daubechies.)

There are many important functionsthat are defined by integrals, theintegration taken over a finite interval, a half-infinite integral, or one

Chapter 9 we develop the necessary techniques for differentiation underthe integral sign of such functions (the Leibniz rule) Although desirable,thischapter isoptional, since the resultsare not used later in the text.Chapter 10 treatsthe Riemann–Stieltjesintegral After an introduction

to functionsof bounded variation, we define the R-S integral and showhow the usual integration-by-parts formula is a special case of this inte-gral The generality of the Riemann–Stieltjesintegral isfurther illustrated

by the fact that an infinite seriescan alwaysbe considered asa specialcase of a Riemann–Stieltjes integral

A subject that is heavily used in both pure and applied mathematicsisthe Lagrange multiplier rule In most casesthisrule isstated withoutproof but with applications discussed However, we establish the rule

in Chapter 11 (Theorem 11.4) after developing the factson the implicit

function theorem needed for the proof

prove the theoremsof Green and Stokesand the divergence theorem,not in full generality but of sufficient scope for most applications The

ambitiousreader can get a more general insight either by referring to the book A First Course in Real Analysis or the text Principles of Mathematical

Analysis by Walter Rudin.

Murray H ProtterBerkeley, CA

Contents Preface vii

Chapter 1 The Real Number System 1

1.1 Axiomsfor a Field 1

1.2 Natural Numbersand Sequences . 7

1.3 Inequalities 11

1.4 Mathematical Induction 19

Chapter 2 Continuity and Limits 23

2.1 Continuity 23

2.2 Limits 28

2.3 One-Sided Limits 33

2.4 Limitsat Infinity; Infinite Limits . 37

2.5 Limitsof Sequences . 42

Chapter 3 Basic Properties of Functions onR1 45

3.1 The Intermediate-Value Theorem 45

3.2 Least Upper Bound; Greatest Lower Bound 48

3.3 The Bolzano–Weierstrass Theorem 54

3.4 The Boundedness and Extreme-Value Theorems 56

3.5 Uniform Continuity 57

3.6 The Cauchy Criterion 61

3.7 The Heine–Borel Theorem 63

ix

x Contents

Chapter 4Elementary Theory of Differentiation 67

4.1 The Derivative inR1 . 67

4.2 Inverse Functions inR1 . 77

Chapter 5 Elementary Theory of Integration 81

5.1 The Darboux Integral for FunctionsonR1 . 81

5.2 The Riemann Integral 91

5.3 The Logarithm and Exponential Functions 96

Chapter 6 Elementary Theory of Metric Spaces 101

6.1 The Schwarz and Triangle Inequalities; Metric Spaces 101

6.2 Topology in Metric Spaces 106

6.3 Countable and Uncountable Sets 114

6.4 Compact Setsand the Heine–Borel Theorem 118

6.5 Functionson Compact Sets 122

Chapter 7 Differentiation and Integration inRN 125

7.1 Partial Derivativesand the Chain Rule 125

7.2 Taylor’sTheorem; Maxima and Minima 130

7.3 The Derivative inRN 136

7.4 The Darboux Integral inRN 141

7.5 The Riemann Integral inRN 145

Chapter 8 Infinite Series 150

8.1 Tests for Convergence and Divergence 150

8.2 Series of Positive and Negative Terms; Power Series 155

8.3 Uniform Convergence 162

8.4 Uniform Convergence of Series; Power Series 168

Chapter 9 The Derivative of an Integral. Improper Integrals 178

9.1 The Derivative of a Function Defined by an Integral The Leibniz Rule 178

9.2 Convergence and Divergence of Improper Integrals 183

Chapter 10 The Riemann–Stieltjes Integral 190

10.1 Functionsof Bounded Variation 190

10.2 The Riemann–StieltjesIntegral 195

Chapter 11 The Implicit Function Theorem. Lagrange Multipliers 205

11.1 The Implicit Function Theorem 205

11.2 Lagrange Multipliers 210

Chapter 12 Vector Functions onRN; The Theorems of

Green and Stokes 214

12.1 Vector FunctionsonRN 214

12.2 Line IntegralsinRN 225

12.3 Green’sTheorem in the Plane 232

12.4 Area of a Surface inR3 . 239

12.5 The StokesTheorem 244

12.6 The Divergence Theorem 253

Answers to Odd-Numbered Problems 261

Index 269

1.1 Axioms for a Field

In thisand the next four chapterswe give a reasonably tion to the processesof calculusof functionsof one variable, a subjectusually studied in a first course in calculus Calculus depends on theproperties of the real number system To give a complete foundation forcalculus we would have to develop the real number system from the be-ginning Since such a development islengthy and would divert usfrom

rigorousfounda-our goal of presenting a crigorousfounda-ourse in analysis, we suppose that the reader is

familiar with the usual properties of real numbers

In this section we present a set of axioms that form a logical basis forthose processes of elementary algebra upon which calculus is based Any

collection of objectssatisfying the axiomsgiven below iscalled a field.

In particular, the system of real numbers satisfies these axioms, and weindicate how the customary laws of elementary algebra concerning addi-tion, subtraction, multiplication, and division are direct consequences ofthe axiomsfor a field

for the words“isthe same as.” The reader should compare thiswith other

two line segments are said to be equal if they have the same length

1

Axioms of Addition

A-1 Closure property

If a and b are numbers,there is one and only one number, denoted a + b,

called their sum.

A-2 Commutative law

For any two numbers a and b,the equality

holds.

A-3 Associative law

For all numbers a,b,and c,the equality

holds.

A-4 Existence of a zero

There is one and only one number 0,called zero,such that a + 0  a for any

number a.

A-5 Existence of a negative

If a is any number,there is one and only one number x such that a + x  0.

This number is called the negative of a and is denoted by −a.

Theorem 1.1

If a and b are any numbers,then there is one and only one number x such that a + x  b This number x is given by x  b + ( − a).

Proof

see that

Therefore, (i) holds To prove (ii), suppose that x is some number such

Now,

 [a + (−a)] + x  0 + x  x.

1.1 Axioms for a Field 3

established

Notation The number b + (−a) isdenoted by b − a.

The next theorem establishes familiar properties of negative numbers

Theorem 1.2

(i) If a is a number,then −( − a)  a.

(ii) If a and b are numbers,then

−(a + b))  ( − a) + ( − b).

Proof

(i) From the definition of negative, we have

−(−a) To establish (ii), we know from the definition of negative that

If a and b are numbers,there is one and only one number,denoted by ab(or

a × b or a · b),called their product.

M-4 Existence of a unit

There is one and only one number u,different from zero, such that au  a

for every number a This number u is called the unit and(as is customary)

is denoted by 1.

M-5 Existence of a reciprocal

For each number a different from zero there is one and only one number x such that ax  1 This number x is called the reciprocal of a(or the inverse

of a)and is denoted by a−1(or 1/a).

AxiomsM-1 through M-4 are the parallelsof AxiomsA-1 through A-4with addition replaced by multiplication However, M-5 isnot the exact

reason for thisisgiven below in Theorem 1.3, where it isshown that theresult of multiplication of any number by zero is zero We are familiarwith the fact that says that division by zero is excluded

Special Axiom on distributivity

point and line are undefined Of course, we have an intuitive idea of the

meaning of these two undefined terms, but in the framework of Euclideangeometry it is not possible to define them In the axioms for algebra given

above, the term number isundefined We shall interpret number to mean

real number (positive, negative, or zero) in the usual sense that we give

to it in elementary courses Actually, the above axioms for a field holdfor many systems, of which the collection of real numbers is only one.For example, all the axioms stated so far hold for the system consisting

of all complex numbers Also, there are many systems, each consisting of

a finite number of elements(finite fields), that satisfy all the axioms we

have stated until now

Additional axioms are needed if we insist that the real number system

be the only collection satisfying all the given axioms The additional axiom

required for this purpose is discussed in Section 1.3

Theorem 1.3

If a is any number,then a · 0  0.

1.1 Axioms for a Field 5

Proof

the distributive law (Axiom D), we find that

The proof of Theorem 1.4 isjust like the proof of Theorem 1.1 with

are left to the reader

Notation The expression “if and only if,” a technical one used

fre-quently in mathematics, requires some explanation Suppose A and B stand for propositions that may be true or false To say that A istrue if B istrue meansthat the truth of B impliesthe truth of A The statement A is true only if B istrue meansthat the truth of A impliesthe truth of B Thus the shorthand statement “A istrue if and only if B istrue” isequivalent

to the double implication that the truth of A impliesand isimplied by the

and only if,” and we write

for the two implicationsstated above The term necessary and sufficient is

used as a synonym for “if and only if.”

We now establish the familiar principle that is the basis for the solution

of quadratic and other algebraic equationsby factoring

Theorem 1.5

(i) We have ab  0 if and only if a  0 or b  0 or both.

(ii) We have a  0 and b  0 if and only if ab  0.

Proof

We must prove two statements in each of parts (i) and (ii) To prove (i),

are two cases: either a  0 or a  0 If a  0, the result follows If a  0,

then we see that

b  1 · b  (a−1a)b  a−1(ab)  a−1· 0  0.

statement “a  0 or b  0 or both.” Thus(i) applies For the second part

Theorem 1.6

(i) If a  0,then a−1 0 and [(a−1)−1] a.

(ii) If a  0 and b  0,then (a · b)−1  (a−1) · (b−1).

The proof of thistheorem islike the proof of Theorem 1.2 with addition

replaced by multiplication, 0 replaced by 1, and (−a), (−b) replaced by

a−1, b−1 We leave the detailsto the reader Note that if a  0, then a−1  0

b  a−1.

Using Theorem 1.3 and the distributive law, we easily prove the laws

of signs stated as Theorem 1.7 below We emphasize that the numbers a

and b may be positive, negative, or zero.

Also, the negative of a · b hasthe property that a · b + [−(a · b)]  0 Hence

(i) by interchanging a and b The proof of (iii) isleft to the reader.

We now show that the laws of fractions, asgiven in elementary algebra,

follow from the axiomsand theoremsabove

Notation We introduce the following standard symbols for a · b−1:

a · b−1 a

These symbols, representing an indicated division, are called fractions.

fraction with denominator zero hasno meaning.

Theorem 1.8

(i) For every number a,the equality a/1  a holds.

(ii) If a  0, then a/a  1.

1.2 Natural Numbers and Sequences 7

Proof

(i) We have a/1  (a · 1−1)  (a · 1−1) · 1  a(1−1· 1)  a · 1  a (ii) If

Problems

(ac) · (bd)  (ab) · (cd).

holds

proof

1.2 Natural Numbers and Sequences

Traditionally we build the real number system by a sequence of ments We start with the positive integers and extend that system toinclude the positive rational numbers (quotients, or ratios, of integers).The system of rational numbers is then enlarged to include all positive

enlarge-real numbers; finally we adjoin the negative numbers and zero to obtainthe collection of all real numbers.

The system of axioms in Section 1.1 does not distinguish (or evenmention) positive numbers To establish the relationship between theseaxioms and the real number system, we begin with a discussion of

natural numbers As we know, these are the same as the positive

integers

We can obtain the totality of natural numbersby starting with the

way the collection of natural numbersisgenerated Actually it ispossible

to give an abstract definition of natural number, one that yields the samecollection and islogically more satisfactory Thisisdone in Section 1.4,where the principle of mathematical induction is established and illus-trated Meanwhile, we shall suppose that the reader is familiar with allthe usual properties of natural numbers

The axiomsfor a field given in Section 1.1 determine addition and

multiplication for any two numbers On the basis of these axioms we were able to define the sum and product of three numbers Before describing

the processof defining sumsand productsfor more than three elements,

we now recall several definitions and give some notations that will beused throughout the book

Definitions

byR3, and so on A relation fromR1toR1isa set of ordered pairsof realnumbers; that is, a relation fromR1toR1isa set inR2 The domain of this

pairs The range of the relation is the set of all the second elements in

the ordered pairs Observe that the range is also a set inR1

such a function The word mapping isa synonym for function.

If D isthe domain of f and S is its range, we shall also use the notation

the pair (x, y) isone of the ordered pairsthat constitute the function.

a given function f , the unique number in the range corresponding to an

this relationship We assume that the reader is familiar with functionalnotation

A sequence isa function that hasasitsdomain some or all of the

natural numbers If the domain consists of a finite number of positive

1.2 Natural Numbers and Sequences 9

integers, we say that the sequence is finite Otherwise, the sequence is called infinite In general, the elementsin the domain of a function do

not have any particular order In a sequence, however, there is a naturalordering of the domain induced by the usual order in terms of size that

we give to the positive integers For example, if the domain of a sequence

consists of the numbers 1, 2, , n, then the elementsof the range, that

is, the terms of the sequence, are usually written in the same order as the natural numbers If the sequence (function) is denoted by a, then the termsof the sequence are denoted by a1, a2, , a n or, sometimes

by a(1), a(2), , a(n) The element a i or a(i) iscalled the ith term of the

sequence If the domain of a sequence a isthe set of all natural numbers

(so that the sequence is infinite), we denote the sequence by

a1, a2, , a n , or {a n }.

Definitions

For all integers n ≥ 1, the sum and product of the numbers a1, a2, , a n

are defined respectively by b n and c n:

isin general use asa compact notation for product

the reader isfamiliar with the lawsof exponentsand the customary rulesfor adding, subtracting, and multiplying polynomials These rules are asimple consequence of the axioms and propositions above.

The decimal system of writing numbers (or the system with any base)

depends on a representation theorem that we now state If n isany natural number, then there is one and only one representation for n of the form

n  d0( 10) k + d1( 10) k−1 + · · · + d k−1( 10) + d k

0, 1, 2, , 9, and d0  0 The numbers0, 1, 2, , 9 are called digits of

the decimal system On the basis of such a representation, the rules of

arithmetic follow from the corresponding rulesfor polynomialsin x with

For completeness, we define the terms integer, rational number, and

irrational number.

Definitions

A real number isan integer if and only if it iseither zero, a natural

number, or the negative of a natural number A real number r issaid to

be rational if and only if there are integers p and q, with q 0, such that

It is clear that the sum and product of a finite sequence of integersisagain an integer, and that the sum, product, or quotient of a finitesequence of rational numbersisa rational number

The rule for multiplication of fractionsisgiven by an extension ofTheorem 1.7 that may be derived by mathematical induction

We emphasize that the axioms for a field given in Section 1.1 implyonly theoremsconcerned with the operationsof addition, subtraction,multiplication, and division The exact nature of the elements in the field

isnot described For example, the axiomsdo not imply the existence of a

number whose square is 2 In fact, if we interpret number to be “rationalnumber” and consider no others, then all the axioms for a field are sat-

isfied The rational number system forms a field An additional axiom is

2

1.3 Inequalities 11

Problems

(x, y) We define the inverse relation of T asthe set of ordered pairs

(x, y) where (y, x) belongsto T Let a function a be given asa nite sequence a1, a2, , a n Under what conditionswill the inverse

fi-relation of a be a function?

3 Show that if a1, a2, , a5isa sequence of 5 terms, then5

i1 a i  0

if and only if at least one term of the sequence is zero

4 If a1, a2, , a n , a n+1is any sequence, show thatn+1

rulesof addition and multiplication:

where a and b are any rational numbers, satisfies all the axioms for

a field if the usual rules for addition and multiplication are used

four elements0, 1, 2, 3 form the elementsof a field? Prove your

statement [Hint: In the multiplication table each row, other than

the one consisting of zeros, must contain the symbols 0, 1, 2, 3 insome order.]

Doesthiscollection satisfy the axiomsfor a field?

1.3 Inequalities

The axioms for a field describe many number systems If we wish todescribe the real number system to the exclusion of other systems, ad-ditional axioms are needed One of these, an axiom that distinguishespositive from negative numbers, is the Axiom of inequality

Axiom I

(Axiom of inequality)Among all the numbers of the system, there is a set called the positive numbers that satisfies the two conditions: (i)for any number

a exactly one of the following three alternatives holds:a is positive or a  0 or

−a is positive; (ii)any finite sum or product of positive numbers is positive.

When we add Axiom I to those of Section 1.1, the resulting system ofaxioms is applicable only to those number systems that have a linear or-der For example, the system of complex numbers does not satisfy Axiom

I but does satisfy all the axioms for a field Similarly, it is easy to see thatthe system described in Problem 6 of Section 1.2 does not satisfy Axiom I.However, both the real number system and the rational number systemsatisfy all the axioms given thus far

Definition

numbers, we say that a > b(read:a is greater than b)whenever a − b is

positive

It is convenient to adopt a geometric point of view and to associate

a horizontal axiswith the totality of real numbers We select any nient point for the origin and call pointsto the right of the origin positivenumbersand pointsto the left negative numbers(Figure 1.1) For everyreal number there will correspond a point on the line, and conversely,

conve-every point will represent a real number Then the inequality a < b may

be read: a is to the left of b Thisgeometric way of looking at inequalities

isfrequently of help in solving problems

Figure 1.1

It ishelpful to introduce the notion of an interval of numbers or points.

If a and b are numbers(asshown in Figure 1.2), then the open interval

from a to b isthe collection of all numbersthat are both larger than a and

smaller than b That is, an open interval consists of all numbers between

x < bare true A compact way of writing these two inequalities is

a < x < b.

The closed interval from a to b consists of all the points between a and

1.3 Inequalities 13

Figure 1.2

Figure 1.3

Suppose a number x iseither equal to a or larger than a, but we don’t

greater than or equal to a Similarly, x ≤ b isread: x is less than or equal

to b, and meansthat x may be either smaller than b or may be b itself A

compact way of designating all points x in the closed interval from a to b

is

An interval that containsthe endpoint b but not a issaid to be half-open

on the left Such an interval consists of all points x that satisfy the double

inequality

a < x ≤ b.

Similarly, an interval containing a but not b iscalled half-open on the

right, and we write

a ≤ x < b.

following way:

(a, b) for the open interval a < x < b,

(a, b] for the interval half-open on the left a < x ≤ b,

[a, b) for the interval half-open on the right a ≤ x < b.

We extend the idea of interval to include the unbounded cases For

exam-ple, consider the set of all numberslarger than 7 Thisset may be thought

of asan interval beginning at 7 and extending to infinity to the right (seeFigure 1.4) Of course, infinity is not a number, but we use the symbol

inequality

7 < x <∞

to represent this set Similarly, the symbol (−∞, 12) stands for all

num-bers less than 12 We also use the inequalities−∞ < x < 12 to represent

thisset

Figure 1.4 Definition

The solution of an equation or inequality in one unknown, say x, is the

collection of all numbersthat make the equation or inequality a true

statement Sometimesthisset of numbersiscalled the solution set For

example, the inequality

3x − 7 < 8

hasasitssolution set all numberslessthan 5 To demonstrate thiswe

argue in the following way If x is a number that satisfies the above

in-equality, we can add 7 to both sides and obtain a true statement That is,the inequality

holds Now, dividing both sides by 3, we obtain x < 5; therefore, if x is

a solution, then it islessthan 5 Strictly speaking, however, we have not

proved that every number that islessthan 5 isa solution In an actual

proof, we begin by supposing that x isany number lessthan 5; that is,

x <5 We multiply both sidesof thisinequality by 3 and then subtract 7from both sides to get

3x − 7 < 8, the original inequality Since the hypothesis that x islessthan 5 implies

the original inequality, we have proved the result The important thing

to notice is that the proof consisted in reversing the steps of the original argument that led to the solution x < 5 in the first place So long as each

step taken is reversible, the above procedure is completely satisfactoryfor obtaining solutions of inequalities

compact form We write

and

The solution set is the interval (−∞, 5).

Notation It is convenient to introduce some terminology and symbols

concerning sets In general, a set isa collection of objects The objects

may have any character (number, points, lines, etc.) so long as we know

which objectsare in a given set and which are not If S isa set and P is

1.3 Inequalities 15

P belongs to S If S1and S2 are two sets, their union, denoted by S1∪ S2,consists of all objects each of which is in at least one of the two sets The

intersection of S1 and S2, denoted by S1∩ S2, consists of all objects each

of which isin both sets

Figure 1.5

1.5 and S2is the vertically shaded set, then S1∪ S2 consists of the entire

may form the union and intersection of any number of sets When we

write S1 ∪ S2 ∪ ∪ S n for the union S of the n sets S1, S2, , S n, then

i1 S i asshorthand for the union of n sets The intersection of n sets S1, S2, , S n iswritten S1 ∩ S2 ∩ ∩ S n or,

i1 S i It may happen that two sets S1and S2 have no elements

in common In such a case their intersection is empty, and we use the

term empty set for the set that is devoid of members.

Most often we will deal with sets each of which is specified by someproperty or propertiesof itselements For example, we may speak of theset of all even integers or the set of all rational numbers between 0 and

1 We employ the special notation

{x : x  2n and n isan integer}

to represent the set of all even integers In this notation the letter x

stands for a generic element of the set, and the properties that determinemembership in the set are listed to the right of the colon The notation

{x : x ∈ (0, 1) and x isrational}

represents the set of rational numbers in the open interval (0, 1) If a s et

hasonly a few elements, it may be specified by listing

are the numbers–2, 0, and 1

The words and and or have precise meanings when used in connection

with sets and their properties The set consisting of elements that have

property A or property B isthe union of the set having property A and the set having property B Symbolically, we write

{x : x hasproperty A or property B}

 {x : x hasproperty A} ∪ {x : x hasproperty B}.

The set consisting of elements that have both property A and property B isthe intersection of the set having property A with the set having property

B In set notation, we write

{x : x hasproperty A and property B}

 {x : x hasproperty A} ∩ {x : x hasproperty B}.

If A and B are two sets and if every element of A isalso an element of

We give two examples that illustrate how set notation is used

Example 1

Solve for x:

3

Solution Since we don’t know in advance whether x ispositive or

nega-tive, we cannot multiply by x unless we impose additional conditions We therefore separate the problem into two cases: (i) x ispositive, and (ii) x

is negative The desired solution set can be written as the union of the

sets S1and S2defined by

For numbersin S2, multiplication of the inequality by the negative

Problems

where a and b are rational Does this field satisfy Axiom I? Justify

your answer

7 where a isa real

ordering in such a way that it satisfies Axiom I Does this set form afield?

satisfies all the axioms for a field

In Problems 4 through 7 find in each case the solution set as an interval,and plot

0 and 10 is not inductive, since it satisfies (a) but not (b) No finite set ofreal numbers can be inductive since (b) will be violated at some stage.

Definition

A real number n issaid to be a natural number if it belongsto every

inductive set of real numbers The set of all natural numberswill be

inductive set, 1 must always be a member of every inductive set As we

know, the set of natural numbers N is identical with the set of positive integers.

Theorem 1.9

The set N of all natural numbers is an inductive set.

in-ductive set As we remarked above, (a) holds Now suppose that k isan

inductive

The principle of mathematical induction iscontained in the next rem, which asserts that any inductive set of natural numbers must consist

theo-of the entire collectionN

Theorem 1.10 (Principle of mathematical induction)

If S is an inductive set of natural numbers,then S  N.

Proof

Since S isan inductive set, we know from the definition of natural number

We now illustrate how the principle of mathematical induction is plied in practice The reader may not recognize Theorem 1.10 asthestatement of the familiar principle of mathematical induction So we shallprove a formula using Theorem 1.10

ap-Example 1

Show that

2

for every natural number n.

Solution Let S be the set of natural numbers n for which the formula (1.1)

holds We shall show that S isan inductive set.

to both sides, we see that

Combining (a) and (b), we conclude that S isan inductive set of natural

numbers and so consists of all natural numbers Therefore formula (1.1)holdsfor all natural numbers

1.4 Mathematical Induction 21

Now mathematical induction can be used to establish Proposition 1.2

of Section 1.2 We state the result in the next theorem

integers, the fact that a natural number ispositive followsfrom the axiom

Theorem 1.12 (The well-ordering principle)

Any nonempty set T of natural numbers contains a smallest element.

Proof

Let k be a member of T We define a set S of natural numbersby the

relation

The set S contains a portion (not necessarily all) of the set consisting of

a smallest element, which we denote by s We now show that s isthe

t isany element of T different from s If t > k, then the inequality k ≥ s

any other element of T, and the proof iscomplete.

Problems

induction to establish the given formula

7 Suppose p, q, and r are natural numberssuch that p + q < p + r Show that q < r.

that q < r.

(c) Isthe set of all complex numbersan inductive set?

In each of Problems10 through 15 use the Principle of mathematicalinduction to establish the given assertion

2.1 Continuity

Most of the functions we study in elementary calculus are described byformulas These functions almost always possess derivatives In fact, aportion of any first course in calculus is devoted to the development ofroutine methodsfor computing derivatives However, not all functions

As we progress in the study of analysis, it is important to enlargesubstantially the class of functions we examine Functions that possessderivatives everywhere form a rather restricted class; extending thisclassto functionsthat are differentiable except at a few isolated pointsdoes not enlarge it greatly We wish to investigate significantly largerclasses of functions, and to do so we introduce the notion of a continuousfunction

Definitions

is continuous at a if (i) the point a isin an open interval I contained in

that

|f(x) − f(a)| < ε whenever |x − a| < δ.

23

If f iscontinuousat each point of a set S, we say that f is continuous on

itsdomain

The geometric significance of continuity at a point a isindicated in

Figure 2.1 We recall that the inequality|f(x) − f(a)| < ε isequivalent to

the double inequality

four lineshasitscenter at the point with coordinates(a, f(a)) The

geo-metric interpretation of continuity at a point may be given in termsof

thisrectangle A function f iscontinuousat a if for each ε > 0 there isa number δ > 0 such that the graph of f remainswithin the rectangle for all x in the interval (a − δ, a + δ).

It isusually very difficult to verify continuity directly from the

defini-tion Such verification requiresthat for every positive number ε we exhibit

a number δ and show that the graph of f liesin the appropriate rectangle However, if the function f is given by a sufficiently simple expression, it

is sometimes possible to obtain an explicit value for the quantity δ

Solution We sketch the graph of f and observe that f isdecreasing for

x > −1 (see Figure 2.2) The equations f(x)−f(1)  0.1, f(x)−f(1)  −0.1 can be solved for x We find

Since f isdecreasing in the interval 23 < x < 32, it isclear that the graph

given quantity ε,then any smaller(positive)value for δ may also be used for the same number ε.

2.1 Continuity 27

Definition

Suppose that a and L are real numbersand f isa function from a domain

function f tends to L as a limit as x tends to a if (i) there isan open

interval I containing a that, except possibly for the point a, iscontained

in D, and (ii) for each positive number ε there isa positive number δ such

(i) We see that a function f iscontinuousat a if and only if a isin the

(ii) The condition 0 < |x − a| < δ (excluding the possibility x  a) is

used rather than the condition|x−a| < δ asin the definition of continuity, since f may not be defined at a itself.

Problems

In Problems1 through 8 the functionsare continuousat the value a given.

In each case find a value δ corresponding to the given value of ε such that

the definition of continuity is satisfied Draw a graph

In Problems9 through 17 the functionsare defined in an interval about the

given value of a but not at a Determine a value δ such that for the given valuesof L and ε, the statement |f(x) − L| < ε whenever 0 < |x − a| < δ

isvalid Sketch the graph of the given function

∗15. f(x)  (x − 1)(3x − 1), a  1, L  3, ε  0.1.

18 Show that limx→0( sin(1/x)) doesnot exist.

x→0xlog|x|  0.

2.2 Limits

The basic theoremsof calculusdepend for their proofson certain standardtheorems on limits These theorems are usually stated without proof in afirst course in calculus In this section we fill the gap by providing proofs

of the customary theorems on limits These theorems are the basis for theformulasfor the derivative of the sum, product, and quotient of functionsaswell asfor the Chain rule

Theorem 2.1 (Limit of a constant)

If c is a number and f (x)  c for all x on R1,then for every real number a

Theorem 2.2 (Obvious limit)

If f (x)  x for all x on R1and a is any real number,then

lim

x →a g(x)  L1+ L2.

2.2 Limits 29

Proof

Let ε > 0 be given Then, using the quantity ε/2, there are positive

|f1(x) − L1| < ε

2 for all x satisfying 0 < |x − a| < δ1

and

|f2(x) − L2| < ε

2 for all x satisfying 0 < |x − a| < δ2.

|g(x) − (L1+ L2) |  |f1(x) − L1+ f2(x) − L2|

≤ |f1(x) − L1| + |f2(x) − L2|, and for 0 < |x − a| < δ, it followsthat

The corollary may be established by induction

Theorem 2.4(Limit of a product)

Suppose that

lim

x→a f1(x)  L1 and lim

x→a f2(x)  L2 Define g(x)  f1(x) · f2(x) Then

If ε1and ε2are positive numbers (their exact selection will be made later),