A course in real analysis

A Course in Real Analysis provides a rigorous treatment of the foundations of differ￾ential and integral calculus at the advanced undergraduate level.

The first part of the text presents the calculus of functions of one variable. This part

covers traditional topics, such as sequences, continuity, differentiability, Riemann inte￾grability, numerical series, and the convergence of sequences and series of functions.

It also includes optional sections on Stirling’s formula, functions of bounded variation,

Riemann–Stieltjes integration, and other topics.

The second part focuses on functions of several variables. It introduces the topological

ideas needed (such as compact and connected sets) to describe analytical properties

of multivariable functions. This part also discusses differentiability and integrability of

multivariable functions and develops the theory of differential forms on surfaces in n

.

The third part consists of appendices on set theory and linear algebra as well as solu￾tions to some of the exercises.

Features

• Provides a detailed axiomatic account of the real number system

• Develops the Lebesgue integral on n

from the beginning

• Gives an in-depth description of the algebra and calculus of differential forms on

surfaces in n

• Offers an easy transition to the more advanced setting of differentiable manifolds

by covering proofs of Stokes’s theorem and the divergence theorem at the

concrete level of compact surfaces in n

• Summarizes relevant results from elementary set theory and linear algebra

• Contains over 90 figures that illustrate the essential ideas behind a concept or

proof

• Includes more than 1,600 exercises throughout the text, with selected solutions

in an appendix

With clear proofs, detailed examples, and numerous exercises, this book gives a thor￾ough treatment of the subject. It progresses from single variable to multivariable func￾tions, providing a logical development of material that will prepare readers for more

advanced analysis-based studies.

K22153

www.crcpress.com

A

COURSE IN

REAL

ANALYSIS

A COURSE IN

REAL ANALYSIS

HUGO D. JUNGHENN

JUNGHENN

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WITH VITALSOURCE®

EBOOK

Mathematics

A

COURSE IN

REAL

ANALYSIS

K22153_FM.indd 1 1/9/15 4:46 PM

K22153_FM.indd 2 1/9/15 4:46 PM

A

COURSE IN

REAL

ANALYSIS

HUGO D. JUNGHENN

The George Washington University

Washington, D.C., USA

K22153_FM.indd 3 1/9/15 4:46 PM

CRC Press

Taylor & Francis Group

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Boca Raton, FL 33487-2742

© 2015 by Taylor & Francis Group, LLC

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TO THE MEMORY OF MY

PARENTS

Rita and Hugo

Contents

Preface xi

List of Figures xiii

List of Tables xvii

List of Symbols xix

I Functions of One Variable 1

1 The Real Number System 3

1.1 From Natural Numbers to Real Numbers . . . . . . . . . . 3

1.2 Algebraic Properties of R . . . . . . . . . . . . . . . . . . . 4

1.3 Order Structure of R . . . . . . . . . . . . . . . . . . . . . 8

1.4 Completeness Property of R . . . . . . . . . . . . . . . . . 12

1.5 Mathematical Induction . . . . . . . . . . . . . . . . . . . . 19

1.6 Euclidean Space . . . . . . . . . . . . . . . . . . . . . . . . 24

2 Numerical Sequences 29

2.1 Limits of Sequences . . . . . . . . . . . . . . . . . . . . . . 29

2.2 Monotone Sequences . . . . . . . . . . . . . . . . . . . . . . 36

2.3 Subsequences and Cauchy Sequences . . . . . . . . . . . . . 38

2.4 Limits Inferior and Superior . . . . . . . . . . . . . . . . . 42

3 Limits and Continuity on R 47

3.1 Limit of a Function . . . . . . . . . . . . . . . . . . . . . . . 47

*3.2 Limits Inferior and Superior . . . . . . . . . . . . . . . . . 55

3.3 Continuous Functions . . . . . . . . . . . . . . . . . . . . . 59

3.4 Properties of Continuous Functions . . . . . . . . . . . . . 63

3.5 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . 67

4 Differentiation on R 73

4.1 Definition of Derivative and Examples . . . . . . . . . . . . 73

4.2 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . 80

*4.3 Convex Functions . . . . . . . . . . . . . . . . . . . . . . . 85

4.4 Inverse Functions . . . . . . . . . . . . . . . . . . . . . . . 88

4.5 L’Hospital’s Rule . . . . . . . . . . . . . . . . . . . . . . . . 94

vii

viii Contents

4.6 Taylor’s Theorem on R . . . . . . . . . . . . . . . . . . . . 100

*4.7 Newton’s Method . . . . . . . . . . . . . . . . . . . . . . . 103

5 Riemann Integration on R 107

5.1 The Riemann–Darboux Integral . . . . . . . . . . . . . . . . 107

5.2 Properties of the Integral . . . . . . . . . . . . . . . . . . . 116

5.3 Evaluation of the Integral . . . . . . . . . . . . . . . . . . . 120

*5.4 Stirling’s Formula . . . . . . . . . . . . . . . . . . . . . . . 129

5.5 Integral Mean Value Theorems . . . . . . . . . . . . . . . . . 131

*5.6 Estimation of the Integral . . . . . . . . . . . . . . . . . . . 134

5.7 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . 143

5.8 A Deeper Look at Riemann Integrability . . . . . . . . . . . 151

*5.9 Functions of Bounded Variation . . . . . . . . . . . . . . . 152

*5.10 The Riemann–Stieltjes Integral . . . . . . . . . . . . . . . . 156

6 Numerical Infinite Series 163

6.1 Definition and Examples . . . . . . . . . . . . . . . . . . . 163

6.2 Series with Nonnegative Terms . . . . . . . . . . . . . . . . 169

6.3 More Refined Convergence Tests . . . . . . . . . . . . . . . 176

6.4 Absolute and Conditional Convergence . . . . . . . . . . . . 181

*6.5 Double Sequences and Series . . . . . . . . . . . . . . . . . 188

7 Sequences and Series of Functions 193

7.1 Convergence of Sequences of Functions . . . . . . . . . . . 193

7.2 Properties of the Limit Function . . . . . . . . . . . . . . . 199

7.3 Convergence of Series of Functions . . . . . . . . . . . . . . 204

7.4 Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . 211

II Functions of Several Variables 229

8 Metric Spaces 231

8.1 Definitions and Examples . . . . . . . . . . . . . . . . . . . . 231

8.2 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . 238

8.3 Closure, Interior, and Boundary . . . . . . . . . . . . . . . 243

8.4 Limits and Continuity . . . . . . . . . . . . . . . . . . . . . 248

8.5 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . 255

*8.6 The Arzelà–Ascoli Theorem . . . . . . . . . . . . . . . . . . 263

8.7 Connected Sets . . . . . . . . . . . . . . . . . . . . . . . . . 268

8.8 The Stone–Weierstrass Theorem . . . . . . . . . . . . . . . 275

*8.9 Baire’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . 282

9 Differentiation on R

n 287

9.1 Definition of the Derivative . . . . . . . . . . . . . . . . . . . 287

9.2 Properties of the Differential . . . . . . . . . . . . . . . . . 295

9.3 Further Properties of the Differential . . . . . . . . . . . . . 301

9.4 Inverse Function Theorem . . . . . . . . . . . . . . . . . . . 306

Contents ix

9.5 Implicit Function Theorem . . . . . . . . . . . . . . . . . . 312

9.6 Higher Order Partial Derivatives . . . . . . . . . . . . . . . 318

9.7 Higher Order Differentials and Taylor’s Theorem . . . . . . 323

*9.8 Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . 330

10 Lebesgue Measure on R

n 343

10.1 General Measure Theory . . . . . . . . . . . . . . . . . . . 343

10.2 Lebesgue Outer Measure . . . . . . . . . . . . . . . . . . . . 347

10.3 Lebesgue Measure . . . . . . . . . . . . . . . . . . . . . . . . 351

10.4 Borel Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356

10.5 Measurable Functions . . . . . . . . . . . . . . . . . . . . . 360

11 Lebesgue Integration on R

n 367

11.1 Riemann Integration on R

n . . . . . . . . . . . . . . . . . . . 367

11.2 The Lebesgue Integral . . . . . . . . . . . . . . . . . . . . . 368

11.3 Convergence Theorems . . . . . . . . . . . . . . . . . . . . 379

11.4 Connections with Riemann Integration . . . . . . . . . . . 385

11.5 Iterated Integrals . . . . . . . . . . . . . . . . . . . . . . . . 388

11.6 Change of Variables . . . . . . . . . . . . . . . . . . . . . . 398

12 Curves and Surfaces in R

n 409

12.1 Parameterized Curves . . . . . . . . . . . . . . . . . . . . . 409

12.2 Integration on Curves . . . . . . . . . . . . . . . . . . . . . 412

12.3 Parameterized Surfaces . . . . . . . . . . . . . . . . . . . . 422

12.4 m-Dimensional Surfaces . . . . . . . . . . . . . . . . . . . . 432

13 Integration on Surfaces 447

13.1 Differential Forms . . . . . . . . . . . . . . . . . . . . . . . . 447

13.2 Integrals on Parameterized Surfaces . . . . . . . . . . . . . . 461

13.3 Partitions of Unity . . . . . . . . . . . . . . . . . . . . . . . 472

13.4 Integration on Compact m-Surfaces . . . . . . . . . . . . . 475

13.5 The Fundamental Theorems of Calculus . . . . . . . . . . . 478

*13.6 Closed Forms in R

n . . . . . . . . . . . . . . . . . . . . . . 495

III Appendices 503

A Set Theory 505

B Linear Algebra 509

C Solutions to Selected Problems 517

Bibliography 581

Index 583

Preface

The purpose of this text is to provide a rigorous treatment of the foundations

of differential and integral calculus at the advanced undergraduate level. It

is assumed that the reader has had the traditional three semester calculus

sequence and some exposure to elementary set theory and linear algebra. As

regards the last two subjects, appendices provide a summary of most of the

results used in the text. Linear algebra will not be needed until Part II.

The book consists of three parts. Part I treats the calculus of functions of

one variable. Here, one can find the traditional topics: sequences, continuity,

differentiability, Riemann integrability, numerical series, and convergence of

sequences and series of functions. Optional sections on Stirling’s formula,

Riemann–Stieltjes integration, and other topics are also included. As the ideas

inherent in these subjects ultimately rest on properties of real numbers, the

book begins with a careful treatment of the real number system. For this we

take an axiomatic rather than a constructive approach, guided as much by

the need for efficiency of exposition as by pedagogical preference. Of course,

presenting the real number system in this way begs the excellent question

as to whether such a system exists. It is a question we do not answer, but

the interested reader may wish to consult a text on the construction of the

real number system from the natural numbers, or even on the philosophy of

mathematics.

Part II treats functions of several variables. Many of the results in Part I,

such as the chain rule, the inverse function theorem, and the change of variables

theorem, have counterparts in Part II. The reader’s exposure to the one-variable

results should make the multivariable versions more meaningful and accessible.

As might be expected, however, some results in Part II have no counterparts in

Part I, the implicit function theorem and the iterated integral (Fubini–Tonelli)

theorem being obvious examples.

Part II begins with a chapter on metric spaces. Here we introduce the

topological ideas needed to describe some of the analytical properties of

multivariable functions. Primary among these are the notions of compact set

and connected set, which, for example, allow the extension to higher dimensions

of the extreme value and intermediate value theorems. The remainder of Part II

covers differentiability and integrability of multivariable functions. As regards

integrability, we have chosen to develop from the beginning the Lebesgue

integral rather than to the extend the Riemann integral to higher dimensions.

The additional time required for this approach is, in my view, more than offset

xi

xii Preface

by the enormous added utility of the Lebesgue integral. The last chapter of

Part II develops the theory of differential forms on surfaces in R

n. The chapter

culminates with proofs of Stokes’s theorem and the divergence theorem for

compact surfaces. It is hoped that exposure to these topics at the concrete

level of surfaces in R

n will ease the transition to more advanced courses such

as calculus on differentiable manifolds.

Part III consists of the aforementioned appendices on set theory and linear

algebra, as well as solutions to some of the over 1600 exercises found in the

text. For convenience, exercises with solutions that appear in the appendix

are marked with a superscript S. Exercises that will find important uses later

are marked with a downward arrow ⇓. Instructors with suitable bona fides

may obtain from the publisher a manual of complete solutions to all of the

exercises.

The book is an outgrowth of notes developed over many years of teaching

real analysis to undergraduates at George Washington University. The more

recent versions of these notes have been specifically tested in classes over the

last three years. During this period, the typical two-semester course closely

followed the non-starred sections of this text: Chapters 1–7 for the first semester

and 8–13 for the second. Given the wealth of material, it was necessary to

leave some proofs for students to read on their own, a not wholly unfortunate

compromise. Material in some starred sections was assigned as optional reading.

I would like to express my gratitude to the many students whose critical

eyes caught errors before they made their way into these pages. Of course, any

remaining errors are my complete responsibility. Special thanks are due to

Zehua Zhang, whose enlightened comments have improved the exposition of

several topics.

Finally, to my wife Mary for her support and understanding during the

writing of this book: thank you!

Hugo D. Junghenn

Washington, D.C.

September 2014

List of Figures

1.1 Supremum and infimum of A . . . . . . . . . . . . . . . . . 12

1.2 Greatest integer function . . . . . . . . . . . . . . . . . . . . 14

2.1 Convergence of a sequence . . . . . . . . . . . . . . . . . . . 30

2.2 Squeeze principle . . . . . . . . . . . . . . . . . . . . . . . . . 31

2.3 Interval halving process . . . . . . . . . . . . . . . . . . . . 39

2.4 Limits supremum and infimum . . . . . . . . . . . . . . . . 42

3.1 Limit of a function . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 L can’t be greater than M . . . . . . . . . . . . . . . . . . . 53

3.3 One-to-one correspondence between D and Q . . . . . . . . . 61

3.4 Intermediate value property . . . . . . . . . . . . . . . . . . 64

4.1 Trigonometric inequality . . . . . . . . . . . . . . . . . . . . 74

4.2 Local extrema . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.3 Mean value theorems . . . . . . . . . . . . . . . . . . . . . . . 81

4.4 Convex function . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.5 Convex function inequalities . . . . . . . . . . . . . . . . . . . 87

4.6 Intermediate value property implies monotonicity . . . . . . 89

4.7 Intermediate value property implies continuity . . . . . . . . 89

4.8 Newton’s method . . . . . . . . . . . . . . . . . . . . . . . . 104

5.1 Upper and lower sums . . . . . . . . . . . . . . . . . . . . . 108

5.2 The partitions P and Q . . . . . . . . . . . . . . . . . . . . 110

5.3 The partition Pn . . . . . . . . . . . . . . . . . . . . . . . . . 111

5.4 The partitions P

0

, P, and P

00

. . . . . . . . . . . . . . . . . 112

5.5 Riemann sum . . . . . . . . . . . . . . . . . . . . . . . . . . 113

5.6 The partitions P

x and P

y

. . . . . . . . . . . . . . . . . . . 122

5.7 Trapezoidal rule approximation . . . . . . . . . . . . . . . . 136

5.8 Midpoint rule approximation . . . . . . . . . . . . . . . . . . . 137

5.9 Simpson’s rule approximation . . . . . . . . . . . . . . . . . 139

5.10 The partition Q . . . . . . . . . . . . . . . . . . . . . . . . . 159

7.1 Uniform convergence . . . . . . . . . . . . . . . . . . . . . . 193

7.2 Pointwise convergence insufficient . . . . . . . . . . . . . . . . 201

xiii

xiv List of Figures

8.1 An open ball is open . . . . . . . . . . . . . . . . . . . . . . 239

8.2 The functions gn and g . . . . . . . . . . . . . . . . . . . . . 240

8.3 Convex and non-convex sets . . . . . . . . . . . . . . . . . . . 241

8.4 The neighborhoods Ux and Vx . . . . . . . . . . . . . . . . . 255

8.5 A 2ε net . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 256

8.6 A bounded set in R

n is totally bounded . . . . . . . . . . . . 257

8.7 A separation (U, V ) of E . . . . . . . . . . . . . . . . . . . . 268

8.8 C1(−1, 0) ∪ C1(1, 0) is path connected . . . . . . . . . . . . . 271

8.9 E is path connected . . . . . . . . . . . . . . . . . . . . . . . 272

8.10 A piecewise linear function . . . . . . . . . . . . . . . . . . . 276

8.11 Sawtooth function . . . . . . . . . . . . . . . . . . . . . . . . 285

9.1 The domain of argθ0

. . . . . . . . . . . . . . . . . . . . . . 310

9.2 Saddle point . . . . . . . . . . . . . . . . . . . . . . . . . . . 330

10.1 Interval grid . . . . . . . . . . . . . . . . . . . . . . . . . . . 348

10.2 Coverings . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349

10.3 Middle thirds . . . . . . . . . . . . . . . . . . . . . . . . . . 353

10.4 Ternary expansion algorithm . . . . . . . . . . . . . . . . . . 354

10.5 Decomposition into half-open intervals . . . . . . . . . . . . . 357

10.6 K = cl(E) \ U . . . . . . . . . . . . . . . . . . . . . . . . . . 358

10.7 The components of fk . . . . . . . . . . . . . . . . . . . . . 363

10.8 The components of fk+1 . . . . . . . . . . . . . . . . . . . . 363

11.1 Partition of an n-dimensional interval . . . . . . . . . . . . . . 367

11.2 Three-dimensional simplex . . . . . . . . . . . . . . . . . . . 390

11.3 Concentric cube and ball . . . . . . . . . . . . . . . . . . . . 402

11.4 The paving Qr . . . . . . . . . . . . . . . . . . . . . . . . . 403

11.5 Theorem of Pappus . . . . . . . . . . . . . . . . . . . . . . . 408

12.1 Curves in R

2

. . . . . . . . . . . . . . . . . . . . . . . . . . . 409

12.2 A piecewise smooth curve with tangent vectors . . . . . . . 410

12.3 Inscribed polygonal line . . . . . . . . . . . . . . . . . . . . 412

12.4 Vector field on E . . . . . . . . . . . . . . . . . . . . . . . . 416

12.5 Closed curve ϕ . . . . . . . . . . . . . . . . . . . . . . . . . 418

12.6 Concatenation of curves . . . . . . . . . . . . . . . . . . . . 419

12.7 Tangent spaces at p . . . . . . . . . . . . . . . . . . . . . . . 422

12.8 Affine space . . . . . . . . . . . . . . . . . . . . . . . . . . . 424

12.9 The inward unit normal . . . . . . . . . . . . . . . . . . . . . 427

12.10 Normal vector to S at p . . . . . . . . . . . . . . . . . . . . . 427

12.11 Surface of revolution . . . . . . . . . . . . . . . . . . . . . . 429

12.12 Möbius strip . . . . . . . . . . . . . . . . . . . . . . . . . . . 430

12.13 The mapping G−1

a . . . . . . . . . . . . . . . . . . . . . . . . 434

12.14 Transition mappings . . . . . . . . . . . . . . . . . . . . . . 435

12.15 Stereographic projection . . . . . . . . . . . . . . . . . . . . 436

12.16 The mapping dψx . . . . . . . . . . . . . . . . . . . . . . . . 438