# An Introduction to Proof through Real Analysis

## Books An engaging and accessible introduction to mathematical proof incorporating ideas from real analysis

A mathematical proof is an inferential argument for a mathematical statement. Since the time of the ancient Greek mathematicians, the proof has been a cornerstone of the science of mathematics. The goal of this book is to help students learn to follow and understand the function and structure of mathematical proof and to produce proofs of their own.

An Introduction to Proof through Real Analysis is based on course material developed and refined over thirty years by Professor Daniel J. Madden and was designed to function as a complete text for both first proofs and first analysis courses. Written in an engaging and accessible narrative style, this book systematically covers the basic techniques of proof writing, beginning with real numbers and progressing to logic, set theory, topology, and continuity. The book proceeds from natural numbers to rational numbers in a familiar way, and justifies the need for a rigorous definition of real numbers. The mathematical climax of the story it tells is the Intermediate Value Theorem, which justifies the notion that the real numbers are sufficient for solving all geometric problems.

• Concentrates solely on designing proofs by placing instruction on proof writing on top of discussions of specific mathematical subjects

• Departs from traditional guides to proofs by incorporating elements of both real analysis and algebraic representation

• Written in an engaging narrative style to tell the story of proof and its meaning, function, and construction

• Uses a particular mathematical idea as the focus of each type of proof presented

• Developed from material that has been class-tested and fine-tuned over thirty years in university introductory courses

An Introduction to Proof through Real Analysis is the ideal introductory text to proofs for second and third-year undergraduate mathematics students, especially those who have completed a calculus sequence, students learning real analysis for the first time, and those learning proofs for the first time.

Daniel J. Madden, PhD, is an Associate Professor of Mathematics at The University of Arizona, Tucson, Arizona, USA. He has taught a junior level course introducing students to the idea of a rigorous proof based on real analysis almost every semester since 1990. Dr. Madden is the winner of the 2015 Southwest Section of the Mathematical Association of America Distinguished Teacher Award.

Jason A. Aubrey, PhD, is Assistant Professor of Mathematics and Director, Mathematics Center of the University of Arizona.

List of Figures xiii

Preface xv

Introduction xvii

Part I A First Pass at Defining ℝ 97

1 Beginnings 3

1.1 A naive approach to the natural numbers 3

1.1.1 Preschool: foundations of the natural numbers 3

1.1.2 Kindergarten: addition and subtraction 5

1.1.3 Grade school: multiplication and division 8

1.1.4 Natural numbers: basic properties and theorems 11

1.2 First steps in proof 12

1.2.1 A direct proof 12

1.2.2 Mathematical induction 14

1.3 Problems 17

2 The Algebra of the Natural Numbers 19

2.1 A more sophisticated look at the basics 19

2.1.1 An algebraic approach 21

2.2 Mathematical induction 22

2.2.1 The theorem of induction 24

2.3 Division 27

2.3.1 The division algorithm 27

2.3.2 Odds and evens 30

2.4 Problems 34

3 Integers 37

3.1 The algebraic properties of ℕ 37

3.1.1 The algebraic definition of the integers 40

3.1.2 Simple results about integers 42

3.1.3 The relationship between ℕ and ℤ 45

3.2 Problems 47

4 Rational Numbers 49

4.1 The algebra 49

4.1.1 Surveying the algebraic properties of ℤ 49

4.1.2 Defining an ordered field 50

4.1.3 Properties of ordered fields 51

4.2 Fractions versus rational numbers 53

4.2.1 In some ways they are different 53

4.2.2 In some ways they are the same 56

4.3 The rational numbers 58

4.3.1 Operations are well defined 58

4.3.2 ℚ is an ordered field 63

4.4 The rational numbers are not enough 67

4.4.1 √2 is irrational 67

4.5 Problems 70

5 Ordered Fields 73

5.1 Other ordered fields 73

5.2 Properties of ordered fields 74

5.2.1 The average theorem 74

5.2.2 Absolute values 75

5.2.3 Picturing number systems 78

5.3 Problems 79

6 The Real Numbers 81

6.1 Completeness 81

6.1.1 Greatest lower bounds 81

6.1.2 So what is complete? 82

6.1.3 An alternate version of completeness 84

6.2 Gaps and caps 86

6.2.1 The Archimedean principle 86

6.2.2 The density theorem 87

6.3 Problems 90

6.4 Appendix 93

Part II Logic, Sets, and Other Basics 97

7 Logic 99

7.1 Propositional logic 99

7.1.1 Logical statements 99

7.1.2 Logical connectives 100

7.1.3 Logical equivalence 104

7.2 Implication 105

7.3 Quantifiers 107

7.3.1 Specification 108

7.3.2 Existence 108

7.3.3 Universal 109

7.3.4 Multiple quantifiers 110

7.4 An application to mathematical definitions 111

7.5 Logic versus English 114

7.6 Problems 116

7.7 Epilogue 118

8 Advice for Constructing Proofs 121

8.1 The structure of a proof 121

8.1.1 Syllogisms 121

8.1.2 The outline of a proof 123

8.2 Methods of proof 127

8.2.1 Direct methods 127

8.2.1.1 Understand how to start 127

8.2.1.2 Parsing logical statements 129

8.2.1.3 Mathematical statements to be proved 131

8.2.1.4 Mathematical statements that are assumed to be true 135

8.2.1.5 What do we know and what do we want? 138

8.2.1.6 Construction of a direct proof 138

8.2.1.7 Compound hypothesis and conclusions 139

8.2.2 Alternate methods of proof 139

8.2.2.1 Contrapositive 139

8.3 An example of a complicated proof 145

8.4 Problems 149

9 Sets 151

9.1 Defining sets 151

9.2 Starting definitions 153

9.3 Set operations 154

9.3.1 Families of sets 157

9.4 Special sets 160

9.4.1 The empty set 160

9.4.2 Intervals 162

9.5 Problems 168

9.6 Epilogue 171

10 Relations 175

10.1 Ordered pairs 175

10.1.1 Relations between and on sets 176

10.2 A total order on a set 179

10.2.1 Definition 179

10.2.2 Definitions that use a total order 179

10.3 Equivalence relations 182

10.3.1 Definitions 182

10.3.2 Equivalence classes 184

10.3.3 Equivalence partitions 185

10.3.3.1 Well defined 187

10.4 Problems 188

11 Functions 193

11.1 Definitions 193

11.1.1 Preliminary ideas 193

11.1.2 The technical definition 194

11.1.2.1 A word about notation 197

11.2 Visualizing functions 202

11.2.1 Graphs in ℝ202

11.2.2 Tables and arrow graphs 202

11.2.3 Generic functions 203

11.3 Composition 204

11.3.1 Definitions and basic results 204

11.4 Inverses 206

11.5 Problems 210

12 Images and preimages 215

12.1 Functions acting on sets 215

12.1.1 Definition of image 215

12.1.2 Examples 217

12.1.3 Definition of preimage 218

12.1.4 Examples 220

12.2 Theorems about images and preimages 222

12.2.1 Basics 222

12.2.2 Unions and intersections 228

12.3 Problems 231

13 Final Basic Notions 235

13.1 Binary operations 235

13.2 Finite and infinite sets 236

13.2.1 Objectives of this discussion 236

13.2.2 Why the fuss? 237

13.2.3 Finite sets 239

13.2.4 Intuitive properties of infinite sets 240

13.2.5 Counting finite sets 241

13.2.6 Finite sets in a set with a total order 243

13.3 Summary 246

13.4 Problems 246

13.5 Appendix 248

13.6 Epilogue 257

Part III A Second Pass at Defining ℝ 261

14 ℕ, ℤ, and ℚ 263

14.0.1 Basic properties of the natural numbers 263

14.0.2 Theorems about the natural numbers 266

14.1 The integers 267

14.1.1 An algebraic definition 267

14.1.2 Results about the integers 268

14.1.3 The relationship between natural numbers and integers 270

14.2 The rational numbers 272

14.3 Problems 279

15 Ordered Fields and the Real Numbers 281

15.1 Ordered fields 281

15.2 The real numbers 284

15.3 Problems 289

15.4 Epilogue 290

15.4.1 Constructing the real numbers 290

16 Topology 293

16.1 Introduction 293

16.1.1 Preliminaries 293

16.1.2 Neighborhoods 295

16.1.3 Interior, exterior, and boundary 298

16.1.4 Isolated points and accumulation points 300

16.1.5 The closure 303

16.2 Examples 305

16.3 Open and closed sets 311

16.3.1 Definitions 311

16.3.2 Examples 315

16.4 Problems 316

17 Theorems in Topology 319

17.1 Summary of basic topology 319

17.2 New results 321

17.2.1 Unions and intersections 321

17.2.2 Open intervals are open 325

17.2.3 Open subsets are in the interior 327

17.2.4 Topology and completeness 328

17.3 Accumulation points 329

17.3.1 Accumulation points are aptly named 329

17.3.2 For all A ⊆ ℝ, A′ is closed 333

17.4 Problems 341

18 Compact Sets 345

18.1 Closed and bounded sets 345

18.1.1 Maximums and minimums 345

18.2 Closed intervals are special 354

18.3 Problems 356

19 Continuous Functions 359

19.1 First semester calculus 359

19.1.1 An intuitive idea of a continuous function 359

19.1.2 The calculus definition of continuity 360

19.1.3 The official mathematical definition of continuity 363

19.1.4 Examples 364

19.2.1 Three specific functions 374

19.2.2 Multiplying a continuous function by a constant 377

19.2.4 Multiplying continuous functions 379

19.2.5 Polynomial functions 382

19.2.6 Composition of continuous functions 382

19.2.7 Dividing continuous functions 384

19.2.8 Gluing functions together 385

19.3 Problems 386

20 Continuity and Topology 389

20.1 Preliminaries 389

20.1.1 Continuous images mess up topology 389

20.2 The topological definitions of continuity 391

20.3 Compact images 397

20.3.1 The main theorem 397

20.3.2 The extreme value theorem 400

20.3.3 The intermediate value theorem 401

20.4 Problems 404

21 A Few Final Observations 407

21.1 Inverses of continuous functions 407

21.1.1 A strange example 408

21.1.2 The theorem about inverses of continuous functions 409

21.2 The intermediate value theorem and continuity 412

21.3 Continuity at discrete points 413

21.4 Conclusion 413

Index 415

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