An Introduction to Proof through Real Analysis

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.

Auteur

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.

Contenu
List of Figures xiii

Preface xv

Introduction xvii

Part I A First Pass at Defining 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 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 and 45

3.2 Problems 47

4 Rational Numbers 49

4.1 The algebra 49

4.1.1 Surveying the algebraic properties of 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 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

8.2.2.2 Contradiction 142

8.3 An example of a complicated proof 145

8.4 Problems 149

9 Sets 151

9.1 Defining sets 151 &l...