An Introduction to Proof through Real Analysis

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 square root 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 TheRealNumbers 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 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 R2 202 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 subset of or equal to 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 Theorems about continuity 374 19.2.1 Three specific functions 374 19.2.2 Multiplying a continuous function by a constant 377 19.2.3 Adding continuous functions 378 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