Kane 2016 Writing Proofs in Analysis

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Jonathan M. Kane

Writing

Proofs in

Analysis

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Jonathan M. Kane

Writing Proofs in Analysis

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Department of Mathematics University of Wisconsin - Madison Madison, WI, USA

ISBN 978-3-319-30965-1 ISBN 978-3-319-30967-5 (eBook) DOI 10.1007/978-3-319-30967-5

Library of Congress Control Number: 2016936668 © Springer International Publishing Switzerland 2016

This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed.

The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

The publisher, the authors and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, express or implied, with respect to the material contained herein or for any errors or omissions that may have been made.

Printed on acid-free paper

This Springer imprint is published by Springer Nature

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To the memory of Sylvan Burgstaller, Duane

E. Anderson, and especially James L. Nelson

who, at the University of Minnesota Duluth,

taught me the fundamentals of writing proofs

in analysis.

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Acknowledgments

I wish to thank Natalya St. Clair for her excellent work creating the illustrations appearing in this textbook. She took my crude sketches and vague ideas and turned them into pleasing artwork and instructive diagrams. I also wish to thank Daniel M. Kane, Alan Gluchoff, Thomas Drucker, and Walter Stromquist for their insightful comments about the presentation, content, and correctness of the text.

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Preface

After learning to solve many types of problems such as those found in the first courses in Algebra, Geometry, Trigonometry, and Calculus, mathematics students are usually exposed to a “transition” course where they are expected to write proofs of various theorems. I taught such a course for a dozen years and was never satisfied with the textbooks available for that course. Although such textbooks often teach the fundamentals of logic (conditionals, biconditionals, negations, truth tables) and give some common proof strategies such as mathematical induction, the textbooks failed to teach what a student needs to be thinking about when trying to construct a proof. Many of these books present a great number of well-written proofs and then ask students to write proofs of similar statements in the hope that the students will be able to mimic what they have seen. Some of these books are also designed to be used as an introductory textbook in Analysis, Abstract Algebra, Topology, Number Theory, or Discrete Mathematics, and, as such, they concentrate more on explaining the fundamentals of those topic areas than on the fundamentals of writing good proofs.

This Book Is Not Your Traditional Transition Textbook The goal of this book is to give the student precise training in the writing of proofs by explaining what elements make up a correct proof, by teaching how to construct an acceptable proof, by explaining what the student is supposed to be thinking about when trying to write a proof, and by warning about pitfalls that result in incorrect proofs. In particular, this book was written with the following directives:

• Unlike many transition books which do not give enough instruction about how to write proofs, most of the proofs presented in this text are preceded by detailed explanations describing the thought process one goes through when constructing the proof. Then a good proof is given that incorporates the elements of that discussion.

• For proofs that share the same general structure such as the proof of lim

x!af.x/ D L

for various functions, proof templates are provided that give a generic approach to writing that type of proof.

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• Many transition books begin with several chapters covering an introduction to logic, set theory, cardinal numbers, and an axiomatic construction of the real numbers. I find that students do not appreciate the details of these discussions when these concepts are presented before they are needed to write a specific proof. For example, truth tables are very helpful in verifying the truth of a complex logical statement, but it is hard for students at that level to see the connection between the truth value of a complex statement and the formation of a proof. Therefore, I introduce many of these ideas as needed within the contexts of writing Analysis proofs and have kept the introductory material to a minimum. • Many books that propose to teach students to write proofs in Analysis get carried away with covering those great topics in Analysis and cut back on the proof writing instruction. The books may start out teaching about proofs, but after a few chapters of introduction, they assume that the students now understand everything they need to know about writing proofs, and the books concentrate entirely on the concepts of Analysis. This book covers plenty of Analysis and can be used as a textbook for a typical beginning Real Analysis course, but it never loses sight of the fact that its primary focus is about proof writing skills. Certainly, one can use this book for a beginning course in Real Analysis because it thoroughly covers the standard theorems, but as a first course in proof writing, it will succeed where others fail.

If the students using this book have already had a thorough background in writing proofs, then this book could be used as a standard one-semester course in Real Analysis. Theses students might begin in Sect.2.5and, depending on their background, be expected to cover the material through Chaps.6,7, or8. On the other hand, if the students are using this book both as an introduction to proof writing and an introduction to Analysis, then the textbook can be used for a two-semester course in Real Analysis and proof writing. The first semester might aim to cover the first five or six chapters, while the second semester aims to complete the book. For most of the topics, it is important that the chapters be covered in their prescribed order. Elements of later chapters do depend on the material covered in earlier chapters.

Madison, Wisconsin, USA Jonathan M. Kane

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7.7 Rearrangement of Terms . . . 224 7.7.1 Addition of Parentheses . . . 224 7.7.2 Order of Terms. . . 225 7.7.3 Exercises . . . 230 7.8 Cauchy Products. . . 231 7.8.1 Exercises . . . 236 8 Sequences of Functions. . . 239 8.1 Pointwise Convergence. . . 239 8.1.1 Exercises . . . 241 8.2 Uniform Convergence. . . 241 8.2.1 Exercises . . . 246 8.3 Monotone Convergence . . . 246 8.4 Series of Functions. . . 252 8.5 Power Series . . . 255 8.5.1 Absolute Convergence. . . 255 8.5.2 Interval of Convergence . . . 256 8.5.3 Differentiability. . . 259 8.5.4 Taylor’s Theorem . . . 263

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CONTENTS xv

8.5.5 Arithmetic of Power Series. . . 265

8.5.6 Exercises . . . 266

8.6 Fundamental Question of Analysis . . . 267

9 Topology of the Real Line . . . 269

9.1 Interior, Exterior, and Boundary . . . 269

9.1.1 Exercises . . . 274

9.2 Open and Closed Sets. . . 274

9.2.1 Exercises . . . 278

9.3 Unions and Intersections . . . 278

9.3.1 Exercises . . . 282

9.4 Continuous Functions Applied to Sets. . . 282

9.4.1 Exercises . . . 285 9.5 Closure . . . 285 9.5.1 Exercises . . . 288 9.6 Compactness. . . 288 9.6.1 Exercises . . . 291 9.7 Connectedness. . . 291 9.7.1 Exercises . . . 293 10 Metric Spaces. . . 295

10.1 Definition of Metric Space. . . 295

10.2 Inequalities. . . 297

10.2.1 Cauchy–Schwarz Inequality. . . 297

10.2.2 Minkowski Inequality. . . 298

10.2.3 Exercises . . . 298

10.3 Examples of Metric Spaces. . . 299

10.3.1 Exercises . . . 305

10.4 Topology of Metric Spaces. . . 306

10.4.1 Exercises . . . 307

10.5 Limits in Metric Spaces . . . 308

10.5.1 Exercises . . . 311

10.6 Continuous Functions on Metric Spaces. . . 311

10.6.1 Exercises . . . 314

10.7 Homeomorphism. . . 315

10.7.1 Exercises . . . 316

10.8 Connected Metric Spaces. . . 316

10.8.1 Exercises . . . 317

10.9 Compact Metric Spaces. . . 317

10.9.1 Exercises . . . 322

10.10 Complete Metric Spaces. . . 323

10.10.1 Exercises . . . 327

10.11 Contraction Mappings. . . 327

10.11.1 Contraction Mapping Theorem. . . 327

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10.11.3 Fractals . . . 333

10.11.4 Exercises . . . 340

Books for Further Reading. . . 341

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List of Figures

Fig. 1.1 Dividing the disk with the chords from n points. . . 6

Fig. 2.1 List of implications for P ! Q. . . 15

Fig. 2.2 .A [ B/c_{is equal to A}c_{\ B}c_{. . . 26}

Fig. 2.3 Showing the least upper bound of S is s Dpr. . . 37

Fig. 2.4 Triangle inequality. . . 39

Fig. 2.5 Composition.f ı g/.x/ D z. . . 43 Fig. 3.1 lim x!af.x/ D L . . . 48 Fig. 3.2 lim x!af.x/ D L . . . 48 Fig. 3.3 lim x!af.x/ D L . . . 48 Fig. 3.4 Graph of f.x/. . . 56

Fig. 3.5 Approaching a limit as x ! 1. . . 57

Fig. 3.6 Proving bounded monotone sequences converge. . . 63

Fig. 3.7 f has no limit at x D2. . . 71

Fig. 3.8 Graph of sin1_x. . . 72

Fig. 3.9 Set with accumulation point a and isolated point b. . . 74

Fig. 3.10 Sequences approaching the lim sup and lim inf. . . 94

Fig. 4.1 Continuity of a function. . . 100

Fig. 4.2 A function equal to2x for rational x and x C 1 for irrational x. . . 104

Fig. 4.3 f.x/ D 1_xis not uniformly continuous. . . 106

Fig. 4.4 Heine–Borel Theorem first proof. . . 112

Fig. 4.5 Heine–Borel Theorem second proof. . . 113

Fig. 4.6 y and z straddle one endpoint but remain in an interval of the open cover. . . 115

Fig. 4.7 Proving that a continuous function onŒa; b is bounded. . . 125

Fig. 4.8 The maximum and minimum of a function f.x/ on an interval . . . 126

Fig. 4.9 f passing through each y between f.c/ and f .d/ . . . 128

Fig. 4.10 A function with a jump discontinuity. . . 130

Fig. 4.11 Graph of sin1_x. . . 130

Fig. 4.12 Graphs of sgn.x/ and bxc. . . 131

Fig. 4.13 Graphs of functions with discontinuities. . . 132

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Fig. 4.14 Graph of Thomae’s function. . . 132

Fig. 5.1 Slope of a Secant Line. . . 134

Fig. 5.2 Tangent Line. . . 134

Fig. 5.3 Restricting sin.x/ to get sin1.x/ . . . 142

Fig. 5.4 Graph showing maxima and minima. . . 145

Fig. 5.5 The proof of Rolle’s Theorem. . . 147

Fig. 5.6 Point c where the tangent line is parallel to the secant line. . . 148

Fig. 5.7 x2sin_x1₂and its derivative. . . 156

Fig. 6.1 The union of countably many countable sets is countable. . . 161

Fig. 6.2 Determining y using a diagonalization argument. . . 161

Fig. 6.3 Construction of the Cantor set. . . 164

Fig. 6.4 Covering a line segment with smaller and smaller squares. . . 165

Fig. 6.5 An8 8 grid of rectangles overlaying a triangle. . . 168

Fig. 6.6 Approximating the area under a curve with narrowing rectangles. . . 170

Fig. 6.7 The step function s.x/. . . 184

Fig. 6.8 Choosingjon.xj1; xj/. . . 185

Fig. 6.9 The mean value theorem for integration. . . 191

Fig. 7.1 Comparing the series with the integral in the Integral Test. . . 216

Fig. 7.2 Converging to ln2 with an alternating series. . . 220

Fig. 7.3 Rearranging terms to converge to L. . . 226

Fig. 8.1 The sequence of functions xn _{converging to a} discontinuous function. . . 240

Fig. 8.2 The sequence of functions jxjnC1n converging to the function f.x/ D jxj. . . 240

Fig. 8.3 A sequence of functions with integral 1 converging to the function f.x/ D 0. . . 241

Fig. 8.4 A sequence of functions converging uniformly. . . 242

Fig. 8.5 If continuous function fnis close to f , then f.x/ is close to f.a/. . . 243

Fig. 9.1 Interior, boundary, and exterior of a set. . . 270

Fig. 9.2 x in@.@S/, y in @S. . . 273

Fig. 9.3 An open set S, its boundary, and its complement Sc_{. . . 275}

Fig. 9.4 Showing boundaries of sets. . . 278

Fig. 9.5 The union of open sets is an open set. . . 279

Fig. 9.6 Mapping sets. . . 284

Fig. 9.7 The closure of a set. . . 287

Fig. 9.8 The setsŒ0; 1 and .4; 5/ are disconnected. . . 292

Fig. 9.9 The set C is a connected set. The set N is not a connected set. . . 293

Fig. 9.10 Graph of sin1_xwith the y-axis. . . 293

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LIST OF FIGURES xix

Fig. 10.2 Euclidean distance isR2 . . . 300

Fig. 10.3 N.0; 1/ in the Euclidean, taxicab, and supremum metrics. . . 301

Fig. 10.4 Some functions in CŒ0; 1 . . . 302

Fig. 10.5 Proving that the union of open sets is open. . . 307

Fig. 10.6 Limit of f W X ! Y as x approaches a is L . . . 308

Fig. 10.7 Limit of a sequence in a metric space. . . 310

Fig. 10.8 A compact set of a metric space is closed and bounded. . . 318

Fig. 10.9 Extrema of a continuous real-valued function on a compact set. . . 320

Fig. 10.10 Continuous bijection on a compact metric space. . . 322

Fig. 10.11 Enclosing a closed bounded set in a grid. . . 326

Fig. 10.12 Contraction mapping. . . 328

Fig. 10.13 The Mandelbrot set. . . 333

Fig. 10.14 Stages in the forming of the Sierpinski triangle. . . 333

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List of Proof Templates

Proof. . . 12

Proving A B for sets A and B . . . 20

Proving A D B for sets A and B . . . 23

Proving a function f is surjective . . . 41

Proving a function f is injective. . . 42

Proving lim x!af.x/ D L . . . 50

Proving a result using mathematical induction . . . 65

Proving lim x_!af.x/ does not exist . . . 70

Proving the function f is continuous at the point a . . . 102

Proving the function f is uniformly continuous on the set A . . . 106

Proving<X; d> is a metric space. . . 296

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Chapter 1 What Are Proofs, and Why Do We Write Them?

1.1 What Is a Proof?

A statement in Mathematics is just a sentence which could be designated as true or false. The sentences “1 C 1 D 2” and “x D 4 implies x2 _{D 16” are true} statements while “All rational numbers are positive” and “There is a real number x such that x2 C 5 D 2” are false statements. Some sentences like “Green is nice” or “Authenticity runs hot” are too ambiguous, a matter of opinion, or are just plain nonsense and cannot be said to be true or false, so mathematicians would not consider them to be statements. Mathematicians have a lot of words for kinds of statements including many that you have heard: definition, axiom, postulate, principle, conjecture, lemma, proposition, law, theorem, contradiction, and others.

You are certainly familiar with the numbers you use for counting items: 1; 2; 3; 4, and so forth. Suppose you wish to investigate statements about these numbers to see which statements hold true for all of these numbers. This is an admirable mathematical pursuit, so how would you get started? Mathematicians know from experience that if you want to begin an investigation, you better start with definitions, that is, you better make some clear statements about the objects you are about to study, because there are examples of mathematicians running off to study something without first making clear what it is they are studying, and later running into problems because they have not been consistent about how they are treating these new objects. This happened, for example, when people investigated the concept of limit before a precise definition of limit was in place. OK, so perhaps you make some statements about the numbers with which you want to work so that you are confident that you understand the collection1; 2; 3; 4; 5; : : : . What are you going to be able to do with these numbers? If you only know the names of these numbers and have a symbolic representation for each, there is not a great deal you can do with them. Perhaps you could get a collection of blocks and paint one number on each block. Then you could have fun rearranging these numbers just as you have seen done by countless children.

J.M. Kane, Writing Proofs in Analysis, DOI 10.1007/978-3-319-30967-5_1

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But more likely you are interested in investigating some properties of these numbers having to do with their order or how they behave when operated on by addition or multiplication. This, of course, would mean that you will need to make clear statements about addition and multiplication operations and a less than relationship, again, so that you do not run into problems later because you were being ambiguous. So, you might write definitions of addition, multiplication, and less than, and then make statements about how these operations behave such as a Commutative Law of Addition (mCn D nCm), a Distributive Law of Multiplication over Addition (a.b C c/ D ab C ac), and an Order Property of Addition (r < s implies r C t < s C t). These statements about how these defined quantities work are called axioms, postulates, or principles. They are statements that you accept as the guiding rules for how your mathematical objects behave and go beyond the definitions to describe and make precise just what the definitions are talking about.

Once you have made definitions and laid out your axioms, you should have the tools necessary to begin an investigation of other properties. Suppose that someone looks at a few examples and notice that1 C 9 D 10 and 10 is 2 times another number, 5. They then notice that4C12 D 16, 3C147 D 150, and 1002C6 D 1008, and all of these results are also numbers equal to 2 times another number. This might lead them to make the statement that “if you add two natural numbers together, the result is always 2 times another number.” Such a statement would be called a conjecture, a statement whose truth has not yet been determined. Of course, you know that this statement is false and came about because the investigator had not yet considered enough examples. Once they stumble upon5 C 8 D 13 and notice that 13 cannot be represented as 2 times another number, they will know that the statement does not hold true in every case.

Other conjectures such as “for every natural number a, the number a2 3a C 12 is a multiple of 2” hold up to more scrutiny. At some point in your investigation you might see a convincing argument that this conjecture is, in fact, a true statement. Such a convincing argument is what is called a proof. Once it is known that a statement has a proof, it is known as a theorem, lemma, corollary, proposition, or law. So, a proof of a statement in mathematics is a convincing argument that establishes the truth of that statement.

Some statements are very easily proved, and certainly mathematicians often set up axioms in order to make particular statements easy to prove. At first this may appear to be cheating or, at best, unproductive and uninteresting because it seems to defeat the purpose of establishing truth by dictating rules that make it trivial to establish the truth. But this is certainly not the case. It is common for mathematicians to have an intuitive idea about how a system should work before they feel that they understand it enough to set down formal definitions and axioms. Perhaps you wanted addition of all natural numbers a and b to satisfy a C b D b C a. Then it would make sense to include this rule among your axioms. The axioms are written with the idea of establishing enough structure so that the statements the mathematicians want to hold true can easily be proved. The richness of mathematics is that after assuring that the obvious can be proved from the axioms, there are many more results that can be proved that are not immediately obvious from the definitions and axioms,

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1.1 What Is a Proof? 3

statements which might never have been apparent to those who set up the system in the first place. For example, Fermat’s Last Theorem (there are no natural numbers a, b, c, and n> 2 such that an_{C b}n_{D c}n_{) is a statement about natural numbers which}

could only be conjectured after investigating a large number of examples, and stood as a conjecture for hundreds of years before a proof was provided.

Occasionally, it is shown that a conjecture is independent of the axioms; that is, neither the truth nor the falseness of the statement follows from the axioms. Two famous examples are the statements about sets known as the Axiom of Choice and the Continuum Hypothesis which have been shown to be independent of the original axioms of Zermelo-Fraenkel Set Theory. The independence of such statements suggests that the axiom system is not rich enough in structure to establish the truth of these statements, and that if one chose to do so, those statements could be added to the list of axioms for the system. The Axiom of Choice or something equivalent to it, for example, is now usually listed along with the Zermelo-Fraenkel axioms.

One certainly hopes that it is not possible to prove two contradictory statements about objects in a system. Such an occurrence would say that the axioms of the system were inconsistent, and this would require the axioms to be changed. After the original ground rules for Set Theory were established by Georg Cantor in the 1870s and 1880s, Bertrand Russell pointed out in 1901 a paradox (contradiction) that is a consequence of those rules. Now commonly known as Russell’s paradox, it stimulated a flurry of activity which resulted in the young field of Set Theory being put on a firm foundation (we hope) with the creation and adoption of the Zermelo–Fraenkel axioms.

The language of a proof can vary depending on who is writing the proof and who is the intended reader. In other words, what makes a convincing argument may well depend on who it is that needs to be convinced. For example, if two experts in Functional Analysis are speaking to each other, one might prove a statement by saying “Oh, that’s just a consequence of the Hahn-Banach Theorem.” That proof might be sufficient since it completely describes the reasoning behind the statement in question due to the shared knowledge of the two experts. On the other hand, if one of these experts were speaking to a beginning mathematics graduate student, the proof would need to include far more detail in order for it to be a convincing argument. If the expert were speaking to a high school student, the proof might need to be a complete book that both introduces the needed concepts and explains many results needed to understand the proof.

It is important to understand that there is a difference between knowing why a statement is true and knowing how to write a good proof of the statement. It is quite possible to learn a great deal of mathematics, to be able to solve many types of mathematical problems, and to understand why particular properties must hold without being able to write coherent proofs of these properties. It is analogous to a police detective who has gathered enough evidence to be convinced which of the many suspects has committed a particular crime, but it is quite another thing to have the criminal successfully prosecuted in a court of law resulting in the criminal’s conviction and eventual punishment for the crime. A student in Analysis needs to learn many strategies that can be brought to bear when writing proofs. Some of these

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strategies are methods or tricks that enter a student’s bag of tricks which can be employed later when solving problems or writing proofs. A student of proof writing needs to learn how to take those strategies and turn them into coherent proofs where the ideas are presented in a logical order, fill in all necessary details, and make clear to the proof reader exactly why the chosen strategies justify the needed result.

This book talks about how you should go about writing proofs of the kinds of statements typically found in the branch of Mathematics called Analysis. The branches of mathematics are not precisely defined. After a new branch arises, some mathematicians begin to combine ideas from older branches with ideas from the new branch to form even newer areas of study. For example, there are branches called Algebra, Geometry, and Topology. During the twentieth century mathematicians began talking about Algebraic Topology, Algebraic Geometry, and Geometric Topology. Very roughly speaking, then, some of the branches of mathematics are

• Set Theory: the study of sets, set operations, functions between sets, orderings of sets, and sizes of sets

• Algebra: the study of sets upon which there are binary operations defined (such as addition or multiplication) and includes Group Theory, Ring Theory, Field Theory, and Linear Algebra

• Topology: the study of continuous functions and properties of sets that are preserved by continuous functions

• Analysis: the study of sets for which there is a measure of distance allowing for the definition of various limiting processes such as those found in the subjects of Calculus, Differential Equations, Functional Analysis, Complex Variables, Measure Theory, and many other areas.

Other areas of study such as Applied Mathematics, Combinatorics, Geometry, Logic, Probability are considered by some mathematicians to be their own branch of mathematics or just as part of one or more of the above four branches. The exact designation is important to some mathematicians and not to others. Although mathematicians learn to write proofs in each of these branches of mathematics, one has to begin the learning process someplace. Many teachers feel that Analysis is a good area to start because students who have completed a study of Calculus will already be familiar with just about all of the theorems discussed in a beginning course in Analysis, and may already have an intuitive feeling for why these results hold. That does not mean that those same students can write convincing proofs of these theorems. It is the goal of this book to provide the training necessary so that a student can learn to write proofs of these and similar theorems. Undergraduate courses in Topology, Group Theory, Advanced Calculus, Graph Theory, and so forth generally present the beginning concepts in each of these fields and try to give students a feel for why the major results in the fields are true. Sometimes this involves having the students learn proofs of these results while other times it only involves a presentation of definitions and known results with the idea that the students will be able to take the why it is true and turn it into a proof themselves. This book is much more interested in turning known strategies into proofs than in introducing a wealth of new strategies.

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1.2 Why We Write Proofs 5

Some arguments in Analysis follow a standard format or template. This book will present several templates for proofs as a tool for teaching how one might approach the writing of a proof. For example, one can learn to prove a statement of the form lim

x!1f.x/ D L by following a standard pattern. This book will display

proof patterns by presenting proof templates, and for each template it will discuss proof examples showing how to use the template and the thought process needed to complete such proofs. After that, a student would be expected to produce similar proofs. There are other theorems in Analysis whose proofs involve the introduction of some clever idea which time has shown to be useful. Beginning students would not be expected to produce proofs using these new ideas on their own, so some of these proofs are presented in order to teach the new proof strategy. The experienced mathematician will have seen a large number of these clever proof techniques and can be expected to reuse these techniques when writing a proof of some new statement. Beginning students do not have this catalog of proof techniques from which to draw, so they are not expected to be able to write proofs for such a wide variety of statements. But one must start someplace when building up this catalog, and it is a goal of this book to get students started in the right direction.

1.2 Why We Write Proofs

There are many reasons why mathematicians put a lot of weight on the writing of proofs. Here are some of the reasons.

Determining Truth Research mathematicians use proofs to determine what math-ematical statements are true. Although many statements in mathematics are obvi-ously true, many remain unproved conjectures for long periods of time before being proved. When a conjecture stands unproved for many years, there is time for more mathematicians to learn about the statement, and the conjecture may attract a great deal of attention. When the conjecture is first stated, some may find it interesting, but finding a suitable proof may not appear to be a difficult problem until many people have tried unsuccessfully to find a proof. As this interesting statement remains a conjecture for a longer and longer period of time, the mathematical community realizes that the problem of finding a proof is much more involved than originally expected. This is exciting partly because a wider community of experts begin to wonder whether the statement under consideration is true and because it becomes clear that new techniques will be needed to find a suitable proof if, in fact, the statement can be proved at all. The problem of determining whether or not the mathematical statement is true takes on the same sort of interest that some people would take in the success of their favorite sports teams; sitting and waiting to see how they will fair in the upcoming contest. When a longstanding conjecture is finally proved, the announcement of the accomplishment will often be covered by the lay press giving mathematics an uncharacteristic brief period of pubic admiration. Perhaps you are familiar with some of these famous problems whose

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Fig. 1.1 Dividing the disk with the chords from n points

1 Point, 1 Region 2 Points, 2 Regions

3 Points, 4 Regions 4 Points, 8 Regions

5 Points, 16 Regions 6 Points, 31 Regions

resolution has alluded mathematicians for years (at least at the time of the writing of this text in January 2016): The Riemann Hypothesis, the Goldbach Conjecture, the Twin Prime Conjecture, the P versus NP Problem, and the Navier–Stokes Equations Existence and Smoothness Problem. During the last 40 years resolutions have been announced for several long-standing problems including the Four Color Theorem, The Bieberbach Conjecture (now called de Branges’s Theorem), Fermat’s Last Theorem, and the Poincaré Conjecture.

Why do mathematicians expend so much effort trying to prove statements, some of which may seem obvious from the start? One reason is that mathematicians are very skeptical of statements that appear obvious, and rightfully so. There is a long history that includes mathematical statements which appear to be true which are eventually shown to be false. Even very clear patterns can be deceptively seductive. Take, for example, the following problem. Select a set of n points along the circumference of a circle, draw the chords between each pair of points, and find out the maximum number of regions into which these segments can divide the disk. Figure1.1shows the results for the first few values of n.

Although from considering n D1; 2; 3; 4; 5 it appears that the chords can divide the disk into2n1regions, this fails to be true when n D6. With a bit more thinking it is not hard to see that2n1could not be the correct answer. With n points there are n

2

chords and at mostn₄intersections of two chords. This number of intersections grows as a fourth-degree polynomial in n suggesting that the number of regions will

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1.2 Why We Write Proofs 7

also grow as a fourth-degree polynomial in n. It would, therefore, be suspicious for the number of regions to grow at the exponential rate of2n1_.

Another well-known example comes from Number Theory. The function.x/ gives the number of positive prime integers less than or equal to the number x. The growth rate of this function has long been of central importance in Number Theory. The Prime Number Theorem says that the function grows at the same rate as the logarithmic integral

li.x/ D Z x

0

dt ln t:

In fact, for many years it was thought that li.x/ > .x/ for all x > 0 because this holds for all small values of x which can be practically checked, for example, all x between 0 and1024. It has now been shown that li.x/ .x/ switches sign infinitely often, although only for extremely large values of x.

It is apparent that sometimes seemingly very obvious patterns do not hold in every case, so mathematicians rely on proofs to convince themselves that the patterns do indeed hold in the general case.

Testing Axiom Systems In the next chapter you will read about the writing of proofs for some very elementary facts in mathematics; so elementary that you may wonder why anyone would bother with these proofs. Clearly, it makes sense to begin any training in the writing of proofs with some very simple results that are easy to understand so that the student can feel confident about all the statements being made in the proofs. But these proofs are not being presented just because they are elementary. When one sets up a mathematical system by making definitions and determining axioms, it is usually with a particular application or example in mind. The desired result is that the new system will include the already partially understood application so that any new discoveries will immediately tell something new about the original application. Suppose someone sets up an axiom system for the real numbers, for example, but is not able to prove that addition of real numbers satisfies the commutative property. Since the commutative property is an important aspect of addition of real numbers, it would appear that the new axiom system does not have enough power to represent all that one would want to show about the real numbers. Perhaps the axiom system will need to be expanded to include an axiom about the commutativity of addition. Thus, if one cannot prove that the expected simple properties hold, then it says that something is missing from the axioms. So mathematicians write proofs to confirm that their axiom systems are representative of the applications they are trying to describe.

Exhibiting Beauty There are no rules about what composers of music need to write, but many composers try to write in standardly accepted formats such as string quartets or symphonies because there are already organizations ready to perform such works and groups of people happy to listen to such works. Scholars of literature compare literary works by writing literary analysis, a form which holds a lot of meaning for those who read and write in that field. Although painters

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choose to make pictures of every sort of object or scene, real or imagined, most painters eventually try their hand at painting some of the standard subjects (still life, nudes, famous religious or historical depictions). Similarly, mathematicians write proofs partly because that is what mathematicians enjoy doing. Although many mathematicians make substantial contributions to the sciences, social sciences, and arts through the application of their mathematical skills, others live in a world of creating and discussing abstract concepts that have no immediate application to real world problems, or at least no application apparent to the mathematicians doing the research. To them, mathematics is studied as part of the humanities and is appreciated for its beauty. And much of the beauty of mathematics lies in the proofs of its theorems. One gets a great deal of pleasure reading a clever proof of a complicated result when the proof can be stated in just a few lines, especially if previous proofs of the same result were considerably longer and more difficult to understand. Many mathematicians like reading articles and attending conferences where they are exposed mainly to proofs of results, partly so that they can learn about new results, but more importantly so they can appreciate the techniques brought to bear to construct the proofs.

Testing Students One should not underestimate the need to educate future mathe-maticians. A good way to test whether a student understands a particular result is to ask the student to present a proof of the result. The presentation of a proof shows a deep understanding of why the result is true and shows an ability to discuss many details about the objects involved. At the graduate school level in mathematics, most test problems require the student to produce a proof of a particular result.

The student who has completed a study of Calculus is likely to have mastered basic skills in Algebra, Geometry, Trigonometry, and Elementary Functions. This is a good point in one’s studies to begin writing proofs. It should not be assumed that one can just begin writing proofs at this stage even if they have had years of experience watching teachers and authors present proofs to them any more than someone can be expected to sit down and begin playing the piano just because they have watched many other people present concerts using the instrument. In this book the reader will be taken through the construction of many proofs in a step-by-step manner that presents the thought process used to write the proofs. Some incorrect proofs are shown and explained so that the student can learn about common pitfalls to avoid. Some students dread the transition to writing proofs because they feel that they do not understand how to write proofs, and are leery of the day when they will be expected to produce what they cannot now do. But the ability to write good proofs is a skill no different from the ability to factor polynomials or integrate rational functions. There is no expectation that the beginner can produce a good proof, but every expectation that the beginner can learn.

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Chapter 2 The Basics of Proofs

2.1 The Language of Proofs

2.1.1 Conditional Statements

Most theorems concern mathematical objects x that satisfy a set of properties P, that is, P.x/ D the properties P hold for object x. The theorem may say that if P.x/ is true, then some additional properties Q.x/ must also be true. Such statements are called conditional statements and can be written P.x/ ! Q.x/. In the context of proving theorems, the “P.x/” portion of the statement is referred to as the hypothesis of the statement, and the “Q.x/” portion of the statement is referred to as the conclusion of the statement. The hypothesis of a conditional statement is often called the antecedent while the conclusion of the conditional statement is often called the consequent. For example, a well-known theorem is that all functions differentiable at a point are also continuous at that point. There are many equivalent ways to express this fact:

• All functions differentiable at a point are also continuous at that point. • If the function f is differentiable at a point, then f is continuous at that point. • The function f is differentiable at a point only if f is continuous at that point. • If the function f is not continuous at a point, then f is not differentiable at that

point.

• There are no functions f such that f is both differentiable at a point and discontinuous at that point.

• The function f is differentiable at a point implies that f is continuous at that point. • The function f is differentiable at x ! f is continuous at x.

All of these statements assert that if a function f satisfies the hypothesis that it has a derivative at a point x, then f must also satisfy the conclusion that f is continuous at x. Note that the truth of a conditional statement, P.x/ ! Q.x/, suggests nothing about the truth of the statement Q.x/ ! P.x/ which is known as the converse

J.M. Kane, Writing Proofs in Analysis, DOI 10.1007/978-3-319-30967-5_2

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of the conditional statement P.x/ ! Q.x/. Indeed, the converse of this theorem is the clearly false statement: “If the function f is continuous at a point, then f is differentiable at that point.” Certainly, there are functions f both continuous and differentiable at a point, but knowing that a function is continuous at a point does not allow one to conclude that it is differentiable at that point. The converse of a conditional statement is not logically equivalent to the original statement, but since the two statements are concerned with the same subject matter, mathematicians are often interested in the converse of a given conditional. If someone succeeds in proving a new theorem expressed as a conditional statement, you might wonder whether the converse of the statement could also be true. Sometimes the truth of the converse statement is a trivial matter because it is well known. But there are many examples where the converse does not hold in every case; that is, there are many known values of x where the converse statement “Q.x/ ! P.x/” is false. Other times, the converse statement is something that has been previously established. But very often, the truth of the converse statement remains an open question, and the proof of the original conditional statement may generate research interest in its converse.

One of the equivalent forms of a conditional statement P.x/ ! Q.x/ is the statement “if Q.x/ is false, then P.x/ must be false.” This can be written as “:Q.x/ ! :P.x/” using the negation symbol : . This form of the statement is called the contrapositive of the original conditional statement. For example, the contrapositive of the statement discussed above is “If the function f is not continuous at a point, then f is not differentiable at that point.” Although logically equivalent to the original conditional statement, the contrapositive often gives you a different way to think about the statement, and you will often see a proof which is a proof of the contrapositive statement instead of a proof of the original conditional statement.

2.1.2 Negation of a Statement

The negation of a statement is a statement with the opposite truth value of the original statement, that is, a statement which is false exactly when the original statement is true. For example, the negation of “n is an integer” is “n is not an integer.” The negation of the statement P.x/ is “not P.x/” or simply “:P.x/.” The conditional statement “P.x/ ! Q.x/” says that every time P.x/ holds it must be the case that Q.x/ also holds. The negation of this statement must, therefore, state that for at least one value of x, P.x/ is true and Q.x/ is false or “P.x/ and :Q.x/.” A proof by contradiction is a proof that assumes both that P.x/ and :Q.x/ are true, and derives a statement that must be false (known as a contradiction) showing that it is impossible to have P.x/ being true at the same time that Q.x/ is false.

The well-known Pythagorean Theorem is a conditional statement: “If a right triangle has legs with lengths a and b and a hypotenuse with length c, then a2C b2D c2.” The converse of the Pythagorean Theorem is also true: “If a triangle has sides with lengths a, b, and c satisfying a2 C b2 D c2, then the triangle is a right triangle.” When a conditional statement, P.x/ ! Q.x/ and its converse

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2.1 The Language of Proofs 11

Q.x/ ! P.x/ are both true, the two statements can be combined into one as P.x/ ! Q.x/. This can also be stated as “P.x/ if and only if Q.x/.” Such statements are called biconditional statements. Thus, the Pythagorean Theorem and its converse could be combined into the single biconditional statement: “A triangle is a right triangle if and only if the triangle has side lengths a, b, and c satisfying a2C b2D c2.”

2.1.3 Proofs of Conditional Statements

Conditional statements often make assertions about a very large number of objects or even an infinite set of objects. Indeed, the statement about differentiable functions being continuous refers to infinitely many functions, and the Pythagorean Theorem refers to an infinite number of triangles. How, then, are you supposed to prove these results since you clearly cannot consider every case individually? A general approach to proving the conditional statement “P.x/ ! Q.x/” is to select a generic element x which could represent any object satisfying P.x/ and then to prove the statement Q.x/. Since a generic object x satisfying P.x/ has been shown to satisfy Q.x/, it follows that every object satisfying P.x/ must also satisfy Q.x/, and the result has been proved. This will be the format of most of the proofs you will ever write in analysis.

If the statement “P.x/ ! Q.x/” is not true, it means that there is at least one value of x that makes “P.x/ ! Q.x/” a false statement. Such an x is called a counterexample to the statement, and exhibiting such a counterexample would be a way to prove that “P.x/ ! Q.x/” is false. A proof of “P.x/ ! Q.x/” is essentially an argument showing that no counterexamples exist.

There are many phrases that occur so frequently when writing proofs, that mathematicians have developed a short hand notation for these phrases. There is little need to use these abbreviations within a textbook such as this or even in a journal article, but the short hand can be useful when writing out a proof by hand on paper or a blackboard. Here is a list of some of the commonly used symbols.

Shorthand Symbols for Proofs

9 there exists 9Š there exists exactly one 8 for all S suppose (or assume) 3 such that

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2.1.4 Exercises

Perform the follows steps for each of the conditional statements in Exercises 1–6. A identify the hypothesis and the conclusion.

B write the converse of the statement.

C decide whether or not the converse of the statement is true. D write the contrapositive of the statement.

E write the negation of the statement. 1. If x D1 and y D 1, then xy D 1.

2. If x is an integer, then2x C 1 is also an integer.

3. f.x/ and g.x/ are both continuous at x D 0 only if f .x/ C g.x/ is continuous at x D0.

4. xy D0 if x D 0 or y D 0.

5. If xy 9y D 0 and y > 0, then x D 9.

6. A rectangle has area xy if two adjacent sides of the rectangle have lengths x and y. 7. Write the following without using shorthand symbols.

(a) 9x 2R 3 x C 4 D 2. (b) 8x 2R 9y 2 R 3 x C y D 10.

2.2 Template for Proofs

Many proofs can be written by following a simple formula or template that suggests guidelines to follow when writing the proof. Mathematicians reading a proof that follows a traditional template will find the proof easier to follow because there will be an expectation about what will be presented in the proof. For example, many proofs will follow the general template given here.

TEMPLATE followed by many proofs • SET THE CONTEXT

• ASSERT THE HYPOTHESIS • LIST IMPLICATIONS • STATE THE CONCLUSION

To illustrate this template, consider this proof of a well-known theorem from elementary Algebra.

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2.2 Template for Proofs 13

PROOF (Quadratic Formula): For constants a, b, and c, the quadratic polynomial ax2C bx C c has roots given by x Db ˙

p

b2 4ac

2a .

• SET THE CONTEXT: Let a, b, and c be constants with a ¤0.

• ASSERT THE HYPOTHESIS: Suppose that x satisfies ax2C bx C c D 0. • LIST IMPLICATIONS: Since a ¤0, it follows that x2C b_axC c_a D 0. • Then x2C b_axC c_aC _4ab2₂ D _4ab2₂.

• Factoring shows thatxC_2ab2C_ac D _4ab22.

• ThenxC _2ab2D _4ab2₂ c_a D b2_4a4ac₂ .

• This means that x C _2ab must be one of the two square roots ofb2_4a4ac₂ . • So, x C b 2a D ˙ s b2 4ac 4a2 D ˙pb2 4ac 2a , and x D b ˙pb2 4ac 2a .

• STATE THE CONCLUSION: Thus, the roots of the quadratic polynomial ax2C bx C c are given by x D b ˙

p

b2 4ac

2a .

The proof template begins with the suggestion to “SET THE CONTEXT” which represents statements designed to tell the reader what is being assumed in the proof. This is usually a sentence or two telling the reader about the properties of the objects that will be encountered in the proof. It may also introduce which variables will appear in the proof and what kinds of objects they represent. So, in the given proof of the Quadratic Formula, the first line tells that the variables a, b, and c are going to represent known constants with a not being 0. Clearly, the fact that a is not 0 needs to be stipulated because if a D 0, the polynomial ax2C bx C c would not be quadratic and would not have the proposed roots. Generally, you are not looking for a lengthy narrative here, and, in fact, brevity is a particularly cherished attribute of a proof. Saying what needs to be said, but only what needs to be said is usually best. Some authors who state a theorem and immediately follow the statement of the theorem with its proof will forgo setting the context at the beginning of the proof because the reader will have just seen the statement of the theorem and may not need to see a repeat of the context for that proof. For example, in the example proof, the statement of the theorem does introduce the constants a, b, and c and polynomial ax2C bx C c, so some authors might just skip the first line of the proof. On the other hand, if the first line of the proof instead introduced the constants r, s, and t, the proof could have proceeded using these variables instead of a, b, and c. The same result would have been proved. So the “SET THE CONTEXT” of the proof makes the proof independent of the statement of the theorem being proved. Thus, for completeness, it is good to establish the habit of including the setting of the context at the beginning of each proof, at least until the student’s experience in proof writing has matured.

Your choices of variables used to represent particular objects in the proof are not critically important to the structure or correctness of the proof, but there are

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certain variables that mathematicians associate with various uses, and sticking to these conventional choices simplifies the understanding of the proof because those variable choices bring with them a history of context that the reader will recognize. There are very few Algebra students who would recognize the Quadratic Formula if you gave them z D s ˙

p s2 4rt

2r . Proofs about limits usually refer to the variables and ı which represent small positive real numbers used in specific ways in the proof. Using these two variables in their traditional contexts makes the proofs easier to understand because the reader will expect these variables to play specific roles, just as they have in many other proofs the reader has seen. Seeing many examples of proofs will familiarize the novice proof writer with these traditional uses of variables.

Suppose that the statement being proved indicates that every object satisfying the properties listed in the hypothesis of the theorem also satisfies some properties listed in the conclusion of the theorem. One generally structures a proof of such a statement by first selecting a generic object satisfying the properties listed in the hypothesis. The “ASSERT THE HYPOTHESIS” part of the proof is where the writer selects an arbitrary element satisfying the hypothesized properties. In the Quadratic Formula proof, it was assumed that x satisfied the quadratic equation ax2C bx C c D 0. Other examples would be statements such as

• Let n be any natural number bigger than 3. • Let x be an element of set A.

• Let y be a root of the polynomial p.x/.

• Assume that the real valued function f has a zero at the point z. • Suppose G and H are any two lines that intersect at a point P.

• Assume that the function f.s/ DR₀sg.x/ dx is a differentiable function of s. In addition assume that0 f .s/ 10 for all s 0.

It is possible that there are infinitely many objects which could play the role of the generically chosen object. But if an argument proves the result is true for this generic object, then the theorem will have been shown to hold for any object that could have played the role of the generic object, and, therefore, the theorem will have been proved for all objects satisfying the hypothesis. The Quadratic Formula proof addresses the one generic polynomial ax2C bx C c and in doing so derives a formula that works for all quadratic polynomials including5x2 17x C 126 and rx2C sx C t. Often the reader of a proof will form a mental picture of the generic object being chosen. For example, after reading “Let n be any natural number bigger than 3,” the reader may think, “OK, how about n D7?” As the proof progresses, the reader may take each statement of the proof and verify that it is valid and makes sense for their choice of n D7. This helps the reader follow the logic of the proof and verifies that they are understanding what the proof is saying.

The proof will be completed when it is shown that the generically chosen element satisfying the hypothesis of the theorem is, in fact, an element satisfying the conclusion of the theorem as stated in the “STATE THE CONCLUSION” part of the template. There will certainly need to be some statements placed between

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2.2 Template for Proofs 15

the original assertion of the hypothesis and the end of the proof that justify the conclusion of the theorem. Those statements make up the “LIST IMPLICATIONS” part of the template. In almost all cases, most of the body of the proof belongs to this list of statements. Each statement in the list should follow from definitions or be simple implications following from previous statements in the proof. In a well-written complete proof, the reader should easily see why each implication follows logically from other statements made earlier in the proof (Fig.2.1). If an implication is not clear on its own, it will need some justification so the reader can follow the logic. The justification may just be a reminder of a key point made earlier in the proof (“as shown earlier, f is continuous at point a”) or a reminder of a well-known definition or theorem (“Since all continuous functions on the interval Œ0; 4 are integrable there, it follows that

4 R 0

f.x/dx exists.”) The given Quadratic Formula proof contains six lines of implications. Each line follows easily from the line before using standard rules of Algebra, and any student familiar with the algebraic manipulations of equations will be able to understand these implications. In the fourth step of the proof, the quantity _4ab22 is added to both sides of an equation. Although this step

surely follows the rules of Algebra, it may not be clear to the reader of the proof why the step is important. As it turns out, this “completing the square” operation prepares for the factoring performed in the fifth step of the proof and is arguably the most clever step of the proof. A proof will often require a clever step such as this. The proof writer may have labored for years looking for the inspiration needed to find such a step, but the proof itself need only make clear the justification for what is being done and does not need to refer to the sweat that went into producing it.

Some implications will be easy for the reader to follow without having to justify the step. Other statements may need some deeper explanation. Here is where the proof writer will need to consider the expertise of the target audience for the proof in order to decide how much detail to provide. How to make your proof “easy to follow” is only clear when you know for whom it is meant to be easy. For example, it made sense to follow the line x2Cb_axC_ac D 0 with the statement x2Cb_axC_acC_4ab22 D

b2

4a2 because this just used the fact that you can add equal quantities to both sides of

an equation to get a new equation that is equivalent. On the other hand, suppose you wish to combine a conditional statement on line 8 of a proof with the fact stated on line 15 of the proof in order to show that the hypothesis of that conditional is satisfied. This would allow the writer to state the conclusion of the conditional

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statement to get line 16 of the proof, but the reader may have to be reminded about which statements are being combined to get that conclusion.

Sometimes the writer of a long or complicated proof will need to make a new definition or point out some new property that will be important later in the proof. Depending on the complexity of the new idea, the proof writer may want to include an example or two of objects satisfying the new definition or property. This will serve to help the reader understand the new concept or to verify that the reader is understanding the new concept. It is admirable to include such examples if the complexity of the proof can be made clearer. But in most other contexts, the proof should be kept short without the inclusion of unnecessary statements. If the intended readers are able to easily construct these examples on their own, then the examples should be left out of the proof.

The remainder of this chapter will discuss proofs that follow this general proof template in contexts that the student should find easy to follow. It will also give an opportunity to present some definitions and notation that will be used in later chapters.

2.2.1 Exercises

1. If you were writing a proof of “All prime numbers greater than 2 are odd,” which of the following would be appropriate ways to begin the proof. (There may be more than one correct answer.)

(a) Let n be an odd prime number.

(b) Assume that all odd prime numbers are greater than 2. (c) Let n be a prime number greater than 2.

(d) Assume that 2 is a prime number.

(e) Assume that n and k are integers with n> k > 2.

(f) The numbers 3, 5, 7, and 11 are prime numbers greater than 2. (g) Let n be a number greater than 2 which is not prime.

2. If you were writing a proof of “The diagonals of a parallelogram bisect each other,” which of the following would be appropriate ways to begin the proof. (There may be more than one correct answer.)

(a) Let ABCD be a parallelogram.

(b) Let ABCD be a quadrilateral whose diagonals bisect each other. (c) Let ABCD be a parallelogram whose diagonals bisect each other. (d) All rectangles are parallelograms.

(e) Assume that the diagonals of a parallelogram bisect each other.

(f) Assume that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

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2.3 Proofs About Sets 17

3. If you were writing a proof of “Every cubic polynomial with real coefficients has at least one real root,” which of the following would be appropriate ways to begin the proof. (There may be more than one correct answer.)

(a) Assume that every cubic polynomial with real coefficients has at least one real root.

(b) Assume that p.x/ is a polynomial with at least one real root.

(c) Assume that a, b, c, and d are real numbers with a ¤ 0, and let p.x/ D ax3C bx2C cx C d.

(c) The polynomial x3 8 has exactly one real root at x D 2.

(e) Let p.x/ be a cubic polynomial with real coefficients and q.x/ be a cubic polynomial with complex coefficients.

(f) Let p.x/ be a cubic polynomial with real coefficients with real root r. Write an appropriate first sentence that would begin proofs of each of the following statements.

4. If m and n are relatively prime integers, then there exist integers x and y such that mxC ny D 1.

5. The three angle bisectors of any triangle intersect at a common point.

6. If a and b are real numbers with a b, and f is a function continuous on the closed intervalŒa; b, then there is a real number M such that jf .x/j M for all x2 Œa; b.

7. If !u , !v , and !w are 3-dimensional vectors, then.!u !v / !w D !u .!v !w/:

2.3 Proofs About Sets

2.3.1 Set Notation

Most courses in mathematics discuss sets: sets of numbers, sets of points, sets of functions, sample spaces, and so forth. This should have given any Calculus student an intuitive understanding of sets. Many theorems in mathematics are statements about sets in disguise. For example, the statement that “If the function f is differentiable at a point, then f is continuous at that point” is equivalent to the statement “The set of functions differentiable at a point is a subset of the set of functions continuous at that point.”

For the purposes of this text, it will be enough to define a set as a collection of elements. That is, elements are those objects that belong to sets, and the notation

x2 A says that x is an element of the set A, and x … A says that x is not an element

of the set A. The set A is a subset of the set B, or A is contained in the set B, if each element of A is also an element of B in which case this fact is written as A B. Two sets, A and B, are equal if they have the same elements, that is, all the elements in the set A are in the set B, and all the elements in the set B are in the set A. Notationally, this says that AD B if and only if both A B and B A.

Kane 2016 Writing Proofs in Analysis

Jonathan M. Kane

Writing

Proofs in

Analysis

Jonathan M. Kane

Writing Proofs in Analysis

To the memory of Sylvan Burgstaller, Duane

E. Anderson, and especially James L. Nelson

who, at the University of Minnesota Duluth,

taught me the fundamentals of writing proofs

in analysis.

Acknowledgments

Preface

Contents

List of Figures

List of Proof Templates

Chapter 1

What Are Proofs, and Why Do We Write Them?

1.1

What Is a Proof?

1.2

Why We Write Proofs

Chapter 2

The Basics of Proofs

2.1

The Language of Proofs

2.1.1

Conditional Statements

2.1.2

Negation of a Statement

2.1.3

Proofs of Conditional Statements

2.1.4

Exercises

2.2

Template for Proofs

2.2.1

Exercises

2.3

Proofs About Sets

2.3.1

Set Notation