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INTRODUCTION to ANALYSIS John Hutchinson email: [email protected]

with earlier revisions by

Richard J. Loy

Pure mathematics have one peculiar advantage, that they occasion no disputes among wrangling disputants, as in other branches of knowledge; and the reason is, because the definitions of the terms are premised, and everybody that reads a proposition has the same idea of every part of it. Hence it is easy to put an end to all mathematical controversies by shewing, either that our adversary has not stuck to his definitions, or has not laid down true premises, or else that he has drawn false conclusions from true principles; and in case we are able to do neither of these, we must acknowledge the truth of what he has proved . . . The mathematics, he [Isaac Barrow] observes, effectually exercise, not vainly delude, nor vexatiously torment, studious minds with obscure subtlities; but plainly demonstrate everything within their reach, draw certain conclusions, instruct by profitable rules, and unfold pleasant questions. These disciplines likewise enure and corroborate the mind to a constant diligence in study; they wholly deliver us from credulous simplicity; most strongly fortify us against the vanity of scepticism, effectually refrain us from a rash presumption, most easily incline us to a due assent, perfectly subject us to the government of right reason. While the mind is abstracted and elevated from sensible matter, distinctly views pure forms, conceives the beauty of ideas and investigates the harmony of proportion; the manners themselves are sensibly corrected and improved, the affections composed and rectified, the fancy calmed and settled, and the understanding raised and excited to more divine contemplations. Encyclopædia Britannica [1771] Philosophy is written in this grand book—I mean the universe—which stands continually open to our gaze, but it cannot be understood unless one first learns to comprehend the language and interpret the characters in which it is written. It is written in the language of mathematics, and its characters are triangles, circles, and other mathematical figures, without which it is humanly impossible to understand a single word of it; without these one is wandering about in a dark labyrinth. Galileo Galilei Il Saggiatore [1623] Mathematics is the queen of the sciences. Carl Friedrich Gauss [1856] Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true. Bertrand Russell Recent Work on the Principles of Mathematics, International Monthly, vol. 4 [1901] Mathematics takes us still further from what is human, into the region of absolute necessity, to which not only the actual world, but every possible world, must conform. Bertrand Russell The Study of Mathematics [1902] Mathematics, rightly viewed, possesses not only truth, but supreme beauty—a beauty cold and austere, like that of a sculpture, without appeal to any part of our weaker nature, without the gorgeous trappings of painting or music, yet sublimely pure, and capable of perfection such as only the greatest art can show. Bertrand Russell The Study of Mathematics [1902] The study of mathematics is apt to commence in disappointment. . . . We are told that by its aid the stars are weighed and the billions of molecules in a drop of water are counted. Yet, like the ghost of Hamlet’s father, this great science eludes the

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efforts of our mental weapons to grasp it. Alfred North Whitehead

An Introduction to Mathematics [1911]

The science of pure mathematics, in its modern developments, may claim to be the most original creation of the human spirit. Alfred North Whitehead Science and the Modern World [1925] All the pictures which science now draws of nature and which alone seem capable of according with observational facts are mathematical pictures . . . . From the intrinsic evidence of his creation, the Great Architect of the Universe now begins to appear as a pure mathematician. Sir James Hopwood Jeans The Mysterious Universe [1930] A mathematician, like a painter or a poet, is a maker of patterns. If his patterns are more permanent than theirs, it is because they are made of ideas. G.H. Hardy A Mathematician’s Apology [1940] The language of mathematics reveals itself unreasonably effective in the natural sciences. . . , a wonderful gift which we neither understand nor deserve. We should be grateful for it and hope that it will remain valid in future research and that it will extend, for better or for worse, to our pleasure even though perhaps to our bafflement, to wide branches of learning. Eugene Wigner [1960] To instruct someone . . . is not a matter of getting him (sic) to commit results to mind. Rather, it is to teach him to participate in the process that makes possible the establishment of knowledge. We teach a subject not to produce little living libraries on that subject, but rather to get a student to think mathematically for himself . . . to take part in the knowledge getting. Knowing is a process, not a product. J. Bruner Towards a theory of instruction [1966] The same pathological structures that the mathematicians invented to break loose from 19-th naturalism turn out to be inherent in familiar objects all around us in nature. Freeman Dyson Characterising Irregularity, Science 200 [1978] Anyone who has been in the least interested in mathematics, or has even observed other people who are interested in it, is aware that mathematical work is work with ideas. Symbols are used as aids to thinking just as musical scores are used in aids to music. The music comes first, the score comes later. Moreover, the score can never be a full embodiment of the musical thoughts of the composer. Just so, we know that a set of axioms and definitions is an attempt to describe the main properties of a mathematical idea. But there may always remain an aspect of the idea which we use implicitly, which we have not formalized because we have not yet seen the counterexample that would make us aware of the possibility of doubting it . . . Mathematics deals with ideas. Not pencil marks or chalk marks, not physical triangles or physical sets, but ideas (which may be represented or suggested by physical objects). What are the main properties of mathematical activity or mathematical knowledge, as known to all of us from daily experience? (1) Mathematical objects are invented or created by humans. (2) They are created, not arbitrarily, but arise from activity with already existing mathematical objects, and from the needs of science and daily life. (3) Once created, mathematical objects have properties which are well-determined, which we may have great difficulty discovering,

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but which are possessed independently of our knowledge of them. Reuben Hersh Advances in Mathematics 31 [1979] Don’t just read it; fight it! Ask your own questions, look for your own examples, discover your own proofs. Is the hypothesis necessary? Is the converse true? What happens in the classical special case? What about the degenerate cases? Where does the proof use the hypothesis? Paul Halmos I Want to be a Mathematician [1985] Mathematics is like a flight of fancy, but one in which the fanciful turns out to be real and to have been present all along. Doing mathematics has the feel of fanciful invention, but it is really a process for sharpening our perception so that we discover patterns that are everywhere around.. . . To share in the delight and the intellectual experience of mathematics – to fly where before we walked – that is the goal of mathematical education. One feature of mathematics which requires special care . . . is its “height”, that is, the extent to which concepts build on previous concepts. Reasoning in mathematics can be very clear and certain, and, once a principle is established, it can be relied upon. This means that it is possible to build conceptual structures at once very tall, very reliable, and extremely powerful. The structure is not like a tree, but more like a scaffolding, with many interconnecting supports. Once the scaffolding is solidly in place, it is not hard to build up higher, but it is impossible to build a layer before the previous layers are in place. William Thurston, Notices Amer. Math. Soc. [1990]

Contents Chapter 1. Introduction 1.1. Preliminary Remarks 1.2. History of Calculus 1.3. Why “Prove” Theorems? 1.4. “Summary and Problems” Book 1.5. The approach to be used 1.6. Acknowledgments

7 7 8 8 8 8 8

Chapter 2. Some Elementary Logic 2.1. Mathematical Statements 2.2. Quantifiers 2.3. Order of Quantifiers 2.4. Connectives 2.4.1. Not 2.4.2. And 2.4.3. Or 2.4.4. Implies 2.4.5. Iff 2.5. Truth Tables 2.6. Proofs 2.6.1. Proofs of Statements Involving Connectives 2.6.2. Proofs of Statements Involving “There Exists” 2.6.3. Proofs of Statements Involving “For Every” 2.6.4. Proof by Cases

9 9 10 11 12 12 14 14 14 15 16 16 16 17 17 18

Chapter 3. The Real Number System 3.1. Introduction 3.2. Algebraic Axioms 3.2.1. Consequences of the Algebraic Axioms 3.2.2. Important Sets of Real Numbers 3.2.3. The Order Axioms 3.2.4. Ordered Fields 3.2.5. Completeness Axiom 3.2.6. Upper and Lower Bounds 3.2.7. *Existence and Uniqueness of the Real Number System 3.2.8. The Archimedean Property

19 19 19 20 21 21 22 23 24 26 26

Chapter 4. Set Theory 4.1. Introduction 4.2. Russell’s Paradox 4.3. Union, Intersection and Difference of Sets 4.4. Functions 4.4.1. Functions as Sets 4.4.2. Notation Associated with Functions

29 29 29 30 33 33 34

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CONTENTS

4.4.3. Elementary Properties of Functions 4.5. Equivalence of Sets 4.6. Denumerable Sets 4.7. Uncountable Sets 4.8. Cardinal Numbers 4.9. More Properties of Sets of Cardinality c and d 4.10. *Further Remarks 4.10.1. The Axiom of choice 4.10.2. Other Cardinal Numbers 4.10.3. The Continuum Hypothesis 4.10.4. Cardinal Arithmetic 4.10.5. Ordinal numbers

35 35 36 37 39 42 43 43 44 45 45 45

Chapter 5. Vector Space Properties of Rn 5.1. Vector Spaces 5.2. Normed Vector Spaces 5.3. Inner Product Spaces

47 47 48 49

Chapter 6. Metric Spaces 6.1. Basic Metric Notions in Rn 6.2. General Metric Spaces 6.3. Interior, Exterior, Boundary and Closure 6.4. Open and Closed Sets 6.5. Metric Subspaces

53 53 53 55 57 60

Chapter 7. Sequences and Convergence 7.1. Notation 7.2. Convergence of Sequences 7.3. Elementary Properties 7.4. Sequences in R 7.5. Sequences and Components in Rk 7.6. Sequences and the Closure of a Set 7.7. Algebraic Properties of Limits

63 63 63 64 65 67 67 68

Chapter 8. Cauchy Sequences 8.1. Cauchy Sequences 8.2. Complete Metric Spaces 8.3. Contraction Mapping Theorem

71 71 73 75

Chapter 9. Sequences and Compactness 9.1. Subsequences 9.2. Existence of Convergent Subsequences 9.3. Compact Sets 9.4. Nearest Points

79 79 79 82 82

Chapter 10.1. 10.2. 10.3. 10.4.

10. Limits of Functions Diagrammatic Representation of Functions Definition of Limit Equivalent Definition Elementary Properties of Limits

85 85 85 91 92

Chapter 11.1. 11.2. 11.3.

11. Continuity Continuity at a Point Basic Consequences of Continuity Lipschitz and H¨older Functions

97 97 98 100

CONTENTS

11.4. Another Definition of Continuity 11.5. Continuous Functions on Compact Sets 11.6. Uniform Continuity Chapter 12.1. 12.2. 12.3. 12.4. 12.5.

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101 102 103

12. Uniform Convergence of Functions Discussion and Definitions The Uniform Metric Uniform Convergence and Continuity Uniform Convergence and Integration Uniform Convergence and Differentiation

107 107 111 113 115 115

Chapter 13. First Order Systems of Differential Equations 13.1. Examples 13.1.1. Predator-Prey Problem 13.1.2. A Simple Spring System 13.2. Reduction to a First Order System 13.3. Initial Value Problems 13.4. Heuristic Justification for the Existence of Solutions 13.5. Phase Space Diagrams 13.6. Examples of Non-Uniqueness and Non-Existence 13.7. A Lipschitz Condition 13.8. Reduction to an Integral Equation 13.9. Local Existence 13.10. Global Existence 13.11. Extension of Results to Systems

117 117 117 118 119 120

Chapter 14. Fractals 14.1. Examples 14.1.1. Koch Curve 14.1.2. Cantor Set 14.1.3. Sierpinski Sponge 14.2. Fractals and Similitudes 14.3. Dimension of Fractals 14.4. Fractals as Fixed Points 14.5. *The Metric Space of Compact Subsets of Rn 14.6. *Random Fractals

135 135 135 136 137 138 139 141 144 148

Chapter 15.1. 15.2. 15.3. 15.4. 15.5. 15.6. 15.7. 15.8.

15. Compactness Definitions Compactness and Sequential Compactness *Lebesgue covering theorem Consequences of Compactness A Criterion for Compactness Equicontinuous Families of Functions Arzela-Ascoli Theorem Peano’s Existence Theorem

151 151 152 154 154 156 158 160 163

Chapter 16.1. 16.2. 16.3. 16.4. 16.5.

16. Connectedness Introduction Connected Sets Connectedness in Rn Path Connected Sets Basic Results

167 167 167 168 169 170

122 123 125 126 128 129 132 133

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CONTENTS

Chapter 17. Differentiation of Real-Valued Functions 17.1. Introduction 17.2. Algebraic Preliminaries 17.3. Partial Derivatives 17.4. Directional Derivatives 17.5. The Differential (or Derivative) 17.6. The Gradient 17.6.1. Geometric Interpretation of the Gradient 17.6.2. Level Sets and the Gradient 17.7. Some Interesting Examples 17.8. Differentiability Implies Continuity 17.9. Mean Value Theorem and Consequences 17.10. Continuously Differentiable Functions 17.11. Higher-Order Partial Derivatives 17.12. Taylor’s Theorem

173 173 173 174 175 175 179 179 180 181 182 182 184 186 188

Chapter 18. Differentiation of Vector-Valued Functions 18.1. Introduction 18.2. Paths in Rm 18.2.1. Arc length 18.3. Partial and Directional Derivatives 18.4. The Differential 18.5. The Chain Rule

193 193 193 196 197 198 201

Chapter 19.1. 19.2. 19.3. 19.4. 19.5. 19.6.

205 205 210 214 218 220 220

19. The Inverse Function Theorem and its Applications Inverse Function Theorem Implicit Function Theorem Manifolds Tangent and Normal vectors Maximum, Minimum, and Critical Points Lagrange Multipliers

Bibliography

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CHAPTER 1

Introduction 1.1. Preliminary Remarks These Notes provide an introduction to the methods of contemporary mathematics, and in particular to Mathematical Analysis, which roughly speaking is the “in depth” study of Calculus. The notes arise from various versions of MATH2320 and previous related courses. They include most of the material from the current MATH2320, and some more. However, the treatment may not always be the same. The notes are not a polished text, and there are undoubtedly a few typos! The mathematics here is basic to most of your subsequent mathematics courses (e.g. differential equations, differential geometry, measure theory, numerical analysis, to name a few), as well as to much of theoretical physics, engineering, probability theory and statistics. Various interesting applications are included; in particular to fractals and to differential and integral equations. There are also a few remarks of a general nature concerning logic and the nature of mathematical proof, and some discussion of set theory. There are a number of Exercises scattered throughout the text. The Exercises are usually simple results, and you should do them all as an aid to your understanding of the material. Sections, Remarks, etc. marked with a * are “extension” material, but you should read them anyway. They often help to set the other material in context as well as indicating further interesting directions. The dependencies of the various chapters are noted in Figure 1.

Figure 1. Chapter Dependencies. There is a list of related books in the Bibliography. The way to learn mathematics is by doing problems and by thinking very carefully about the material as you read it. Always ask yourself why the various assumptions in a theorem are made. It is almost always the case that if any particular assumption is dropped, then the conclusion of the theorem will no longer be true. Try to think of examples where the conclusion of the theorem is no longer valid if the various assumptions are changed. Try to see where each assumption is used in the proof of the theorem. Think of various interesting examples of the theorem. 7

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1. INTRODUCTION

1.2. History of Calculus Calculus developed in the seventeenth and eighteenth centuries as a tool to describe various physical phenomena such as occur in astronomy, mechanics, and electrodynamics. But it was not until the nineteenth century that a proper understanding was obtained of the fundamental notions of limit, continuity, derivative, and integral. This understanding is important in both its own right and as a foundation for further deep applications to all of the topics mentioned in Section 1.1. 1.3. Why “Prove” Theorems? A full understanding of a theorem, and in most cases the ability to apply it and to modify it in other directions as needed, comes only from knowing what really “makes it work”, i.e. from an understanding of its proof. 1.4. “Summary and Problems” Book There is an accompanying set of notes which contains a summary of all definitions, theorems, corollaries, etc. You should look through this at various stages to gain an overview of the material. There is also a separate selection of problems and solutions available. The problems are at the level of the assignments which you will be required to do. They are not necessarily in order of difficulty. You should attempt, or at the very least think about, the problems before you look at the solutions. You will learn much more this way, and will in fact find the solutions easier to follow if you have already thought enough about the problems in order to realise where the main difficulties lie. You should also think of the solutions as examples of how to set out your own answers to other problems. 1.5. The approach to be used Mathematics can be presented in a precise, logically ordered manner closely following a text. This may be an efficient way to cover the content, but bears little resemblance to how mathematics is actually done. In the words of Saunders Maclane (one of the developers of category theory) “intuition–trial–error– speculation–conjecture–proof is a sequence for understanding of mathematics.” It is this approach which will be taken here, at least in part. 1.6. Acknowledgments Thanks are due to many past students for suggestions and corrections, including Paulius Stepanas and Simon Stephenson, and to Maciej Kocan for supplying problems for some of the later chapters.

CHAPTER 2

Some Elementary Logic In this Chapter we will discuss in an informal way some notions of logic and their importance in mathematical proofs. A very good reference is [Mo, Chapter I]. 2.1. Mathematical Statements In a mathematical proof or discussion one makes various assertions, often called statements or sentences.1 For example: (1) (x + y)2 = x2 + 2xy + y 2 . (2) 3x2 + 2x − 1 = 0. (3) if n (≥ 3) is an integer then an + bn = c n has no positive integer solutions. (4) the derivative of the function x2 is 2x. Although a mathematical statement always has a very precise meaning, certain things are often assumed from the context in which the statement is made. For example, depending on the context in which statement (1) is made, it is probably an abbreviation for the statement for all real numbers x and y, (x + y)2 = x2 + 2xy + y 2 . However, it may also be an abbreviation for the statement for all complex numbers x and y, (x + y)2 = x2 + 2xy + y 2 . The precise meaning should always be clear from context; if it is not then more information should be pro...