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Introduction to Mathematical Analysis...

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Introduction To Mathematical Analysis

John E. Hutchinson 1994

Revised by Richard J. Loy 1995/6/7

Department of Mathematics School of Mathematical Sciences ANU

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 subtilties; 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 exited to more divine contemplations.

Encyclopædia Britannica

[1771]

i 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 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]

ii 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 as 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, 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 1 Introduction

1

1.1 Preliminary Remarks . . . . . . . . . . . . . . . . . . . . . . .

1

1.2 History of Calculus . . . . . . . . . . . . . . . . . . . . . . . .

2

1.3 Why “Prove” Theorems? . . . . . . . . . . . . . . . . . . . . .

2

1.4 “Summary and Problems” Book . . . . . . . . . . . . . . . . .

2

1.5 The approach to be used . . . . . . . . . . . . . . . . . . . . .

3

1.6 Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . .

3

2 Some Elementary Logic

5

2.1 Mathematical Statements . . . . . . . . . . . . . . . . . . . .

5

2.2 Quantifiers . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

7

2.3 Order of Quantifiers

. . . . . . . . . . . . . . . . . . . . . . .

8

2.4 Connectives . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9

2.4.1

Not . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.4.2

And . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.3

Or . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.4.4

Implies . . . . . . . . . . . . . . . . . . . . . . . . . . . 13

2.4.5

Iff . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.5 Truth Tables

9

. . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.6 Proofs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 2.6.1

Proofs of Statements Involving Connectives . . . . . . 16

2.6.2

Proofs of Statements Involving “There Exists” . . . . . 16

2.6.3

Proofs of Statements Involving “For Every” . . . . . . 17

2.6.4

Proof by Cases . . . . . . . . . . . . . . . . . . . . . . 18

3 The Real Number System

19

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2 Algebraic Axioms . . . . . . . . . . . . . . . . . . . . . . . . . 19 i

ii 3.2.1

Consequences of the Algebraic Axioms . . . . . . . . . 21

3.2.2

Important Sets of Real Numbers . . . . . . . . . . . . . 22

3.2.3

The Order Axioms . . . . . . . . . . . . . . . . . . . . 23

3.2.4

Ordered Fields . . . . . . . . . . . . . . . . . . . . . . 24

3.2.5

Completeness Axiom . . . . . . . . . . . . . . . . . . . 25

3.2.6

Upper and Lower Bounds . . . . . . . . . . . . . . . . 26

3.2.7

*Existence and Uniqueness of the Real Number System 29

3.2.8

The Archimedean Property . . . . . . . . . . . . . . . 30

4 Set Theory

33

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.2 Russell’s Paradox . . . . . . . . . . . . . . . . . . . . . . . . . 33 4.3 Union, Intersection and Difference of Sets . . . . . . . . . . . . 35 4.4 Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 4.4.1

Functions as Sets . . . . . . . . . . . . . . . . . . . . . 38

4.4.2

Notation Associated with Functions . . . . . . . . . . . 39

4.4.3

Elementary Properties of Functions . . . . . . . . . . . 40

4.5 Equivalence of Sets . . . . . . . . . . . . . . . . . . . . . . . . 41 4.6 Denumerable Sets . . . . . . . . . . . . . . . . . . . . . . . . . 42 4.7 Uncountable Sets . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.8 Cardinal Numbers

. . . . . . . . . . . . . . . . . . . . . . . . 46

4.9 More Properties of Sets of Cardinality c and d . . . . . . . . . 50 4.10 *Further Remarks . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.10.1 The Axiom of choice . . . . . . . . . . . . . . . . . . . 52 4.10.2 Other Cardinal Numbers . . . . . . . . . . . . . . . . . 53 4.10.3 The Continuum Hypothesis . . . . . . . . . . . . . . . 54 4.10.4 Cardinal Arithmetic . . . . . . . . . . . . . . . . . . . 55 4.10.5 Ordinal numbers . . . . . . . . . . . . . . . . . . . . . 55 5 Vector Space Properties of Rn

57

5.1 Vector Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 5.2 Normed Vector Spaces . . . . . . . . . . . . . . . . . . . . . . 59 5.3 Inner Product Spaces . . . . . . . . . . . . . . . . . . . . . . . 60 6 Metric Spaces 6.1 Basic Metric Notions in R

63 n

. . . . . . . . . . . . . . . . . . . 63

iii 6.2 General Metric Spaces . . . . . . . . . . . . . . . . . . . . . . 63 6.3 Interior, Exterior, Boundary and Closure . . . . . . . . . . . . 66 6.4 Open and Closed Sets . . . . . . . . . . . . . . . . . . . . . . 69 6.5 Metric Subspaces . . . . . . . . . . . . . . . . . . . . . . . . . 73 7 Sequences and Convergence

77

7.1 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 7.2 Convergence of Sequences . . . . . . . . . . . . . . . . . . . . 77 7.3 Elementary Properties . . . . . . . . . . . . . . . . . . . . . . 79 7.4 Sequences in R . . . . . . . . . . . . . . . . . . . . . . . . . . 80 7.5 Sequences and Components in Rk . . . . . . . . . . . . . . . . 83 7.6 Sequences and the Closure of a Set . . . . . . . . . . . . . . . 83 7.7 Algebraic Properties of Limits . . . . . . . . . . . . . . . . . . 84 8 Cauchy Sequences

87

8.1 Cauchy Sequences . . . . . . . . . . . . . . . . . . . . . . . . . 87 8.2 Complete Metric Spaces . . . . . . . . . . . . . . . . . . . . . 90 8.3 Contraction Mapping Theorem . . . . . . . . . . . . . . . . . 92 9 Sequences and Compactness

97

9.1 Subsequences . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 9.2 Existence of Convergent Subsequences . . . . . . . . . . . . . 98 9.3 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 9.4 Nearest Points . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 10 Limits of Functions

103

10.1 Diagrammatic Representation of Functions . . . . . . . . . . . 103 10.2 Definition of Limit . . . . . . . . . . . . . . . . . . . . . . . . 106 10.3 Equivalent Definition . . . . . . . . . . . . . . . . . . . . . . . 111 10.4 Elementary Properties of Limits . . . . . . . . . . . . . . . . . 112 11 Continuity

117

11.1 Continuity at a Point . . . . . . . . . . . . . . . . . . . . . . . 117 11.2 Basic Consequences of Continuity . . . . . . . . . . . . . . . . 119 11.3 Lipschitz and H¨ older Functions . . . . . . . . . . . . . . . . . 121 11.4 Another Definition of Continuity . . . . . . . . . . . . . . . . 122 11.5 Continuous Functions on Compact Sets . . . . . . . . . . . . . 124

iv 11.6 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . . 125 12 Uniform Convergence of Functions

129

12.1 Discussion and Definitions . . . . . . . . . . . . . . . . . . . . 129 12.2 The Uniform Metric . . . . . . . . . . . . . . . . . . . . . . . 135 12.3 Uniform Convergence and Continuity . . . . . . . . . . . . . . 138 12.4 Uniform Convergence and Integration . . . . . . . . . . . . . . 140 12.5 Uniform Convergence and Differentiation . . . . . . . . . . . . 141 13 First Order Systems of Differential Equations

143

13.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143 13.1.1 Predator-Prey Problem . . . . . . . . . . . . . . . . . . 143 13.1.2 A Simple Spring System . . . . . . . . . . . . . . . . . 144 13.2 Reduction to a First Order System . . . . . . . . . . . . . . . 145 13.3 Initial Value Problems . . . . . . . . . . . . . . . . . . . . . . 147 13.4 Heuristic Justification for the Existence of Solutions . . . . . . . . . . . . . . . . . . . . . . 149 13.5 Phase Space Diagrams . . . . . . . . . . . . . . . . . . . . . . 151 13.6 Examples of Non-Uniqueness and Non-Existence . . . . . . . . . . . . . . . . . . . . . . . . 153 13.7 A Lipschitz Condition . . . . . . . . . . . . . . . . . . . . . . 155 13.8 Reduction to an Integral Equation . . . . . . . . . . . . . . . . 157 13.9 Local Existence . . . . . . . . . . . . . . . . . . . . . . . . . . 158 13.10Global Existence . . . . . . . . . . . . . . . . . . . . . . . . . 162 13.11Extension of Results to Systems . . . . . . . . . . . . . . . . . 163 14 Fractals

165

14.1 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166 14.1.1 Koch Curve . . . . . . . . . . . . . . . . . . . . . . . . 166 14.1.2 Cantor Set . . . . . . . . . . . . . . . . . . . . . . . . . 167 14.1.3 Sierpinski Sponge . . . . . . . . . . . . . . . . . . . . . 168 14.2 Fractals and Similitudes . . . . . . . . . . . . . . . . . . . . . 170 14.3 Dimension of Fractals . . . . . . . . . . . . . . . . . . . . . . . 171 14.4 Fractals as Fixed Points . . . . . . . . . . . . . . . . . . . . . 174 14.5 *The Metric Space of Compact Subsets of Rn . . . . . . . . . 177 14.6 *Random Fractals . . . . . . . . . . . . . . . . . . . . . . . . . 182

v 15 Compactness

185

15.1 Definitions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185 15.2 Compactness and Sequential Compactness . . . . . . . . . . . 186 15.3 *Lebesgue covering theorem . . . . . . . . . . . . . . . . . . . 189 15.4 Consequences of Compactness . . . . . . . . . . . . . . . . . . 190 15.5 A Criterion for Compactness . . . . . . . . . . . . . . . . . . . 191 15.6 Equicontinuous Families of Functions . . . . . . . . . . . . . . 194 15.7 Arzela-Ascoli Theorem . . . . . . . . . . . . . . . . . . . . . . 197 15.8 Peano’s Existence Theorem . . . . . . . . . . . . . . . . . . . 201 16 Connectedness

207

16.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 16.2 Connected Sets . . . . . . . . . . . . . . . . . . . . . . . . . . 207 16.3 Connectedness in R . . . . . . . . . . . . . . . . . . . . . . . . 209 16.4 Path Connected Sets . . . . . . . . . . . . . . . . . . . . . . . 210 16.5 Basic Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 212 17 Differentiation of Real-Valued Functions

213

17.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213 17.2 Algebraic Preliminaries . . . . . . . . . . . . . . . . . . . . . . 214 17.3 Partial Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 215 17.4 Directional Derivatives . . . . . . . . . . . . . . . . . . . . . . 215 17.5 The Differential (or Derivative) . . . . . . . . . . . . . . . . . 216 17.6 The Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . 220 17.6.1 Geometric Interpretation of the Gradient . . . . . . . . 221 17.6.2 Level Sets and the Gradient . . . . . . . . . . . . . . . 221 17.7 Some Interesting Examples . . . . . . . . . . . . . . . . . . . . 223 17.8 Differentiability Implies Continuity . . . . . . . . . . . . . . . 224 17.9 Mean Value Theorem and Consequences . . . . . . . . . . . . 224 17.10Continuously Differentiable Functions . . . . . . . . . . . . . . 227 17.11Higher-Order Partial Derivatives . . . . . . . . . . . . . . . . . 229 17.12Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . 231 18 Differentiation of Vector-Valued Functions

237

18.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237 18.2 Paths in Rm . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

vi 18.2.1 Arc length . . . . . . . ....