Tao_T.-Analysis_I_(Volume_1)__-Hindustan_Book_Agency(2006)

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2.1. The Pean? axioms However, we shall stick with the Peano axiomatic approach for now. How are we to define what the natural numbers are? Informally, we could say Definition 2.1.1. (Informal) A natural number is any element of the set N := {0,1,2,3,4, ... }, which is the set of all the numbers created by starting with 0 and then counting forward indefinitely. We call N the set of natural numbers. Remark 2.1.2. In some texts the natural numbers start at 1 instead of 0,. but this is a matter of notational convention more than anything else. In this text we shall refer to the set {1, 2, 3, ... } as the positive integers z+ rather than the natural numbers. Natural numbers are sometimes also known as whole numbers. In a sense, this definition solves the problem of what the natural numbers are: a natural number is any element of the set 1 N. However, it is not really that satisfactory, because it begs the question of what N is. This definition of "start at 0 and count indefinitely" seems like an intuitive enough definition of N, but it is not entirely acceptable, because it leaves many questions unanswered. For instance: how do we know we can keep counting indefinitely, without cycling back to 0? Also, how do you perform operations such as addition, multiplication, or exponentiation? We can answer the latter question first: we can define complicated operations in terms of simpler operations. Exponentiation is nothing more than repeated multiplication: 53 is nothing more than three fives multiplied together. Multiplication is nothing more than repeated addition; 5 x 3 is nothing more than three fives added together. (Subtraction and division will not be covered here, because they are not operations which are well-suited 1Strictly speaking, there is another problem with this informal definition: we have not yet defined what a "set" is, or what "element of" is. Thus for the rest of this chapter we shall avoid mention of sets and their elements as much as possible, except in informal discussion. 17
18 2. The natural numbers to the natural numbers; they will have to wait for the integers and rationals, respectively.) And addition? It is nothing more than the repeated operation of counting forward, or incrementing. If you add three to five, what you are doing is incrementing five three times. On the other hand, incrementing seems to be a fundamental operation, not reducible to any simpler operation; indeed, it is the first operation one learns on numbers, even before learning to add. Thus, to define the natural numbers, we will use two fundamental concepts: the zero number 0, and the increment operation. In deference to modern computer languages, we will use n++ to denote the increment or successor of n, thus for instance 3++ = 4, (3++ )++ = 5, etc. This is a slightly different usage from that in computer languages such as C, where n++ actually redefines the value of n to be its successor; however in mathematics we try not to define a variable more than once in any given setting, as it can often lead to confusion; many of the statements which were true for the old value of the variable can now become false, and vice versa. So, it seems like we want to say that N consists of 0 and everything which can be obtained from 0 by incrementing: N should consist of the objects O,O++,(O++)++,((O++)++)++,etc. If we start writing down what this means about the natural numbers, we thus see that we should have the following axioms concerning 0 and the increment operation ++: Axiom 2.1. 0 is a natural number. Axiom 2.2. If n is a natural number, then n++ is also a natural number. Thus for instance, from Axiom 2.1 and two applications of Axiom 2.2, we see that (0++ )++ is a natural number. Of course, this notation will begin to get unwieldy, so we adopt a convention to write these numbers in more familiar notation:
Page 4: ll:JQ1@ HINDUS TAN Ul!!.rU BOOK AGE
Page 7 and 8: To my parents, for everything
Page 9: viii 4.4 Gaps in the rational numbe
Page 12 and 13: CONTENTS 15 Power series 15.1 Forma
Page 14 and 15: Preface This text originated from t
Page 16 and 17: Preface XV more advanced facts abou
Page 18 and 19: Preface xvii tive; much of the key
Page 21 and 22: 2 1. Introduction 1. What is a real
Page 24 and 25: 1.2. Why do analysis? 5 At this poi
Page 26: 1.2. Why do analysis? 7 However, de
Page 33 and 34: Chapter 2 Starting at the beginning
Page 39: 20 2. The natuml numbers Now we can
Page 42 and 43: 2.1. The Peano axioms 23 from appea
Page 45 and 46: 26 2. The natural numbers understoo
Page 48 and 49: 2.2. Addition 29 m)++· But by defi
Page 52 and 53: 2.3. Multiplication 33 Exercise 2.2
Page 55 and 56: 36 2. The natural numbers Proof. Se
Page 57 and 58: 38 3. Set theory that some of the a
Page 59: 40 3. Set theory A.7). Because of t
Page 62: 3.1. Fundamentals 43 proof We prove
Page 68: 3.1. Fundamentals We have now accum
Page 73: 54 3. Set theory Instead, we shall
Page 76: 3.3. Functions 57 One common way to
Page 79: 60 3. Set theory Proof. Since g o h
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9.4. Images and inverse images 67 N
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70 3. Set theory This should be com
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4 .1. The integers 85 tage of our f
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4.1. The integers 87 Lemma 4.1.3 (A
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4.1. The integers 89 Jfn is a posit
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92 4. Integers and rationals Proof.
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96 4. Integers and rationals xx-1 =
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4-4· Gaps in the rational numbers
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108 5. The real numbers or more of
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5.5. The least upper bound property
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5.6. Real exponentiation, part I 14
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164 6. Limits of sequences Example
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8.1. Countability 209 Jtelllark 8.1
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216 8. Infinite sets Exercise 8.1.4
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8.5. Ordered sets 241 yet another c
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9.2. The algebra of real-valued fun
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g, 1. The intermediate value theore
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Chapter 10 Differentiation of funct
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292 10. Differentiation of function
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300 10. Differentiation of function
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308 11. The Riemann integral (a) X
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318 11. The Riemann integral Proof.
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322 11. The Riemann integral Proof.
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324 11. The Riemann integral for ev
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11.5. Riemann integrability of cont
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Chapter A Appendix: the basics of m
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A.1. Mathematical statements 351 :E
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A.2. Implication 357 Sometimes it i
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364 A. Appendix: the basics of math
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;1.4. Variables and quantifiers 367
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A.4· Variables and quantifiers 371
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374 A. Appendix: the basics of math
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A. 7. Equality 377 takes the value
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A. 7. Equality 379 sin(y) = sin(z2)
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B.l. The decimal representation of
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Index ++ (increment), 18, 56 on int
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INDEX C, C 0 , C 1 , C 2 , Ck, 556
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INDEX of functions, 75, 425 discont
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VIII equivalence of in finite dimen
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X negative: see negation, positive
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INDEX sup norm: see supremum as nor
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