TBB - Dripped - Work In Progress - June2008, Notas de estudo de Matemática

Baixe TBB - Dripped - Work In Progress - June2008 e outras Notas de estudo em PDF para Matemática, somente na Docsity! i ELEMENTARY REAL ANALYSIS: DRIPPED VERSION ————————————— thomson·bruckner2 ————————————— Brian S. Thomson Judith B. Bruckner Andrew M. Bruckner www.classicalrealanalysis.com (2008) ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner ii D.R.I.P. = Dump the Riemann Integral Project. This version of the text includes an account of the natural integral on the real line and removes the development of the Riemann integral and the improper Riemann integral. It should be considered experimental. c© Work in Progress: As a work in progress this will change frequently over the next year. Your input and feedback can help make this project work. Please contact us with suggestions. This PDF file includes material from the text Elementary Real Analysis originally published by Prentice Hall (Pearson) in 2001. The authors retain the copyright and all commercial uses. [2008] Date PDF file compiled: June 16, 2008 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner v 3.5.2 Cauchy Criterion 133 3.5.3 Absolute Convergence 135 3.6 Tests for Convergence 139 3.6.1 Trivial Test 140 3.6.2 Direct Comparison Tests 140 3.6.3 Limit Comparison Tests 143 3.6.4 Ratio Comparison Test 145 3.6.5 d’Alembert’s Ratio Test 146 3.6.6 Cauchy’s Root Test 149 3.6.7 Cauchy’s Condensation Test 150 3.6.8 Integral Test 152 3.6.9 Kummer’s Tests 154 3.6.10 Raabe’s Ratio Test 157 3.6.11 Gauss’s Ratio Test 158 3.6.12 Alternating Series Test 162 3.6.13 Dirichlet’s Test 163 3.6.14 Abel’s Test 165 3.7 Rearrangements 172 3.7.1 Unconditional Convergence 173 3.7.2 Conditional Convergence 175 3.7.3 Comparison of ∑∞ i=1 ai and ∑ i∈IN ai 177 3.8 Products of Series 180 3.8.1 Products of Absolutely Convergent Series 183 3.8.2 Products of Nonabsolutely Convergent Series 185 3.9 Summability Methods 188 3.9.1 Cesàro’s Method 189 3.9.2 Abel’s Method 191 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner vi 3.10 More on Infinite Sums 196 3.11 Infinite Products 199 3.12 Challenging Problems for Chapter 3 205 Notes 210 4 SETS OF REAL NUMBERS 216 4.1 Introduction 216 4.2 Points 217 4.2.1 Interior Points 218 4.2.2 Isolated Points 220 4.2.3 Points of Accumulation 221 4.2.4 Boundary Points 222 4.3 Sets 225 4.3.1 Closed Sets 226 4.3.2 Open Sets 227 4.4 Elementary Topology 235 4.5 Compactness Arguments 238 4.5.1 Bolzano-Weierstrass Property 240 4.5.2 Cantor’s Intersection Property 242 4.5.3 Cousin’s Property 244 4.5.4 Heine-Borel Property 246 4.5.5 Compact Sets 251 4.6 Countable Sets 254 4.7 Challenging Problems for Chapter 4 256 Notes 259 5 CONTINUOUS FUNCTIONS 262 5.1 Introduction to Limits 262 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner vii 5.1.1 Limits (ε-δ Definition) 263 5.1.2 Limits (Sequential Definition) 268 5.1.3 Limits (Mapping Definition) 271 5.1.4 One-Sided Limits 273 5.1.5 Infinite Limits 275 5.2 Properties of Limits 278 5.2.1 Uniqueness of Limits 278 5.2.2 Boundedness of Limits 279 5.2.3 Algebra of Limits 281 5.2.4 Order Properties 285 5.2.5 Composition of Functions 290 5.2.6 Examples 293 5.3 Limits Superior and Inferior 301 5.4 Continuity 304 5.4.1 How to Define Continuity 304 5.4.2 Continuity at a Point 308 5.4.3 Continuity at an Arbitrary Point 312 5.4.4 Continuity on a Set 315 5.5 Properties of Continuous Functions 319 5.6 Uniform Continuity 320 5.7 Extremal Properties 325 5.8 Darboux Property 327 5.9 Points of Discontinuity 329 5.9.1 Types of Discontinuity 330 5.9.2 Monotonic Functions 332 5.9.3 How Many Points of Discontinuity? 337 5.10 Challenging Problems for Chapter 5 339 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner x 8.1.3 The “improper” calculus integral 485 8.2 The infinite integral 488 8.3 Cauchy’s analysis of the integral 491 8.3.1 First mean-value theorem for integrals 491 8.3.2 The method of exhaustion 492 8.3.3 Riemann sums 495 8.3.4 The integral of continuous functions as a limit of Riemann sums 496 8.3.5 The calculus integral as a limit of Riemann sums 498 8.4 Extensions of the integral 503 8.4.1 Riemann’s integral 504 8.4.2 Lebesgue’s integral 504 8.4.3 The Henstock-Kurzweil extension 506 8.4.4 An extended calculus integral 506 8.5 Challenging Problems for Chapter 8 509 Notes 510 9 COVERING RELATIONS 512 9.1 Partitions and subpartitions 512 9.2 Covering relations 513 9.2.1 Prunings 513 9.2.2 Full covers 513 9.2.3 Fine covers 514 9.2.4 Uniformly full covers 514 9.3 Cousin covering lemma 517 9.4 Riemann sums 518 Notes 521 10 THE INTEGRAL 523 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner xi 10.1 Upper and lower integrals 524 10.1.1 The integral and integrable functions 525 10.2 Integrability criteria 527 10.2.1 First Cauchy criterion 528 10.2.2 Second Cauchy criterion 529 10.2.3 Integrability on subintervals 531 10.2.4 The indefinite integral 531 10.2.5 Absolutely integrable functions 532 10.2.6 Henstock’s zero variation criterion 534 10.3 Continuous functions are absolutely integrable 538 10.4 Elementary properties of the integral 538 10.4.1 Integration and order 539 10.4.2 Integration of linear combinations 539 10.4.3 The integral as an additive interval function 539 10.4.4 Change of variable 540 10.5 The fundamental theorem of the calculus 542 10.5.1 Derivative of the integral of continuous functions 542 10.5.2 Relation to the calculus integral 543 10.5.3 Integral of the derivative 543 10.5.4 Relation to the Newton integral 544 Notes 546 11 NULL SETS AND NULL FUNCTIONS 548 11.1 Sets of measure zero 548 11.1.1 Lebesgue measure of open sets 550 11.1.2 Sets of measure zero 551 11.1.3 Sequences of measure zero sets 553 11.1.4 Compact sets of measure zero 553 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner xii 11.2 Full null sets 556 11.3 Fine null sets 558 11.4 The Mini-Vitali Covering Theorem 558 11.4.1 Covering lemmas for families of compact intervals 559 11.4.2 Proof of the Mini-Vitali covering theorem 561 11.5 Null functions 564 11.6 Integral of null functions 564 11.7 Functions with a zero integral 565 11.8 Almost everywhere language 567 11.9 Integration conventions on ignoring points 568 11.10Bounded a.e. continuous functions are absolutely integrable 570 Notes 572 12 VARIATION OF A FUNCTION 574 12.1 Functions having zero variation 575 12.2 Zero variation and zero derivatives 577 12.2.1 Proof of the zero variation/derivative theorem 577 12.2.2 Generalization of the zero derivative/variation 579 12.3 Functions of bounded variation 580 12.4 Lebesgue differentiation theorem 581 12.4.1 Upper and lower derivates 581 12.4.2 Geometrical lemmas 582 12.4.3 Proof of the Lebesgue differentiation theorem 584 12.5 Continuity and absolute continuity 587 12.5.1 Decompositions of monotone functions 590 12.6 Absolute continuity of the indefinite integral 591 12.7 Lipschitz functions 593 12.8 Monotonicity theorems 593 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner xv 17 LEBESGUE’S PROGRAM 720 17.1 Lebesgue measure 721 17.1.1 Basic property of Lebesgue measure 722 17.2 Vitali covering theorem 723 17.2.1 Classical version of Vitali’s theorem 723 17.2.2 Proof that L = L∗ = L∗. 726 17.3 Density theorem 727 17.4 Additivity 728 17.5 Measurable sets 731 17.6 Measurable functions 733 17.6.1 Continuous functions are measurable 734 17.6.2 Derivatives and integrable functions are measurable 734 17.6.3 Simple functions 736 17.6.4 Series of simple functions 737 17.6.5 Limits of measurable functions 738 17.7 Construction of the integral 739 17.7.1 Characteristic functions of measurable sets 739 17.7.2 Characterizations of measurable sets 741 17.7.3 Integral of simple functions 742 17.7.4 Integral of nonnegative measurable functions 742 17.7.5 Derivatives of functions of bounded variation 743 17.7.6 Integral of absolutely integrable functions 745 17.7.7 McShane’s Criterion 746 17.7.8 Nonabsolutely integrable functions 748 17.8 Characterizations of the indefinite integral 749 17.8.1 Integral of nonnegative, integrable functions 751 17.8.2 Integral of absolutely integrable functions 751 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner xvi 17.8.3 Integral of nonabsolutely integrable functions 752 17.8.4 Proofs 752 17.9 Denjoy’s program 755 17.10Challenging Problems for Chapter 17 756 Notes 758 18 STIELTJES INTEGRALS 760 18.1 Stieltjes integrals 761 18.1.1 Definition of the Stieltjes integral 762 18.1.2 Henstock’s zero variation criterion 766 18.2 Regulated functions 766 18.3 Variation expressed as an integral 769 18.4 Representation theorems for functions of bounded variation 771 18.4.1 Jordan decomposition 771 18.4.2 Jordan decomposition theorem: differentiation 772 18.4.3 Representation by saltus functions 774 18.4.4 Representation by singular functions 774 18.5 Reducing a Stieltjes integral to an ordinary integral 774 18.6 Properties of the indefinite integral 778 18.6.1 Existence of the integral from derivative statements 781 18.7 Existence of the Stieltjes integral for continuous functions 782 18.8 Integration by parts 783 18.9 Mutually singular functions 785 18.10Singular functions 788 18.11Length of curves 789 18.11.1Formula for the length of curves 790 18.12Challenging Problems for Chapter 18 792 Notes 793 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner xvii A BACKGROUND A-1 A.1 Should I Read This Chapter? A-1 A.2 Notation A-1 A.2.1 Set Notation A-1 A.2.2 Function Notation A-5 A.3 What Is Analysis? A-13 A.4 Why Proofs? A-13 A.5 Indirect Proof A-15 A.6 Contraposition A-17 A.7 Counterexamples A-18 A.8 Induction A-19 A.9 Quantifiers A-23 Notes A-25 SUBJECT INDEX A-28 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner xx Preface • [Full drip ] Add parts of the advanced material as desired [Chapters 15, 18, and 17]. This brings the student up to a serious level in integration theory but not in the usual direction. The culmination, in Chapter 17, is the Lebesgue program for the measure-theoretic development of his integral. The standard approach starts with measure theory and slowly (some say painfully) develops the integral and its theory; this easier version starts with the integral and the measure theory develops naturally from it. Summaries of the added drip chapters Chapter 8 just recounts the calculus version of the integral and suggests that a study of Riemann sums should lead to an adequate theory of integration. This is at an entirely elementary level and, given a minimal ambition for the course, could be used alone (with no further drip material) for a simple course. Chapter 9 gives the basics of the covering argument approach to elementary analysis, the use of partitions, Riemann sums, full and fine covers, and the Cousin covering lemma. Chapter 10 contains the integration theory. There is enough theory on integrability criteria so that one can demonstrate just how large the class of integrable functions are. This chapter includes proofs that continuous functions and derivatives are integrable, in fact that the integral includes a fairly general version of the Newton integral. The usual (and some unusual) simple properties of integrals are proved. Chapter 11 gives the theory of sets of measure zero and the usual applications for the integral. Sets of measure zero play a peculiar (but important) role in the theory of the Riemann integral. Since we have “dumped” that integral the measure zero sets now play a natural and compelling role. We include an elementary proof of the Lebesgue differentiation theorem, that functions of bounded variation are a.e. dif- ferentiable. The chapter contains a narrow version of the Vitali covering theorem, showing that null sets can be characterized by fine covers. The proof is elementary and is a convenient way to introduce Vitali covering arguments at an elementary level. Chapter 13 gives a complete account of the fundamental theorem of the calculus for this integral. This goes far beyond what would be done in a traditional undergraduate class, and even exceeds somewhat what is done in many graduate classes, albeit using here fairly elementary methods. Indeed, the elementary ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Preface xxi version of the fundamental theorem of the calculus in Chapter 10 is already beyond what most courses would attempt. Chapter 15 completes the previous chapter on integration of sequences and series by proving the mono- tone convergence theorem. This is done with no measure theory which many consider one of the strongest reasons for dumping the Riemann integral in favor of this integral. Chapter 18 develops basic material on functions of bounded variation and Stieltjes integral. The Jordan decomposition theorem is proved. This material is more frequently reserved for advanced courses but there is little trouble in presenting this at an undergraduate level if you have sufficient reason to do so. It can be entirely skipped without interfering with any later chapters. Finally, Chapter 17 gives Lesbesgue’s measure-theoretic program for the integral. It starts with a proof of the Vitali covering theorem that should be accessible, especially since it is well anticipated by the mini-Vitali version of Chapter 11. One main difference with the standard treatment is that the integral is not defined by the measure methods, but is characterized by them. This would be a suitable elementary introduction to measure theory, preparatory to the student taking an abstract course in the subject. This chapter, too, can be entirely skipped without interfering with any later chapters. Original Preface (2001) University mathematics departments have for many years offered courses with titles such as Advanced Calculus or Introductory Real Analysis. These courses are taken by a variety of students, serve a number of purposes, and are written at various levels of sophistication. The students range from ones who have just completed a course in elementary calculus to beginning graduate students in mathematics. The purposes are multifold: 1. To present familiar concepts from calculus at a more rigorous level. 2. To introduce concepts that are not studied in elementary calculus but that are needed in more advanced undergraduate courses. This would include such topics as point set theory, uniform continuity of functions, and uniform convergence of sequences of functions. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner xxii Preface 3. To provide students with a level of mathematical sophistication that will prepare them for graduate work in mathematical analysis, or for graduate work in several applied fields such as engineering or economics. 4. To develop many of the topics that the authors feel all students of mathematics should know. There are now many texts that address some or all of these objectives. These books range from ones that do little more than address objective (1) to ones that try to address all four objectives. The books of the first extreme are generally aimed at one-term courses for students with minimal background. Books at the other extreme often contain substantially more material than can be covered in a one-year course. The level of rigor varies considerably from one book to another, as does the style of presentation. Some books endeavor to give a very efficient streamlined development; others try to be more user friendly. We have opted for the user-friendly approach. We feel this approach makes the concepts more meaningful to the student. Our experience with students at various levels has shown that most students have difficulties when topics that are entirely new to them first appear. For some students that might occur almost immediately when rigorous proofs are required, for example, ones needing ε-δ arguments. For others, the difficulties begin with elementary point set theory, compactness arguments, and the like. To help students with the transition from elementary calculus to a more rigorous course, we have included motivation for concepts most students have not seen before and provided more details in proofs when we introduce new methods. In addition, we have tried to give students ample opportunity to see the new tools in action. For example, students often feel uneasy when they first encounter the various compactness arguments (Heine-Borel theorem, Bolzano-Weierstrass theorem, Cousin’s lemma, introduced in Section 4.5). To help the student see why such theorems are useful, we pose the problem of determining circumstances under which local boundedness of a function f on a set E implies global boundedness of f on E. We show by example that some conditions on E are needed, namely that E be closed and bounded, and then show how each of several theorems could be used to show that closed and boundedness of the set E suffices. Thus we introduce students to the theorems by showing how the theorems can be used in natural ways to solve a problem. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Preface xxv chapter 1 ? chapter 2 - chapter 4 - chapter 5 chapter 6 6 ? chapter 3 ? chapter 7chapter 8 ? chapter 19 ? chapter 20 chapter 14 ? chapter 16 ? chapter 21  ∗ Figure 0.1. Chapter Dependencies (Unmarked Sections). Chapter numbers refer to this dripped edition, not the original. Designing a Course We have attempted to write this book in a manner sufficiently flexible to make it possible to use the book for courses of various lengths and a variety of levels of mathematical sophistication. Much of the material in the book involves rigorous development of topics of a relatively elementary nature, topics that most students have studied at a nonrigorous level in a calculus course. A short course of moderate mathematical sophistication intended for students of minimal background can be based entirely on this material. Such a course might meet objective (1). We have written this book in a leisurely style. This allows us to provide motivational discussions and historical perspective in a number of places. Even though the book is relatively large (in terms of number ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner xxvi Preface of pages), we can comfortably cover most of the main sections in a full-year course, including many of the interesting exercises. Instructors teaching a short course have several options. They can base a course entirely on the unmarked material of Chapters 1, 2, 4, 5, and 7. As time permits, they can add the early parts of Chapters 3 and 8 or parts of Chapters 11 and 12 and some of the enrichment material. Background We should make one more point about this book. We do assume that students are familiar with nonrigorous calculus. In particular, we assume familiarity with the elementary functions and their elementary properties. We also assume some familiarity with computing derivatives and integrals. This allows us to illustrate various concepts using examples familiar to the students. For example, we begin Chapter 2, on sequences, with a discussion of approximating √ 2 using Newton’s method. This is merely a motivational discussion, so we are not bothered by the fact that we don’t treat the derivative formally until Chapter 7 and haven’t yet proved that ddx(x 2−2) = 2x. For students with minimal background we provide an appendix that informally covers such topics as notation, elementary set theory, functions, and proofs. Acknowledgments A number of friends, colleagues, and students have made helpful comments and suggestions while the text was being written. We are grateful to the reviewers of the text: Professors Eugene Allgower (Colorado State University), Stephen Breen (California State University, Northridge), Robert E. Fennell (Clemson Univer- sity), Jan E. Kucera (Washington State University), and Robert F. Lax (Louisiana State University). The authors are particularly grateful to Professors Steve Agronsky (California Polytechnic State University), Peter Borwein (Simon Fraser University), Paul Humke (St. Olaf College), T. H. Steele (Weber State Uni- versity), and Clifford Weil (Michigan State University) for using preliminary versions of the book in their classes. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Preface xxvii Additional acknowledgments, added January 2008 We are grateful to many users of the original 2001 version of the text for spotting errors and oversights. Special mention should be made of Richard Delaware (University of Missouri–Kansas City) for considerable help in this. A. M. B. J. B. B. B. S. T. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 1.2. The Real Number System 3 Large amounts of time in elementary school are devoted to an understanding of these operations and the order relation. (Subtraction and division can also be defined, but not for all pairs in IN. While 7 − 5 and 10/5 are assigned a meaning [we say x = 7 − 5 if x + 5 = 7 and we say x = 10/5 if 5 · x = 10] there is no meaning that can be attached to 5 − 7 and 5/10 in this number system.) The Integers For various reasons, usually well motivated in the lower grades, the natural numbers prove to be rather limited in representing problems that arise in applications of mathematics to the real world. Thus they are enlarged by adjoining the negative integers and zero. Thus the collection . . . ,−4,−3,−2,−1, 0, 1, 2, 3, 4, . . . is denoted Z and called the integers. (The symbol IN seems obvious enough [N for “natural”] but the symbol Z for the integers originates in the German word for whole number.) Once again, there are two operations on Z, addition and multiplication: m + n and m · n. Again there is an order relation m < n. Fortunately, the rules of arithmetic and order learned for the simpler system IN continue to hold for Z, and young students extend their abilities perhaps painlessly. (Subtraction can now be defined in this larger number system, but division still may not be defined. For example, −9/3 is defined but 3/(−9) is not.) The Rational Numbers At some point the problem of the failure of division in the sets IN and Z becomes acute and the student must progress to an understanding of fractions. This larger number system is denoted Q, where the symbol chosen is meant to suggest quotients, which is after all what fractions are. The collection of all “numbers” of the form m n , ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 4 Properties of the Real Numbers Chapter 1 where m ∈ Z and n ∈ IN is called the set of rational numbers and is denoted Q. A higher level of sophistication is demanded at this stage. Equality has a new meaning. In IN or Z a statement m = n meant merely that m and n were the same object. Now m n = a b for m, a ∈ Z and n, b ∈ IN means that m · b = a · n. Addition and multiplication present major challenges too. Ultimately the students must learn that m n + a b = mb + na nb and m n · a b = ma nb . Subtraction and division are similarly defined. Fortunately, once again the rules of arithmetic are unchanged. The associative rule, distributive rule, etc. all remain true even in this number system. Again, too, an order relation m n < a b is available. It can be defined by requiring, for m, a ∈ Z and n, b ∈ IN, mb < na. The same rules for inequalities learned for integers and natural numbers are valid for rationals. The Real Numbers Up to this point in developing the real numbers we have encountered only arithmetic operations. The progression from IN to Z to Q is simply algebraic. All this algebra might have been a burden to the weaker students in the lower grades, but conceptually the steps are easy to grasp with a bit of familiarity. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 1.2. The Real Number System 5 The next step, needed for all calculus students, is to develop the still larger system of real numbers, denoted as R. We often refer to the real number system as the real line and think about it as a geometrical object, even though nothing in our definitions would seem at first sight to allow this. Most calculus students would be hard pressed to say exactly what these numbers are. They recognize that R includes all of IN, Z, and Q and also many new numbers, such as √ 2, e, and π. But asked what a real number is, many would return a blank stare. Even just asked what √ 2, e, or π are often produces puzzlement. Well, √ 2 is a number whose square is 2. But is there a number whose square is 2? A calculator might oblige with 1.4142136, but (1.4142136)2 6= 2. So what exactly “is” this number √ 2? If we are unable to write down a number whose square is 2, why can we claim that there is a number whose square is 2? And π and e are worse. Some calculus texts handle this by proclaiming that real numbers are obtained by infinite decimal expansions. Thus while rational numbers have infinite decimal expansions that terminate (e.g., 1/4 = 0.25) or repeat (e.g., 1/3 = 0.333333 . . . ), the collection of real numbers would include all infinite decimal expansions whether repeating, terminating, or not. In that case the claim would be that there is some infinite decimal expansion 1.414213 . . . whose square really is 2 and that infinite decimal expansion is the number we mean by the symbol √ 2. This approach is adequate for applications of calculus and is a useful way to avoid doing any hard mathematics in introductory calculus courses. But you should recall that, at certain stages in the calculus textbook that you used, appeared a phrase such as “the proof of this next theorem is beyond the level of this text.” It was beyond the level of the text only because the real numbers had not been properly treated and so there was no way that a proof could have been attempted. We need to construct such proofs and so we need to abandon this loose, descriptive way of thinking about the real numbers. Instead we will define the real numbers to be a complete, ordered field. In the next sections each of these terms is defined. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 8 Properties of the Real Numbers Chapter 1 Note that we have labeled the axioms with letters indicating which operations are affected, thus A for addition and M for multiplication. The distributive rule AM1 connects addition and multiplication. How are we to use these axioms? The answer likely is that, in an analysis course, you would not. You might try some of the exercises to understand what a field is and why the real numbers form a field. In an algebra course it would be most interesting to consider many other examples of fields and some of their applications. For an analysis course, understand that we are trying to specify exactly what we mean by the real number system, and these axioms are just the beginning of that process. The first step in that process is to declare that the real numbers form a field under the two operations of addition and multiplication. Exercises 1.3.1 The field axioms include rules known often as associative rules, commutative rules and distributive rules. Which are which and why do they have these names? 1.3.2 To be precise we would have to say what is meant by the operations of addition and multiplication. Let S be a set and let S × S be the set of all ordered pairs (s1, s2) for s1, s2 ∈ S. A binary operation on S is a function B : S × S → S. Thus the operation takes the pair (s1, s2) and outputs the element B(s1, s2). For example, addition is a binary operation. We could write (s1, s2) → A(s1, s2) rather than the more familiar (s1, s2) → s1 + s2. (a) Rewrite axioms A1–A4 using this notation A(s1, s2) instead of the sum notation. (b) Define a binary operation on R different from addition, subtraction, multiplication, or division and determine some of its properties. (c) For a binary operation B define what you might mean by the commutative, associative, and distributive rules. (d) Does the binary operation of subtraction satisfy any one of the commutative, associative, or distributive rules? 1.3.3 If in the field axioms for R we replace R by any other set with two operations + and · that satisfy these nine properties, then we say that that structure is a field. For example, Q is a field. The rules are valid since Q ⊂ R. The only thing that needs to be checked is that a + b and a · b are in Q if both a and b are. For this reason Q is called a subfield of R. Find another subfield. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 1.3. Algebraic Structure 9 See Note 1 1.3.4 Let S be a set consisting of two elements labeled as A and B. Define A+A = A, B+B = A, A+B = B+A = B, A ·A = A, A ·B = B ·A = A, and B ·B = B. Show that all nine of the axioms of a field hold for this structure. 1.3.5 Using just the field axioms, show that (x + 1)2 = x2 + 2x + 1 for all x ∈ R. Would this identity be true in any field? See Note 2 1.3.6 Define operations of addition and multiplication on Z5 = {0, 1, 2, 3, 4} as follows: + 0 1 2 3 4 0 0 1 2 3 4 1 1 2 3 4 0 2 2 3 4 0 1 3 3 4 0 1 2 4 4 0 1 2 3 × 0 1 2 3 4 0 0 0 0 0 0 1 0 1 2 3 4 2 0 2 4 1 3 3 0 3 1 4 2 4 0 4 3 2 1 Show that Z5 satisfies all the field axioms. 1.3.7 Define operations of addition and multiplication on Z6 = {0, 1, 2, 3, 4, 5} as follows: + 0 1 2 3 4 5 0 0 1 2 3 4 5 1 1 2 3 4 5 0 2 2 3 4 5 0 1 3 3 4 5 0 1 2 4 4 5 0 1 2 3 5 5 0 1 2 3 4 × 0 1 2 3 4 5 0 0 0 0 0 0 0 1 0 1 2 3 4 5 2 0 2 4 0 2 4 3 0 3 0 3 0 3 4 0 4 2 0 4 2 5 0 5 4 3 2 1 Which of the field axioms does Z6 fail to satisfy? ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 10 Properties of the Real Numbers Chapter 1 1.4 Order Structure The real number system also enjoys an order structure. Part of our usual picture of the reals is the sense that some numbers are “bigger” than others or more to the “right” than others. We express this by using inequalities x < y or x ≤ y. The order structure is closely related to the field structure. For example, when we use inequalities in elementary courses we frequently use the fact that if x < y and 0 < z, then xz < yz (i.e., that inequalities can be multiplied through by positive numbers). This structure, too, can be axiomatized and reduced to a small set of rules. Once again, these same rules can be found in other applications of mathematics. When these rules are added to the field axioms the result is called an ordered field. The real number system is an ordered field, satisfying the four additional axioms. Here a < b is now a statement that is either true or false. (Before a + b and a · b were not statements, but elements of R.) O1 For any a, b ∈ R exactly one of the statements a = b, a < b or b < a is true. O2 For any a, b, c ∈ R if a < b is true and b < c is true, then a < c is true. O3 For any a, b ∈ R if a < b is true, then a + c < b + c is also true for any c ∈ R. O4 For any a, b ∈ R if a < b is true, then a · c < b · c is also true for any c ∈ R for which c > 0. Exercises 1.4.1 Using just the axioms, prove that ad + bc < ac + bd if a < b and c < d. 1.4.2 Show for every n ∈ IN that n2 ≥ n. 1.4.3 Using just the axioms, prove the arithmetic-geometric mean inequality: √ ab ≤ a + b 2 for any a, b ∈ R with a > 0 and b > 0. (Assume, for the moment, the existence of square roots.) See Note 3 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 1.6. Sups and Infs 13 Definition 1.9: (Least Upper Bound/Supremum) Let E be a set of real numbers that is bounded above and nonempty. If M is the least of all the upper bounds, then M is said to be the least upper bound of E or the supremum of E and we write M = supE. Definition 1.10: (Greatest Lower Bound/Infimum) Let E be a set of real numbers that is bounded below and nonempty. If m is the greatest of all the lower bounds of E, then m is said to be the greatest lower bound of E or the infimum of E and we write M = inf E. To complete the definition of inf E and supE it is most convenient to be able write this expression even for E = ∅ or for unbounded sets. Thus we write 1. inf ∅ = ∞ and sup ∅ = −∞. 2. If E is unbounded above, then supE = ∞. 3. If E is unbounded below, then inf E = −∞. The Axiom of Completeness Any example of a nonempty set that you are able to visualize that has an upper bound will also have a least upper bound. Pages of examples might convince you that all nonempty sets bounded above must have a least upper bound. Indeed your intuition will forbid you to accept the idea that this could not always be the case. To prove such an assertion is not possible using only the axioms for an ordered field. Thus we shall assume one further axiom, known as the axiom of completeness. Completeness Axiom A nonempty set of real numbers that is bounded above has a least upper bound (i.e., if E is nonempty and bounded above, then supE exists and is a real number). This now is the totality of all the axioms we need to assume. We have assumed that R is a field with two operations of addition and multiplication, that R is an ordered field with an inequality relation “<”, and finally that R is a complete ordered field. This is enough to characterize the real numbers and the phrase “complete ordered field” refers to the system of real numbers and to no other system. (We shall not prove this statement; see Exercise 1.11.3 for a discussion.) ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 14 Properties of the Real Numbers Chapter 1 Exercises 1.6.1 Show that a set of real numbers E is bounded if and only if there is a positive number r so that |x| < r for all x ∈ E. 1.6.2 Find supE and inf E and (where possible) maxE and minE for the following examples of sets: (a) E = IN (b) E = Z (c) E = Q (d) E = R (e) E = {−3, 2, 5, 7} (f) E = {x : x2 < 2} (g) E = {x : x2 − x − 1 < 0} (h) E = {1/n : n ∈ IN} (i) E = { n√n : n ∈ IN} 1.6.3 Under what conditions does supE = max E? 1.6.4 Show for every nonempty, finite set E that supE = max E. See Note 4 1.6.5 For every x ∈ R define [x] = max{n ∈ Z : n ≤ x} called the greatest integer function. Show that this is well defined and sketch the graph of the function. 1.6.6 Let A be a set of real numbers and let B = {−x : x ∈ A}. Find a relation between maxA and minB and between minA and maxB. 1.6.7 Let A be a set of real numbers and let B = {−x : x ∈ A}. Find a relation between supA and inf B and between inf A and supB. 1.6.8 Let A be a set of real numbers and let B = {x+ r : x ∈ A} for some number r. Find a relation between supA and supB. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 1.6. Sups and Infs 15 1.6.9 Let A be a set of real numbers and let B = {xr : x ∈ A} for some positive number r. Find a relation between supA and supB. (What happens if r is negative?) 1.6.10 Let A and B be sets of real numbers such that A ⊂ B. Find a relation among inf A, inf B, supA, and supB. 1.6.11 Let A and B be sets of real numbers and write C = A ∪ B. Find a relation among supA, supB, and supC. 1.6.12 Let A and B be sets of real numbers and write C = A ∩ B. Find a relation among supA, supB, and supC. 1.6.13 Let A and B be sets of real numbers and write C = {x + y : x ∈ A, y ∈ B}. Find a relation among supA, supB, and supC. 1.6.14 Let A and B be sets of real numbers and write C = {x + y : x ∈ A, y ∈ B}. Find a relation among inf A, inf B, and inf C. 1.6.15 Let A be a set of real numbers and write A2 = {x2 : x ∈ A}. Are there any relations you can find between the infs and sups of the two sets? 1.6.16 Let E be a set of real numbers. Show that x is not an upper bound of E if and only if there exists a number e ∈ E such that e > x. 1.6.17 Let A be a set of real numbers. Show that a real number x is the supremum of A if and only if a ≤ x for all a ∈ A and for every positive number ε there is an element a′ ∈ A such that x − ε < a′. 1.6.18 Formulate a condition analogous to the preceding exercise for an infimum. 1.6.19 Using the completeness axiom, show that every nonempty set E of real numbers that is bounded below has a greatest lower bound (i.e., inf E exists and is a real number). 1.6.20 A function is said to be bounded if its range is a bounded set. Give examples of functions f : R → R that are bounded and examples of such functions that are unbounded. Give an example of one that has the property that sup{f(x) : x ∈ R} is finite but max{f(x) : x ∈ R} does not exist. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 18 Properties of the Real Numbers Chapter 1 1.7.2 Suppose that it is true that for each x > 0 there is an n ∈ IN so that 1/n < x. Prove the archimedean theorem using this assumption. 1.7.3 Without using the archimedean theorem, show that for each x > 0 there is an n ∈ IN so that 1/n < x. See Note 5 1.7.4 Let x be any real number. Show that there is a m ∈ Z so that m ≤ x < m + 1. Show that m is unique. 1.7.5 The mathematician Leibniz based his calculus on the assumption that there were “infinitesimals,” positive real numbers that are extremely small—smaller than all positive rational numbers certainly. Some calculus students also believe, apparently, in the existence of such numbers since they can imagine a number that is “just next to zero.” Is there a positive real number smaller than all positive rational numbers? 1.7.6 The archimedean property asserts that if x > 0, then there is a natural number N so that 1/N < x. The proof requires the completeness axiom. Give a proof that does not use the completeness axiom that works for x rational. Find a proof that is valid for x = √ y, where y is rational. 1.7.7 In Section 1.2 we made much of the fact that there is a number whose square is 2 and so √ 2 does exist as a real number. Show that α = sup{x ∈ R : x2 < 2} exists as a real number and that α2 = 2. See Note 6 1.8 Inductive Property of IN Since the natural numbers are included in the set of real numbers there are further important properties of IN that can be deduced from the axioms. The most important of these is the principle of induction. This is the basis for the technique of proof known as induction, which is often used in this text. For an elementary account and some practice, see Section A.8 in the appendix. We first prove a statement that is equivalent. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 1.8. Inductive Property of IN 19 Theorem 1.12 (Well-Ordering Property) Every nonempty subset of IN has a smallest element. Proof. Let S ⊂ IN and S 6= ∅. Then α = inf S must exist and be a real number since S is bounded below. If α ∈ S, then we are done since we have found a minimal element. Suppose not. Then, while α is the greatest lower bound of S, α is not a minimum. There must be an element of S that is smaller than α + 1 since α is the greatest lower bound of S. That element cannot be α since we have assumed that α 6∈ S. Thus we have found x ∈ S with α < x < α + 1. Now x is not a lower bound of S, since it is greater than the greatest lower bound of S, so there must be yet another element y of S such that α < y < x < α + 1. But now we have reached an impossibility, for x and y are in S and both natural numbers, but 0 < x−y < 1, which cannot happen. From this contradiction the proof now follows.  Now we can state and prove the principle of induction. Theorem 1.13 (Principle of Induction) Let S ⊂ IN so that 1 ∈ S and, for every natural number n, if n ∈ S then so also is n + 1. Then S = IN. Proof. Let E = IN \ S. We claim that E = ∅ and then it follows that S = IN proving the theorem. Suppose not (i.e., suppose E 6= ∅). By Theorem 1.12 there is a first element α of E. Can α = 1? No, because 1 ∈ S by hypothesis. Thus α − 1 is also a natural number and, since it cannot be in E it must be in S. By hypothesis it follows that α = (α − 1) + 1 must be in S. But it is in E. This is impossible and so we have obtained a contradiction, proving our theorem.  Exercises 1.8.1 Show that any bounded, nonempty set of natural numbers has a maximal element. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 20 Properties of the Real Numbers Chapter 1 1.8.2 Show that any bounded, nonempty subset of Z has a maximum and a minimum. 1.8.3 For further exercises on proving statements using induction as a method, see Section A.8. 1.9 The Rational Numbers Are Dense There is an important relationship holding between the set of rational numbers Q and the larger set of real numbers R. The rational numbers are dense. They make an appearance in every interval; there are no gaps, no intervals that miss having rational numbers. For practical purposes this has great consequences. We need never actually compute with arbitrary real numbers, since close by are rational numbers that can be used. Thus, while π is irrational, in routine computations with a practical view any nearby fraction might do. At various times people have used 3, 22/7, and 3.14159, for example. For theoretical reasons this fact is of great importance too. It allows many arguments to replace a consideration of the set of real numbers with the smaller set of rationals. Since every real is as close as we please to a rational and since the rationals can be carefully described and easily worked with, many simplifications are allowed. Definition 1.14: (Dense Sets) A set E of real numbers is said to be dense (or dense in R) if every interval (a, b) contains a point of E. Theorem 1.15: The set Q of rational numbers is dense. Proof. Let x < y and consider the interval (x, y). We must find a rational number inside this interval. By the archimedean theorem, Theorem 1.11, there is a natural number n > 1 y − x. This means that ny > nx + 1. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 1.10. The Metric Structure of R 23 inequalities. These properties are used routinely and the student will need to have a complete mastery of them. Theorem 1.17: The absolute value function has the following properties: 1. For any x ∈ R, −|x| ≤ x ≤ |x|. 2. For any x, y ∈ R, |xy| = |x| |y|. 3. For any x, y ∈ R, |x + y| ≤ |x| + |y|. 4. For any x, y ∈ R, |x| − |y| ≤ |x − y| and |y| − |x| ≤ |x − y|. Distances on the Real Line Using the absolute value function we can define the distance function or metric. Definition 1.18: (Distance) The distance between two real numbers x and y is d(x, y) = |x − y|. We hardly ever use the notation d(x, y) in elementary analysis, preferring to write |x− y| even while we are thinking of this as the distance between the two points. Thus if a sequence of points x1, x2, x3, . . . is growing ever closer to a point c, we should perhaps describe d(xn, c) as getting smaller and smaller, thus emphasizing that the distances are shrinking; more often we would simply write |xn − c| and expect you to interpret this as a distance. Properties of the Distance Function The main properties of the distance function are just interpretations of the absolute value function. Expressed in the language of a distance function, they are geometrically very intuitive: 1. d(x, y) ≥ 0 (all distances are positive or zero). ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 24 Properties of the Real Numbers Chapter 1 2. d(x, y) = 0 if and only if x = y (different points are at positive distance apart). 3. d(x, y) = d(y, x) (distance is symmetric, that is the distance from x to y is the same as from y to x)). 4. d(x, y) ≤ d(x, z) + d(z, y) (the triangle inequality, that is it is no longer to go directly from x to y than to go from x to z and then to y). In Chapter ?? we will study general structures called metric spaces, where exactly such a notion of distance satisfying these four properties is used. For now we prefer to rewrite these properties in the language of the absolute value, where they lose some of their intuitive appeal. But it is in this form that we are likely to use them. 1. |a| ≥ 0. 2. |a| = 0 if and only if a = 0. 3. |a| = | − a|. 4. |a + b| ≤ |a| + |b| (the triangle inequality). Exercises 1.10.1 Show that |x| = max{x,−x}. 1.10.2 Show that max{x, y} = |x − y|/2 + (x + y)/2. What expression would give min{x, y}? 1.10.3 Show that the inequalities |x − a| < ε and a − ε < x < a + ε are equivalent. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 1.11. Challenging Problems for Chapter 1 25 1.10.4 Show that if α < x < β and α < y < β, then |x − y| < β − α and interpret this geometrically as a statement about the interval (α, β). 1.10.5 Show that ||x| − |y|| ≤ |x − y| assuming the triangle inequality (i.e., that |a + b| ≤ |a| + |b|). This inequality is also called the triangle inequality. 1.10.6 Under what conditions is it true that |x + y| = |x| + |y|? 1.10.7 Under what conditions is it true that |x − y| + |y − z| = |x − z|? 1.10.8 Show that |x1 + x2 + · · · + xn| ≤ |x1| + |x2| + · · · + |xn| for any numbers x1, x2, . . . , xn. 1.10.9 Let E be a set of real numbers and let A = {|x| : x ∈ E}. What relations can you find between the infs and sups of the two sets? 1.10.10 Find the inf and sup of the set {x : |2x + π| < √ 2}. 1.11 Challenging Problems for Chapter 1 1.11.1 The complex numbers C are defined as equal to the set of all ordered pairs of real numbers subject to these operations: (a1, b1) + (a2, b2) = (a1 + a2, b1 + b2) and (a1, b1) · (a2, b2) = (a1a2 − b1b2, a1b2 + a2b1). (a) Show that C is a field. (b) What are the additive and multiplicative identity elements? (c) What are the additive and multiplicative inverses of an element (a, b)? (d) Solve (a, b)2 = (1, 0) in C. (e) We identify R with a subset of C by identifying the elements x ∈ R with the element (x, 0) in C. Explain how this can be interpreted as saying that “R is a subfield of C.” ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 28 NOTES Carry on. What have you proved? Now what if a = b? 4Exercise 1.6.4. You can use induction on the size of E, that is, prove for every natural number n that if E has n elements, then sup E = max E. 5Exercise 1.7.3. Suppose not, then the set {1/n : n = 1, 2, 3, . . . } has a positive lower bound, etc. You will have to use the existence of a greatest lower bound. 6Exercise 1.7.7. Not that easy to show. Rule out the possibilities α2 < 2 and α2 > 2 using the archimedean property to assist. 7Exercise 1.9.8. To find a number in (x, y), find a rational in (x// √ 2, y// √ 2). Conclude from this that the set of all (irrational) numbers of the form ±m √ 2/n is dense. 8Exercise 1.11.6. If G = {0}, then take α = 0. If not, let α = inf G ∩ (0,∞). Case 1: If α = 0 show that G is dense. Case 2: If α > 0 show that G = {nα : n = 0,±1,±2,±3, . . . }. For case 1 consider an interval (r, s) with r < s. We wish to find a member of G in that interval. To keep the argument simple just consider, for the moment, the situation in which 0 < r < s. Choose g ∈ G with 0 < g < s − r. The set M = {n ∈ IN : ng ≥ s} is nonempty (why?) and so there is a minimal element m in M (why?). Now check that (m − 1)g is in G and inside the interval (r, s). ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Chapter 2 SEQUENCES 2.1 Introduction Let us start our discussion with a method for solving equations that originated with Newton in 1669. To solve an equation f(x) = 0 the method proposes the introduction of a new function F (x) = x − f(x) f ′(x) . We begin with a guess at a solution of f(x) = 0, say x1 and compute x2 = F (x1) in the hopes that x2 is closer to a solution than x1 was. The process is repeated so that x3 = F (x2), x4 = F (x3), x5 = F (x4), . . . and so on until the desired accuracy is reached. Processes of this type have been known for at least 3500 years although not in such a modern notation. We illustrate by finding an approximate value for √ 2 this way. We solve the equation f(x) = x2 − 2 = 0 by computing the function F (x) = x − f(x) f ′(x) = x − x 2 − 2 2x and using it to improve our guess. A first (very crude) guess of x1 = 1 will produce the following list of values for our subsequent steps in the procedure. We have retained 60 digits in the decimal expansions to 29 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 30 Sequences Chapter 2 show how this is working: x1 = 1.00000000000000000000000000000000000000000000000000000000000 x2 = 1.50000000000000000000000000000000000000000000000000000000000 x3 = 1.41666666666666666666666666666666666666666666666666666666667 x4 = 1.41421568627450980392156862745098039215686274509803921568628 x5 = 1.41421356237468991062629557889013491011655962211574404458490 x6 = 1.41421356237309504880168962350253024361498192577619742849829 x7 = 1.41421356237309504880168872420969807856967187537723400156101. To compare, here is the value of the true solution √ 2, computed in a different fashion to the same number of digits: √ 2 = 1.41421356237309504880168872420969807856967187537694807317668. Note that after only four steps the procedure gives a value differing from the true value only in the sixth decimal place, and all subsequent values remain this close. A convenient way of expressing this is to write that |xn − √ 2| < 10−5 for all n ≥ 4. By the seventh step, things are going even better and we can claim that |xn − √ 2| < 10−47 for all n ≥ 7. It is inconceivable that anyone would require any further accuracy for any practical considerations. The error after the sixth step cannot exceed 10−47, which is a tiny number. Even so, as mathematicians we can ask what may seem an entirely impractical sort of question. Can this accuracy of approximation continue forever? Is it possible that, if we wait long enough, we can find an approximation to √ 2 with any degree of accuracy? ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.2. Sequences 33 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 x 9 x 10 Figure 2.1. An arithmetic progression. 2.2.1 Sequence Examples In order to specify some sequence we need to communicate what every term in the sequence is. For example, the sequence of even integers 2, 4, 6, 8, 10, . . . could be communicated in precisely that way: “Consider the sequence of even integers.” Perhaps more direct would be to give a formula for all of the terms in the sequence: “Consider the sequence whose nth term is xn = 2n.” Or we could note that the sequence starts with 2 and then all the rest of the terms are obtained by adding 2 to the previous term: “Consider the sequence whose first term is 2 and whose nth term is 2 added to the (n − 1)st term,” that is, xn = 2 + xn−1. Often an explicit formula is best. Frequently though, a formula relating the nth term to some preceding term is preferable. Such formulas are called recursion formulas and would usually be more efficient if a computer is used to generate the terms. Arithmetic Progressions The simplest types of sequences are those in which each term is obtained from the preceding by adding a fixed amount. These are called arithmetic progressions. The sequence c, c + d, c + 2d, c + 3d, c + 4d, . . . , c + (n − 1)d, . . . is the most general arithmetic progression. The number d is called the common difference. Every arithmetic progression could be given by a formula xn = c + (n − 1)d ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 34 Sequences Chapter 2 or a recursion formula x1 = c xn = xn−1 + d. Note that the explicit formula is of the form xn = f(n), where f is a linear function, f(x) = dx + b for some b. Figure 2.1 shows the points of an arithmetic progression plotted on the line. If, instead, you plot the points (n, xn) you will find that they all lie on a straight line with slope d. Geometric Progressions. A variant on the arithmetic progression is obtained by replacing the addition of a fixed amount by the multiplication by a fixed amount. These sequences are called geometric progressions. The sequence c, cr, cr2, cr3, cr4, . . . , crn−1, . . . is the most general geometric progression. The number r is called the common ratio. Every geometric progression could be given by a formula xn = cr n−1 or a recursion formula x1 = c xn = rxn−1. Note that the explicit formula is of the form xn = f(n), where f is an exponential function f(x) = br x for some b. Figure 2.2 shows the points of a geometric progression plotted on the line. Alternatively, plot the points (n, xn) and you will find that they all lie on the graph of an exponential function. If c > 0 and the common ratio r is larger than 1, the terms increase in size, becoming extremely large. If 0 < r < 1, the terms decrease in size, getting smaller and smaller. (See Figure 2.2.) Iteration The examples of an arithmetic progression and a geometric progression are special cases of a process called iteration. So too is the sequence generated by Newton’s method in the introduction to this chapter. Let f be some function. Start the sequence x1, x2, x3, . . . by assigning some value in the domain of f , say x1 = c. All subsequent values are now obtained by feeding these values through the function repeatedly: c, f(c), f(f(c)), f(f(f(c))), f(f(f(f(c)))), . . . . ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.2. Sequences 35 x 1 x 2 x 3 x 4 x 5 x 6 x 7 x 8 Figure 2.2. A geometric progression. As long as all these values remain in the domain of the function f , the process can continue indefinitely and defines a sequence. If f is a function of the form f(x) = x + b, then the result is an arithmetic progression. If f is a function of the form f(x) = ax, then the result is a geometric progression. A recursion formula best expresses this process and would offer the best way of writing a computer program to compute the sequence: x1 = c xn = f(xn−1). Sequence of Partial Sums. If a sequence x1, x2, x3, x4, . . . is given, we can construct a new sequence by adding the terms of the old one: s1 = x1 s2 = x1 + x2 s3 = x1 + x2 + x3 s4 = x1 + x2 + x3 + x4 and continuing in this way. The process can also be described by a recursion formula: s1 = x1 , sn = sn−1 + xn. The new sequence is called the sequence of partial sums of the old sequence {xn}. We shall study such sequences in considerable depth in the next chapter. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 38 Sequences Chapter 2 The arrangement of a collection of objects into a list is sometimes called an enumeration. Thus another way of phrasing this question is to ask what sets of real numbers can be enumerated? The set of natural numbers is already arranged into a list in its natural order. The set of integers (including 0 and the negative integers) is not usually presented in the form of a list but can easily be so presented, as the following scheme suggests: 0, 1,−1, 2,−2, 3,−3, 4,−4, 5,−5, 6,−6, 7,−7, . . . . Example 2.3: The rational numbers can also be listed but this is quite remarkable, for at first sight no reasonable way of ordering them into a sequence seems likely to be possible. The usual order of the rationals in the reals is of little help. To find such a scheme define the “rank” of a rational number m/n in its lowest terms (with n ≥ 1) to be |m|+ n. Now begin making a finite list of all the rational numbers at each rank; list these from smallest to largest. For example, at rank 1 we would have only the rational number 0/1. At rank 2 we would have only the rational numbers −1/1, 1/1. At rank 3 we would have only the rational numbers −2/1, −1/2, 1/2, 2/1. Carry on in this fashion through all the ranks. Now construct the final list by concatenating these shorter lists in order of the ranks: 0/1,−1/1, 1/1,−2/1,−1/2, 1/2, 2/1, . . . . The range of this sequence is the set of all rational numbers. ◭ Your first impression might be that few sets would be able to be the range of a sequence. But having seen in Example 2.3 that even the set of rational numbers Q that is seemingly so large can be listed, it might then appear that all sets can be so listed. After all, can you conceive of a set that is “larger” than the rationals in some way that would stop it being listed? The remarkable fact that there are sets that cannot be arranged to form the elements of some sequence was proved by Georg Cantor (1845–1918). This proof is essentially his original proof. (Note that this requires some familiarity with infinite decimal expansions; the exercises review what is needed.) Theorem 2.4 (Cantor) No interval (a, b) of real numbers can be the range of some sequence. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.3. Countable Sets 39 Proof. It is enough to prove this for the interval (0, 1) since there is nothing special about it (see Exercise 2.3.1). The proof is a proof by contradiction. We suppose that the theorem is false and that there is a sequence {sn} so that every number in the interval (0, 1) appears at least once in the sequence. We obtain a contradiction by showing that this cannot be so. We shall use the sequence {sn} to find a number c in the interval (0, 1) so that sn 6= c for all n. Each of the points s1, s2, s3 . . . in our sequence is a number between 0 and 1 and so can be written as a decimal fraction. If we write this sequence out in decimal notation it might look like s1 = 0.x11x12x13x14x15x16 . . . s2 = 0.x21x22x23x24x25x26 . . . s3 = 0.x31x32x33x34x35x36 . . . etc. Now it is easy to find a number that is not in the list. Construct c = 0.c1c2c3c4c5c6 . . . by choosing ci to be either 5 or 6 whichever is different from xii. This number cannot be equal to any of the listed numbers s1, s2, s3 . . . since c and si differ in the ith position of their decimal expansions. This gives us our contradiction and so proves the theorem.  Definition 2.5: (Countable) A nonempty set S of real numbers is said to be countable if there is a sequence of real numbers whose range is the set S. In the language of this definition then we can see that (1) any finite set is countable, (2) the natural numbers and the integers are countable, (3) the rational numbers are countable, and (4) no interval of real numbers is countable. By convention we also say that the empty set ∅ is countable. Exercises 2.3.1 Show that, once it is known that the interval (0, 1) cannot be expressed as the range of some sequence, it follows that any interval (a, b), [a, b), (a, b], or [a, b] has the same property. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 40 Sequences Chapter 2 See Note 12 2.3.2 Some novices, on reading the proof of Cantor’s theorem, say “Why can’t you just put the number c that you found at the front of the list.” What is your rejoinder? 2.3.3 A set (any set of objects) is said to be countable if it is either finite or there is an enumeration (list) of the set. Show that the following properties hold for arbitrary countable sets: (a) All subsets of countable sets are countable. (b) Any union of a pair of countable sets is countable. (c) All finite sets are countable. 2.3.4 Show that the following property holds for countable sets: If S1, S2, S3, . . . is a sequence of countable sets of real numbers, then the set S formed by taking all elements that belong to at least one of the sets Si is also a countable set. See Note 13 2.3.5 Show that if a nonempty set is contained in the range of some sequence of real numbers, then there is a sequence whose range is precisely that set. 2.3.6 In Cantor’s proof presented in this section we took for granted material about infinite decimal expansions. This is entirely justified by the theory of sequences studied later. Explain what it is that we need to prove about infinite decimal expansions to be sure that this proof is valid. See Note 14 2.3.7 Define a relation on the family of subsets of R as follows. Say that A ∼ B, where A and B are subsets of R, if there is a function f : A → B that is one-to-one and onto. (If A ∼ B we would say that A and B are “cardinally equivalent.”) Show that this is an equivalence relation, that is, show that (a) A ∼ A for any set A. (b) If A ∼ B then B ∼ A. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.4. Convergence 43 Note. In the definition the N depends on ε. If ε is particularly small, then N might have to be chosen large. In fact, then N is really a function of ε. Sometimes it is best to emphasize this and write N(ε) rather than N . Note, too, that if an N is found, then any larger N would also be able to be used. Thus the definition requires us to find some N but not necessarily the smallest N that would work. While the definition does not say this, the real force of the definition is that the N can be determined no matter how small a number ε is chosen. If ε is given as rather large there may be no trouble finding the N value. If you find an N that works for ε = .1 that same N would work for all larger values of ε. Example 2.7: Let us use the definition to prove that lim n→∞ n2 2n2 + 1 = 1 2 . It is by no means clear from the definition how to obtain that the limit is the number L = 12 . Indeed the definition is not intended as a method of finding limits. It assigns a precise meaning to the statement about the limit but offers no way of computing that limit. Fortunately most of us remember some calculus devices that can be used to first obtain the limit before attempting a proof of its validity. lim n→∞ n2 2n2 + 1 = lim n→∞ 1 2 + 1/n2 = 1 limn→∞(2 + 1/n2) = 1 2 + limn→∞(1/n2) = 1 2 . Indeed this would be a proof that the limit is 1/2 provided that we could prove the validity of each of these steps. Later on we will prove this and so can avoid the ε, N arguments that we now use. Let any positive ε be given. We need to find a number N [or N(ε) if you prefer] so that every term in the sequence on and after the Nth term is closer to 1/2 than ε, that is, so that ∣ ∣ ∣ ∣ n2 2n2 + 1 − 1 2 ∣ ∣ ∣ ∣ < ε ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 44 Sequences Chapter 2 for n = N , n = N + 1, n = N + 2, . . . . It is easiest to work backward and discover just how large n should be for this. A little work shows that this will happen if 1 2(2n2 + 1) < ε or 4n2 + 2 > 1 ε . The smallest n for which this statement is true could be our N . Thus we could use any integer N with N2 > 1 4 ( 1 ε − 2 ) . There is no obligation to find the smallest N that works and so, perhaps, the most convenient one here might be a bit larger, say take any integer N larger than N > 1 2 √ ε . ◭ The real lesson of the example, perhaps, is that we wish never to have to use the definition to check any limit computation. The definition offers a rigorous way to develop a theory of limits but an impractical method of computation of limits and a clumsy method of verification. Only rarely do we have to do a computation of this sort to verify a limit. Uniqueness of Sequence Limits Let us take the first step in developing a theory of limits. This is to ensure that our definition has defined limit unambiguously. Is it possible that the definition allows for a sequence to converge to two different limits? If we have established that sn → L is it possible that sn → L1 for a different number L1? ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.4. Convergence 45 Theorem 2.8 (Uniqueness of Limits) Suppose that lim n→∞ sn = L1 and lim n→∞ sn = L2 are both true. Then L1 = L2. Proof. Let ε be any positive number. Then, by definition, we must be able to find a number N1 so that |sn − L1| < ε whenever n ≥ N1. We must also be able to find a number N2 so that |sn − L2| < ε whenever n ≥ N2. Take m to be the maximum of N1 and N2. Then both assertions |sm − L1| < ε and |sm − L2| < ε are true. This allows us to conclude that |L1 − L2| ≤ |L1 − sm| + |sm − L2| < 2ε so that |L1 − L2| < 2ε. But ε can be any positive number whatsoever. This could only be true if L1 = L2, which is what we wished to show.  Exercises 2.4.1 Give a precise ε, N argument to prove that limn→∞ 1 n = 0. 2.4.2 Give a precise ε, N argument to prove the existence of lim n→∞ 2n + 3 3n + 4 . ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 48 Sequences Chapter 2 Note. The definition does not announce this, but the force of the definition is that the choice of N is possible no matter how large M is chosen. There may be no difficulty in finding an N if the M given is not big. Example 2.10: Let us prove that n2 + 1 n + 1 → ∞ using the definition. If M is any positive number we need to find some point in the sequence after which all terms exceed M . Thus we need to consider the inequality n2 + 1 n + 1 ≥ M. After some arithmetic we see that this is equivalent to n + 1 n + 1 − n n + 1 ≥ M. Since n n + 1 < 1 we see that, as long as n ≥ M + 1 this will be true. Thus take any integer N ≥ M + 1 and it will be true that n2 + 1 n + 1 ≥ M for all n ≥ N . (Any larger value of N would work too.) ◭ Exercises 2.5.1 Formulate the definition of a sequence diverging to −∞. 2.5.2 Show, using the definition, that limn→∞ n 2 = ∞. 2.5.3 Show, using the definition, that limn→∞ n3+1 n2+1 = ∞. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.6. Boundedness Properties of Limits 49 2.5.4 Prove that if sn → ∞ then −sn → −∞. 2.5.5 Prove that if sn → ∞ then (sn)2 → ∞ also. 2.5.6 Prove that if xn → ∞ then the sequence sn = xnxn+1 is convergent. Is the converse true? See Note 18 2.5.7 Suppose that a sequence {sn} of positive numbers satisfies limn→∞ sn = 0. Show that limn→∞ 1/sn = ∞. Is the converse true? 2.5.8 Suppose that a sequence {sn} of positive numbers satisfies the condition sn+1 > αsn for all n where α > 1. Show that sn → ∞. 2.5.9 The sequence sn = (−1)n does not diverge to ∞. For what values of M is it nonetheless true that there is an integer N so that sn > M whenever n ≥ N? 2.5.10 Show that the sequence np + α1n p−1 + α2n p−2 + · · · + αp diverges to ∞, where here p is a positive integer and α1, α2, . . . , αp are real numbers (positive or negative). 2.6 Boundedness Properties of Limits A sequence is said to be bounded if its range is a bounded set. Thus a sequence {sn} is bounded if there is a number M so that every term in the sequence satisfies |sn| ≤ M. For such a sequence, every term belongs to the interval [−M, M ]. It is fairly evident that a sequence that is not bounded could not converge. This is important enough to state and prove as a theorem. Theorem 2.11: Every convergent sequence is bounded. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 50 Sequences Chapter 2 Proof. Suppose that sn → L. Then for every number ε > 0 there is an integer N so that |sn − L| < ε whenever n ≥ N . In particular we could take just one value of ε, say ε = 1, and find a number N so that |sn − L| < 1 whenever n ≥ N . From this we see that |sn| = |sn − L + L| ≤ |sn − L| + |L| < |L| + 1 for all n ≥ N . This number |L| + 1 would be an upper bound for all the numbers |sn| except that we have no indication of the values for |s1|, |s2|, . . . , |sN−1|. Thus if we write M = max{|s1|, |s2|, . . . , |sN−1|, |L| + 1} we must have |sn| ≤ M for every value of n. This is an upper bound, proving the theorem.  As a consequence of this theorem we can conclude that an unbounded sequence must diverge. Thus, even though it is a rather crude test, we can prove the divergence of a sequence if we are able somehow to show that it is unbounded. The next example illustrates this technique. Example 2.12: We shall show that the sequence sn = 1 + 1 2 + 1 3 + 1 4 + · · · + 1 n diverges. The easiest proof of this is to show that it is unbounded and hence, by Theorem 2.11, could not converge. We watch only at the steps 1, 2, 4, 8, . . . and make a rough lower estimate of s1, s2, s4, s8, . . . in order to show that there can be no bound on the sequence. After a bit of arithmetic we see that s1 = 1 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.7. Algebra of Limits 53 = 1 2 + limn→∞ 1/n2 = 1 2 could be justified. Note how this sequence has been obtained from simpler ones by ordinary processes of arithmetic. To justify such a method we need to investigate how the limit operation is influenced by algebraic operations. Suppose that sn → S and tn → T. Then we would expect Csn → CS sn + tn → S + T sn − tn → S − T sntn → ST and sn/tn → S/T. Each of these statements must be justified, however, solely on the basis of the definition of convergence, not on intuitive feelings that this should be the case. Thus we need to develop what could be called the “algebra of limits.” Theorem 2.14 (Multiples of Limits) Suppose that {sn} is a convergent sequence and C a real number. Then lim n→∞ Csn = C ( lim n→∞ sn ) . Proof. Let S = limn→∞ sn. In order to prove that limn→∞ Csn = CS we need to prove that, no matter what positive number ε is given, we can find an integer N so that, for all n ≥ N , |Csn − CS| < ε. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 54 Sequences Chapter 2 Note that |Csn − CS| = |C| |sn − S| by properties of absolute values. This gives us our clue. Suppose first that C 6= 0 and let ε > 0. Choose N so that |sn − S| < ε/|C| if n ≥ N . Then if n ≥ N we must have |Csn − CS| = |C| |sn − S| < |C| (ε/|C|) = ε. This is precisely the statement that lim n→∞ Csn = CS and the theorem is proved in the case C 6= 0. The case C = 0 is obvious. (Now we should probably delete our first paragraph since it does not contribute to the proof; it only serves to motivate us in finding the correct proof.)  Theorem 2.15 (Sums/Differences of Limits) Suppose that the sequences {sn} and {tn} are conver- gent. Then lim n→∞ (sn + tn) = lim n→∞ sn + lim n→∞ tn and lim n→∞ (sn − tn) = lim n→∞ sn − lim n→∞ tn. Proof. Let S = limn→∞ sn and T = limn→∞ tn. In order to prove that lim n→∞ (sn + tn) = S + T we need to prove that no matter what positive number ε is given we can find an integer N so that |(sn + tn) − (S + T )| < ε ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.7. Algebra of Limits 55 if n ≥ N . Note that |(sn + tn) − (S + T )| ≤ |sn − S| + |tn − T | by the triangle inequality. Thus we can make this expression smaller than ε by making each of the two expressions on the right smaller than ε/2. This provides the method. Suppose that ε > 0. Choose N1 so that |sn − S| < ε/2 if n ≥ N1 and also choose N2 so that |tn − T | < ε/2 if n ≥ N2. Then if n is greater than both N1 and N2 both of these inequalities will be true. Set N = max{N1, N2} and note that if n ≥ N we must have |(sn + tn) − (S + T )| ≤ |sn − S| + |tn − T | < ε/2 + ε/2 = ε. This is precisely the statement that lim n→∞ (sn + tn) = S + T and the first statement of the theorem is proved. The second statement is similar and is left as an exercise. (Once again, for a more formal presentation, we would delete the first paragraph.)  Theorem 2.16 (Products of Limits) Suppose that {sn} and {tn} are convergent sequences. Then lim n→∞ (sntn) = ( lim n→∞ sn )( lim n→∞ tn ) . Proof. Let S = limn→∞ sn and T = limn→∞ tn. In order to prove that limn→∞(sntn) = ST we need to prove that no matter what positive number ε is given we can find an integer N so that, for all n ≥ N , |sntn − ST | < ε. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 58 Sequences Chapter 2 if n ≥ N2. From the first inequality we see that |T | − |tn| ≤ |T − tn| < |T |/2 and so |tn| ≥ |T |/2 if n ≥ N1. Set N = max{N1, N2} and note that if n ≥ N we must have ∣ ∣ ∣ ∣ 1 tn − 1 T ∣ ∣ ∣ ∣ = |tn − T | |tn| |T | < ε|T |2/2 |T |2/2 = ε. This is precisely the statement that limn→∞ (1/tn) = 1/T . We now complete the proof of the theorem by applying the product theorem along with what we have just proved to obtain lim n→∞ ( sn tn ) = ( lim n→∞ sn ) ( lim n→∞ 1 tn ) = limn→∞ sn limn→∞ tn as required.  Exercises 2.7.1 By imitating the proof given for the first part of Theorem 2.15 show that lim n→∞ (sn − tn) = lim n→∞ sn − lim n→∞ tn. 2.7.2 Show that limn→∞ (sn) 2 = (limn→∞ sn) 2 using the theorem on products and also directly from the definition of limit. 2.7.3 Explain which theorems are needed to justify the computation of the limit lim n→∞ n2 2n2 + 1 that introduced this section. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.7. Algebra of Limits 59 2.7.4 Prove Theorem 2.16 but verifying and using the inequality |sntn − ST | ≤ |(sn − S)(tn − T )| + |S(tn − T )| + |T (sn − S)| in place of the inequality (1). Which proof do you prefer? 2.7.5 Which statements are true? (a) If {sn} and {tn} are both divergent then so is {sn + tn}. (b) If {sn} and {tn} are both divergent then so is {sntn}. (c) If {sn} and {sn + tn} are both convergent then so is {tn}. (d) If {sn} and {sntn} are both convergent then so is {tn}. (e) If {sn} is convergent so too is {1/sn}. (f) If {sn} is convergent so too is {(sn)2}. (g) If {(sn)2} is convergent so too is {sn}. 2.7.6 Note that there are extra hypotheses in the quotient theorem (Theorem 2.17) that were not in the product theorem (Theorem 2.16). Explain why both of these hypotheses are needed. 2.7.7 A careless student gives the following as a proof of Theorem 2.16. Find the flaw: “Suppose that ε > 0. Choose N1 so that |sn − S| < ε 2|T | + 1 if n ≥ N1 and also choose N2 so that |tn − T | < ε 2|sn| + 1 if n ≥ N2. If n ≥ N = max{N1, N2} then |sntn − ST | ≤ |sn| |tn − T | + |T | |sn − S| ≤ |sn| ( ε 2|sn| + 1 ) + |T | ( ε 2|T | + 1 ) < ε. Well, that works!” ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 60 Sequences Chapter 2 2.7.8 Why are Theorems 2.15 and 2.16 no help in dealing with the limits lim n→∞ (√ n + 1 −√n ) and lim n→∞ √ n (√ n + 1 −√n ) ? What else can you do? 2.7.9 In calculus courses one learns that a function f : R → R is continuous at y if for every ε > 0 there is a δ > 0 so that |f(x) − f(y)| < ε for all |x − y| < δ. Show that if f is continuous at y and sn → y then f(sn) → f(y). Use this to prove that limn→∞(sn) 2 = (limn→∞ sn) 2. 2.8 Order Properties of Limits In the preceding section we discussed the algebraic structure of limits. It is a natural mathematical question to ask how the algebraic operations are preserved under limits. As it happens, these natural mathematical questions usually are important in applications. We have seen that the algebraic properties of limits can be used to great advantage in computations of limits. There is another aspect of structure of the real number system that plays an equally important role as the algebraic structure and that is the order structure. Does the limit operation preserve that order structure the same way that it preserves the algebraic structure? For example, if sn ≤ tn for all n, can we conclude that lim n→∞ sn ≤ lim n→∞ tn? In this section we solve this problem and several others related to the order structure. These results, too, will prove to be most useful in handling limits. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.8. Order Properties of Limits 63 Proof. Let L be the limit of the two sequences. Choose N1 so that |sn − L| < ε if n ≥ N1 and also choose N2 so that |tn − L| < ε if n ≥ N2. Set N = max{N1, N2}. Note that sn − L ≤ xn − L ≤ tn − L for all n and so −ε < sn − L ≤ xn − L ≤ tn − L < ε if n ≥ N . From this we see that −ε < xn − L < ε or, to put it in a more familiar form, |xn − L| < ε proving the statement of the theorem.  Example 2.21: Let θ be some real number and consider the computation of lim n→∞ sinnθ n . While this might seem hopeless at first sight since the values of sinnθ are quite unpredictable, we recall that none of these values lies outside the interval [−1, 1]. Hence − 1 n ≤ sinnθ n ≤ 1 n . The two outer sequences converge to the same value 0 and so the inside sequence (the “squeezed” one) must converge to 0 as well. ◭ ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 64 Sequences Chapter 2 Absolute Values A further theorem on the theme of order structure is often needed. The absolute value, we recall, is defined directly in terms of the order structure. Is absolute value preserved by the limit operation? Theorem 2.22 (Limits of Absolute Values) Suppose that {sn} is a convergent sequence. Then the sequence {|sn|} is also a convergent sequence and lim n→∞ |sn| = ∣ ∣ ∣ lim n→∞ sn ∣ ∣ ∣ . Proof. Let S = limn→∞ sn and suppose that ε > 0. Choose N so that |sn − S| < ε if n ≥ N . Observe that, because of the triangle inequality, this means that ||sn| − |S|| ≤ |sn − S| < ε for all n ≥ N . By definition lim n→∞ |sn| = |S| as required.  Maxima and Minima Since maxima and minima can be expressed in terms of absolute values, there is a corollary that is sometimes useful. Corollary 2.23 (Max/Min of Limits) Suppose that {sn} and {tn} are convergent sequences. Then the sequences {max{sn, tn}} and {min{sn, tn}} are also convergent and lim n→∞ max{sn, tn} = max{ lim n→∞ sn, lim n→∞ tn} and lim n→∞ min{sn, tn} = min{ lim n→∞ sn, lim n→∞ tn}. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.8. Order Properties of Limits 65 Proof. The first of these follows from the identity max{sn, tn} = sn + tn 2 + |sn − tn| 2 and the theorem on limits of sums and the theorem on limits of absolute values. In the same way the second assertion follows from min{sn, tn} = sn + tn 2 − |sn − tn| 2 .  Exercises 2.8.1 Show that the condition sn < tn does not imply that lim n→∞ sn < lim n→∞ tn. (If the proof of Theorem 2.18 were modified in an attempt to prove this false statement, where would the modifications fail?) See Note 20 2.8.2 If {sn} is a sequence all of whose values lie inside an interval [a, b] prove that {sn/n} is convergent. 2.8.3 A careless student gives the following as a proof of the squeeze theorem. Find the flaw: “If limn→∞ sn = limn→∞ tn = L, then take limits in the inequality sn ≤ xn ≤ tn to get L ≤ limn→∞ xn ≤ L. This can only be true if limn→∞ xn = L.” 2.8.4 Suppose that sn ≤ tn for all n and that sn → ∞. What can you conclude? 2.8.5 Suppose that limn→∞ sn n > 0 Show that sn → ∞. 2.8.6 Suppose that {sn} and {tn} are sequences of positive numbers, that lim n→∞ sn tn = α and that sn → ∞. What can you conclude? ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 68 Sequences Chapter 2 Theorem 2.28 (Monotone Convergence Theorem) Suppose that {sn} is a monotonic sequence. Then {sn} is convergent if and only if {sn} is bounded. More specifically, 1. If {sn} is nondecreasing then either {sn} is bounded and converges to sup{sn} or else {sn} is unbounded and sn → ∞. 2. If {sn} is nonincreasing then either {sn} is bounded and converges to inf{sn} or else {sn} is unbounded and sn → −∞. Proof. If the sequence is unbounded then it diverges. This is true for any sequence, not merely monotonic sequences. Thus the proof is complete if we can show that for any bounded monotonic sequence {sn} the limit is sup{sn} in case the sequence is nondecreasing, or it is inf{sn} in case the sequence is nonincreasing. Let us prove the first of these cases. Let {sn} be assumed to be nondecreasing and bounded, and let L = sup{sn}. Then sn ≤ L for all n and if β < L there must be some term sm say, with sm > β. Let ε > 0. We know that there is an m so that sn ≥ sm > L − ε for all n ≥ m. But we already know that every term sn ≤ L. Putting these together we have that L − ε < sn ≤ L < L + ε or |sn − L| < ε for all n ≥ m. By definition then sn → L as required.  How would we normally apply this theorem? Suppose a sequence {sn} were given that we recognize as increasing (or maybe just nondecreasing). Then to establish that {sn} converges we need only show that ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner Section 2.9. Monotone Convergence Criterion 69 the sequence is bounded above, that is, we need to find just one number M with sn ≤ M for all n. Any crude upper estimate would verify convergence. Example 2.29: Let us show that the sequence sn = 1/ √ n converges. This sequence is evidently decreasing. Can we find a lower bound? Yes, all of the terms are positive so that 0 is a lower bound. Consequently, the sequence must converge. If we wish to show that lim n→∞ 1√ n = 0 we need to do more. But to conclude convergence we needed only to make a crude estimate on how low the terms might go. ◭ Example 2.30: Let us examine the sequence sn = 1 + 1 2 + 1 3 + 1 4 + · · · + 1 n . This sequence is evidently increasing. Can we find an upper bound? If we can then the series does converge. If we cannot then the series diverges. We have already (earlier) checked this sequence. It is unbounded and so limn→∞ sn = ∞. ◭ Example 2.31: Let us examine the sequence √ 2, √ 2 + √ 2 , √ 2 + √ 2 + √ 2 , √ 2 + √ 2 + √ 2 + √ 2, . . . . Handling such a sequence directly by the limit definition seems quite impossible. This sequence can be defined recursively by x1 = √ 2 xn = √ 2 + xn−1. ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner 70 Sequences Chapter 2 The computation of a few terms suggests that the sequence is increasing and so should be accessible by the methods of this section. We prove this by induction. That x1 < x2 is just an easy computation (do it). Let us suppose that xn−1 < xn for some n and show that it must follow that xn < xn+1. But xn = √ 2 + xn−1 < √ 2 + xn = xn+1 where the middle step is the induction hypothesis (i.e., that xn−1 < xn). It follows by induction that the sequence is increasing. Now we show inductively that the sequence is bounded above. Any crude upper bound will suffice. It is clear that x1 < 10. If xn−1 < 10 then xn = √ 2 + xn−1 < √ 2 + 10 < 10 and so it follows, again by induction, that all terms of the sequence are smaller than 10. We conclude from the monotone convergence theorem that this sequence is convergent. But to what? (Certainly it does not converges to 10 since that estimate was extremely crude.) That is not so easy to sort out, it seems. But perhaps it is, since we know that the sequence converges to something, say L. In the equation (xn) 2 = 2 + xn−1, obtained by squaring the recursion formula given to us, we can take limits as n → ∞. Since xn → L so too does xn−1 → L and (xn)2 → L2. Hence L2 = 2 + L. The only possibilities for L in this quadratic equation are L = −1 and L = 2. We know the limit L exists and we know that it is either −1 or 2. We can clearly rule out −1 as all of the numbers in our sequence were positive. Hence xn → 2. ◭ Exercises 2.9.1 Define a sequence {sn} recursively by setting s1 = α and sn = (sn−1) 2 + β 2sn−1 ClassicalRealAnalysis.com [TBB-Dripped] Elementary Real Analysis - Dripped Version Thomson*Bruckner*Bruckner