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Main Real Functions First Edition

# Real Functions First Edition

Categories: Mathematics\\Analysis
Year: 1953
Publisher: Rinehart & Company
Language: english
Pages: 271
File: DJVU, 4.79 MB

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Year: 2009
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Real Functions by Casper Gqffman ASSOCIATE PROFESSOR OP MATHEMATICS, WAYNE UNIVERSITY

Thtrd Frtnung Aprtl 1958 Co&nght91953

In memory of Henry Blumberg

Preface This book is an outgrowth of my course in functions of a real variable given to first-year graduate students at the University of Oklahoma. It can be covered in~a year's course, in a class meeting three times a week. The material has been strongly influenced by the classical volumes of F. Hausdorff, H. Hahn, and С Caratheodory, by papers of various authors, and by personal contacts, especially those with Ή. Blumberg and L. W. Cohen. This book is written as a textbook. Accordingly, there are many exercises Some of them, the simpler exercises, are included to help fix certain concepts in the minds of the students. Other exercises require various degrees of insight and ingenuity. The exercises at the end of each chapter are numbered according to the section to which they refer. For example, Exercise 3 of Section 4 is numbered 4.3. I am grateful to Professor L. W. Cohen of Queens College, Professor C. J. Pipes of Southern Methodist University, and Dr. С V. Newsom, mathematics editor for Rinehart & Company, as well as to several members of the mathematics department of the University of Oklahoma, for many helpful suggestions. Finally, I should like to thank Mrs. Marjorie Coker, former secretary "for the mathematics department at the University of Oklahoma, for typing the work. GASPER GOFFMAN February, 1953 vfl

Contents 1 Sets and Operations 1 1. Sets 2. Subsets 3. The positive integers 4. The union of sets 5. The intersection of sets 6. Limits of sequences of sets 7. A connection with algebra 8. Equivalence relations 2 Equivalence of Sets. Cardinal Numbers 16 1. Equivalence between sets 2. Examples of equivalent sets 3. Denumerable sets 4. Nondenumerable sets 5. Cardinal number 6.' Arithmetic of cardinal numbers 7. The set 2A 3 The Real Numbers 28 1. The rational numbers 2, Equivalence classes of fundamental sequences of rational numbers 3. Sum of equivalence classes 4. Product of equivalence classes 5. Order between equivalence classes 6. Convergence and limit 7. Real numbers 8. The Dedekind method 4 Limit Theorems 48 1. Introduction 2. Least upper bound and greatest lower bound of a set 3. Nests of intervals 4. Metric space 5. Limit points of a set 6. Bolzano-Weierstrass Theorem in metric space 5 Simple Properties of Sets 64 1. Closed sets 2. Further properties of closed sets 3. Closed to

χ Contents sets in metric space 4 Perfect sets 5. Open sets 6. Structure of open sets 7. Open and closed sets relative to a set 6 The Cantor Ternary Set 78 7 Functions 81 1. Definition 2. Sets associated with a function 3. Continuous functions 4. Properties of continuous functions 5. Uniform continuity 6. Sets of points of discontinuity 7. Semicontinous functions 8. Interval functions 9. Special semicontinuous functions 10. Unbounded functions 8 Sequences of Functions 100 1. Convergence and uniform convergence 2. Sequences of continuous functions 3. Equicontinuity 4. Limits of sequences of continuous functions 5. Properties of uniform convergence 9 The Derivative of a Function 114 1. The Dini derivates 2. Simple properties of the derivative 3. A theorem of G. C. Young 4. Continuous functions whose derivatives do not exist 10 Order Types and Ordinals 122 1. Introduction 2. Special kinds of orders 3. Well-ordered sets 4. Comparability of well-ordered sets 5. Comparability of cardinals 6. The smallest nondenumerable ordinal number 7. Transfinite induction 8. Paradoxes 11 Borel Sets and Baire Functions 134 1. Introduction 2. Borel sets 3. Baire functions 4. Relation between Borel sets and Baire functions 12 Applications.of Well-Ordering 146 1. Introduction 2. Existence ot a nondenumerable set whose intersection with every nowhere dense set is denumerable 3. Existence of a function which is discontinuous on every nondenumerable set relative to the set 4. Existence of a sequence of functions which converges on a nondenumerable set but converges uniformly on none of its nondenumerable subsets 5. Decomposition of the plane into a denumerable number of curves 6. Ex-

Contents xi istence of a function/(x) such that/(* +j>) ** /(*) +/O0 for every χ and у but for which/(*) is not of the form ex 13 Measure and Measurable Sets 153 1. What is measure? 2. Definition of Lebesgue measure 3. Special measurable sets 4. The structure of a measurable set 5. Properties of measurable sets 6. Nonmeasurable sets 14 Metric Properties of Sets 170 1. Introduction 2. The Vitali Covering Theorem 3. Metric density 4. Converse of the Lebesgue Density Theorem 5. Derivative of a monotone function 15 Measurable Functions 182 1. Definition 2. Simple properties of measurable functions 3. Functions defined on measurable sets 4. Approximate uniform convergence 16 Approximation of Measurable Functions 189 1. Introduction 2. Approximate continuity 3. Approximation by continuous functions 4. Approximation by Baire "functions 5. Continuity of a measurable function 17 The Ubesgue Integral and the Riemann Integral 197 1. Introduction 2. Definition of Lebesgue integral for bounded measurable simple functions 3. Unbounded measurable simple functions 4. Summability of arbitrary measurable functions 5. The Riemann integral 6. A condition for the existence of the Riemann integral 7. First defect of the Riemann integral 8. Second defectof the Riemann integral 18 The Ubesgue Integral as a Set Function 214 1. Introduction 2. Lebesgue integral on a measurable set 3. Integral of a sum 4. Absolute continuity 5. Complete additivity 6. Limits of convergent sequences of integrable functions (uniformly bounded case) 7: Limits of sequences of nonnegative integrable functions 8. Theorem of Lebesgue 19 The Fundamental Theorem of the Calculus 230 1. Introduction 2. The formula F'(x) «/(*) for bounded simple functions 3. The formula F'(x) */(*) for arbitrary

χίί Contents summable functions 4. Bounded variation and absolute continuity 5. / f'(i)di for nondecreasing functions 6. Г/'({У! for absolutely continuous functions 7. Definition of length 8. The formula f [(/'(0)2 + (ϊ(*))ψ* Jo 20 Planar Measure and Double Integration 249 1. Introduction 2. Definition of measure Э. Relation between planar measure and linear measure 4. Measurable and summable functions of two variables 5. Theorem of Fubini

1 Sets and Operations 1. Sets Mathematics deals with special kinds of sets. In arithmetic we are concerned with the set of positive integers, the set of even positive integers, the set of all integers, the set of rational numbers, etc. The elements of these sets are numbers. In elementary geometry we are concerned with the set of all triangles, the set of equilateral triangles, the set of all circles, the set of all polygons, etc. The elements of these sets are figures. In matters concerning functions of a real variable, interest is centered on certain kinds of sets of real numbers. The investigation of the properties of these sets depends upon a rigorous definition of the real numbers. This will be given in Chapter 3. Although this book will be concerned mainly with special sets, certain relations between sets which will be used consistently hold for general sets. It is the purpose of this chapter to present these relations. 2. Subsets We use the notation xuAto denote that χ is an element of the set A. This notation has become standard. Sometimes, instead of using the words "x is an element of 4," we shall say that χ is a member of A or that χ belongs to A. That χ does not belong to Л is denoted by χ 4 A. For example: 1. If A is the set of integers and χ « 2, then x*A. 2. If A is the set of integers and χ « 1/2, then χ 4 A. 3. If A is the set of isosceles triangles and χ is a triangle the lengths of whose sides are 3, 4, and 5, then χ 4 A. 4. If A is the set of right triangles and χ is the triangle of example 3, then ХйА.

2 Set $and Operations 5. If A is the set of polynomials and χ is the function /(0 ■» f3, then χ € A. 6. If A is the set of polynomials and χ is the functional) * sin *, then χ 4 A. DmNmoN i. Л is said to be а дебде/ of В if every member of A is a member o£B. А С В is the notation used for XtA te a subset of B." Evidently А С В if and only if for every χ € A it is true that χ € B. И А С В and В С -4, the sets Л and 2? are identical; i.e., they are composed of the same elements. DiraimoN 2. If А С В and it is not true that В С A then A is said to be a proper subset of B. Evidently, Л is a proper subset of В if and only if (1) for every χ € A it is true that χ € В and (2) there is an χ € В such that χ 4 A. The notation A D В is often used, as a convenience, to denote tfctfBisa subset of A. We give several examples of these concepts: 1. If A is the set of even positive integers and В is the set of all positive integers, then А С В, but it is not true that В С Л, so that A is a proper subset of B. 2. If Л is the set of even positive integers and В is the set of odd positive integers, then neither А С В nor В С A so that neither A nor В is a subset of the other. 3. If A is the set of positive integers and В is the set of even integers, then neither А С В nor В С Л so that neither A nor В is a subset of the other. 4. If A is the set of equilateral triangles and В is the set of isosceles triangles, then Л is a proper subset of B. 3· The positive integers Since we shall use the set of positive integers consistently in this book, it is appropriate that we introduce it now. The positive integers form a set Ρ which is an order; i.e., there is a relation "<" which holds between pairs of elements of В such that 1. ifacP, b*P, aoday* b, then either β <botb < a, 2. Ma < by then it is not true that * < д, 3. if a < b and b < c, then a < c. The statement "4 < b" is read "a is less than b*%Ua <b it is sometimes convenient to say b > a and to read this as "b is greater than a." The order Ρ of positive integers satisfies two additional conditions: The Union of Sets 3 1. For every S С Ρ, S has a first element; i.e., there is an element a € S such that b > a for every δ € 5. An immediate consequence of this condition is that Ρ itself has a first element which we denote by "1." Another consequence of (1) is that every element a e Ρ has an immediate successor: i.e., there is an element a'e Ρ such that a1 > a and a' is the first element greater than л. 2. If α € Ρ and a ^ 1, then a has an immediate predecessor; i.e., there is an element btPsuch that α » У. We can now prove that the so-called principle of finite induction holds for the system Ρ of positive integers. theorem ι. (Principle of Finite Induction.) If S С Ρ, 1 € S, and if for every a uSit follows that a' € S, then S «- P. Proof. Let 5 consist of all elements in Ρ which are not in S. Suppose that S is a proper subset of P. Then there are elements in 3. By condition 1, 5 has a first element a. Since, by hypothesis, \ eS, α ψ* \. Hence, by condition 2, a has an immediate predecessor b. Since α is the first element in 3 it follows that b € S. But У » d. Accordingly, by hypothesis, a =* b' *S. The assumption that S is a proper subset of Ρ has thus led to a contradiction, so S » P. We shall use the standard notation Г»2,2'»3,··· and shall write 1, 2, 3, ···,«, · · · for the set of all positive integers. definition Э. A set S of positive integers is called finite if there is a positive integer η such that η > a for every a e S. S will be called infinite if it is not finite; i.e., for every positive integer η there is an α € £ such that a > n. 4. The union of sets We now define the basic operations for sets. It will be convenient to illustrate these operations by means of intervals in the "real-number continuum.'' The intuitive idea of real numbers as being represented by the points on a line is adequate for our present purpose. A rigorous definition of the real numbers will be given in Chapter 3 and a full treatment of the operations on intervals will be given in Chapter 4. By an open interval (л, b\ where a < b, we mean all numbers χ such that a < χ < b; by a closed interval [α, δ] we mean all χ such that a £ χ й b. The notations [>, b) and (a, 6] designate the sets for which a £ χ < b and a < χ й b, respectively. As a first operation on sets we consider the union. The union of two sets A and В is the set whose elements are all the members of A together with all the members of B. The union of Л and В is denoted by A u JB. For example, if 4 Sets and Opera* tons A » CO, 2] and В = [1,3] then A и В = [0, 3], It is evident that for any two sets A and В Α ν Β = В \J A. We may just as well define the union for any finite number of sets, or for an infinite sequence of sets, or for that matter, for an arbitrary system of sets. We shall give the definition for all three cases. The student will observe, however, that the third case includes the other two as special cases. DEFMmoN 4«. If Au Л2, * * * , An is a finite number of sets (i.e., a set Ak associated with every positive integer к < я), their union я Л* w Ak » Αχ vi Α2 vi · · · vi An is the set consisting of all elements belonging to at least one of the sets Alt A2, · · · , An* Thus η A ■» vi Ah is the set consisting of all χ e Ak for some k> where 1 < к < я. We have already given one example of this concept. Further examples: 1. If A * [0,1] and В — [2, 3] the set Л и Я consists of all points in [0,1] and all points in [[2, 3]. 2. If Αχ is the set of integers which are not multiples of 2 and A2 is the set of integers which are not multiples of 3, then Αχ vi Аг is the set of integers which are not multiples of 6. definition 4b. If Αχ, Лг, * * · , Ащ * * * is an infinite sequence of sets (i.e., a set Ak associated with every positive integer /c)> their union «0 Л « vi Ак is the set consisting of all elements belonging to at least one of the sets At, к « 1, 2, · · · , η · · · . We now give the definition of union for an arbitrary system of sets Aa, α € A. The first case above is that for which A is a finite set of positive integers 1, 2, · · · , «, and the second case is that for which A is the set of all positive integers 1, 2, · · ·, я, · · · . In the general case, A can be any set, and there is associated with every element a € A a set Л*. For example, A could be the set of all real numbers. defmition 4c. If Au> α б A, is a system of sets, their union A « и Ал The Intersection of Sets 5 is the set consisting of all elements belonging to at least one A*9 where <*€ A. In order to make Definition 4c more meaningful at this time, we consider the case where A is the set of real numbers. Suppose that for every α « A there is a set Л« of real numbers. We may give a simple interpretation to the set A · u A„ otA as follows. Let S be any set in the cartesian plane. For each α с A, let Aa be the set of β such that (α, β) «& The set A « и А* «A is the projection of S on the tf-axis. 5. The intersection of sete The second basic operation on sets is the intersection* The intersection of two sets A and В is the set consisting of all elements belonging to both A and B. The intersection of A and В is denoted by А л В, It is evident that for any two sets A and В Α η Β » J? л А. We now define the intersection of an arbitrary system Aai α € A, of sets. definition 5. If A«, α с A, is a system of sets, their intersection A = л Aa <mA is the set consisting of those elements belonging to all the sets 4«, a € A. Thus xeA=* л Лв «A if and only tixtA* for every α € A. For the special case where A is the finite set of positive integers 1,2, · · · , it, the definition states that the intersection η A» л Ah of the sets Au A2, · · · , An is the set consisting of all elements belonging to all the sets Au Л2, * · · , An. For the case where A is the infinite sequence 1, 2, ···,«,···. is the set of elements belonging to all the sets Αι9Α%···9ΑΛ···. 6 Sets and Operations Some examples of set intersections are 1. If Ax « CO, 2] and A2 - [1, 31 then Л1 л Л2 « [1,2]. 2. If A\ is the set of all integers which are multiples of 2 and Ai is the set of all integers which are multiples of 3, then Ax η Аг is the set of all integers which are multiples of 6. 3. If A\ » [0,1] and A* « [2, 3], then Αχ л Аг is the empty set; i.e., it has no members. More will be said regarding the empty set later. 4. Let S be the set of points in the cartesian plane, let A be the set of real numbers, and let Aa be the set of β such that («, β) с S. The set A » η Л* is the projection on the 0-axis of that subset of S composed <r«A of lines parallel to the α-axis which belong to S. In other words, fie A if and only if («, β) tSfor every a. A third useful operation, meaningful only for pairs of sets, is the difference. definition 6. For two sets A and 2?, the difference С ■» A — В is the set consisting of all elements in A which are not in B.~ Thus С is the set consisting of all χ € Л such that xiB. While the union and intersection of two sets are commutative operations, the difference is not; indeed, A — В » В — A only HA » B. If A » B, then A — В and В — Л are the empty set, so that, of course, i-B»B-il, Suppose Aj*B. Suppose, first, that there is an χ e A such that χ 4 В. Then χ € A — В and χ 4 В — Л so that A — Bj&B — A. However, if every χ с A is such that* €# then since Л ?* В there is an xe В such that χ 4 Л. But, then, дг *B — Л and ж € Л — В, so that, again, Л — Β ρ* Β — Л. If 2? С Л, the difference С = Л — 5 is usually called the complement of В relative to Л. In nearly all the considerations of this book, all sets which arise are subsets of some given set U. In such a case, the difference С » U — Л, Л С U9 is simply called the complement of Л and is denoted by C(A). In order that every subset of U should have a complement, the empty set φ, which contains no elements, must be admitted as a subset of U. Then φ is the complement of U. definition 7. Л and В are d&ptfil if A r\ В =ф. Our first proposition gives a relation between all the operations which have been introduced thus far. proposition i. For every system Л«, a € A, of subsets of i/, we have 1. и Л« - С(л С(Лв)), 2. л Аа « C(v С(Л«)). «A «A Umfts of Sequences of Seh 7 Proof. We prove only (1). The proof of (2) is quite similar and is left to the student. Our method consists of showing that ν Л« С С( л С(Аа)) and that C( л С(Аа)) С и Л«, so that u Aa « C( л C{Aa)). α«Α ocA аеА аиА For every χ € υ Л« there is an a t A such that χ € Л«. Hence ж * С(Л«) «иА so that *« л С(Аа). Accordingly, xeC(n C(Aa))- We thus have shown that и Лл С С(л C(Aa)). лсА <mA For every χ e C( л С(Л«)), л * л* С(Л«). Hence there is an α € A such ouA o<A that х#С(Ла). But then xeAa. Accordingly, xe ν Aa. It follows that С(л С(Лв)) С и Л«. β«Α βκΑ The two inclusions show that u Л« = С( л С(Аа)). oreA <mA Examples 2, §4, and 2, §5, furnish an illustration of this proposition. For let Л be the set of integers which are multiples of 2, В the set of integers which are multiples of 3, and U the set of all integers. Then C(A) and C(B) are the sets of integers which are not multiples of 2 and of 3, respectively. Now, Л л В is the set of integers which are multiples of 6, and C(A) и С(£), the set of integers which are not multiples of 6, so that Л л В ш С(С(А) и С(В)), which is in accordance with Proposition 1. 6. Limits of sequences of sets If Ль Лг, · · ·, An> · · · is any sequence of sets, the set of all elements which belong to any infinite number of sets in the sequence is called its limit superior, and the set of all elements which belong to all but a finite number of sets in the sequence is called its limit inferior. The limit inferior is always a subset of the limit superior. If the limit inferior is the same set as the limit superior, the limit of the sequence of sets is said to exist. In other words, the limit of Αχ, Ля, · · ·, Л„ · · * exists if every element which belongs to an infinite number of the sets belongs to all but a finite number of them. dcmnition ·. If A\, Аг, · · ·, Лп, · · · is a sequence of sets, its limit superior, Em A n, is the set consisting of all elements belonging to an infinite number of the sets. Its limit inferior, lim Л», consists of all elements belonging to all but a finite number of the sets. Now if χ с Em An, then χ is in an infinite number of sets in the sequence ·» Αχ, Аг> ···, Ащ ···. Hence χ* и Л* for every positive integer я. For if 8 Sets and Operations there were an m such that xi и Л*, then χ would belong to only a finite fcatfl» number of the sets in the sequence» It follows that xe r% (u Ak). пш1 4-n «0 «0 Conversely, suppose χ e л ( u Л*)· Then for every positive integer я, П ml к**П to x€ и Ak· In other words, for ever positive integer я there is an integer m(n) > η such that xeA^n)· Accordingly, χ is in an infinite number of the sets Au Л2, · · · , An, · · · so that χ с lim Л*. · · «» %<· We have now shown that lim А» С η (ν Л*) and that л ( и Л*) ^^ » «o С lim Ля. Hence lim An — л (и Л*). Henceforth, we shall write n«l J5»»n «0 «0 «0 «0 л .( и Л*) simply as л u Л*. The above remarks constitute a proof of the first part of the following proposition. proposition 2. If A\9 Аъ · · · , Ani · · · is a sequence of sets, lim An β л к/ Ak and lim Л* * и л Л*. TKe proof of the second part is left for the student. definition 9. If A\, Аг, · · · , л *, · · · is a sequence of sets and fim A n — Urn Л n then the limit, lim An, of Ль Лг, · · · , Л*, · · · is said to exist and lim An «= lim An — Hm A*. Accordingly, lim An exists if every element which belongs to an infinite number of sets in the sequence Л1, Л2, · · · , An, · * * belongs to all but a «о «о со β finite number of them. Or, if one prefers, lim A« exists \t η и Л* ■» и п Л*. «о «о «о «о We leave it for the student to show that u η Л* С η -и Л* always, so π»1 Λ—η π*<1 *«η •ο m that to prove that lim Л« exists one needs only to show that c\ \j Аь η-It-η о» «о Си л Л*. η—1 Jb—π Examples: 1. U An - [0, я/(я + 1)1 ««1,2, ···, then* Л* - СО, я/(* + 1)] *«п «о and u4t = [0,1), where the interval [0,1) is the set of all numbers *-n limits of Sequences of Sets 9 00 «0 00 00 * such that 0 й x < 1. Then u η 4* = [0,1) and л ν Ak n»l &-л n=l Jb«n β [0,1) so that lim 4Λ β lim 4ft. Wc have accordingly shown that the limit of the sequence exists and is the interval [0,1). 2. If An = [-1/и, (2я - 1)/я], η ш 1, 2, · ·· , then л Лй - Jfe«n •0 «β β» [0, (2я - 1)/я] and υ^ = [-1/я, 2). Then и л Лй = [0, 2) and η υ 4& — [0, 2) so that Km 4Я — lim An and lim An exists n»l fc=n and is the interval [0, 2). 3. If An в [0,1] for odd values of η and An = £—1, 0] for even values of я, then л Л& consists of the single point 0 for all values of я, while • «. «» yj Ак- С—1,1] for all n. Hence lim 4Я ■» π ν 4Λ = [-1,1] and t«n n«U«n •0 00 lim An— u л Л* = [0], the set consisting of the single element 0. In this case, the limit of the sequence Ль ^2, · · · , A», · · · does not exist. There is an important class of sequences of sets, the (monotonically) nonincreasing and (monotonically) nondecreasing sequences for which the limit exists. definition io. A sequence A\, Л2, · · ·, Л«, · · · of sets is called (monotonically) nondecreasing if Αχ С Аъ С · · · С Ап С · · ·, i.e., if Ая is a subset of An+i for every positive integer я. It is called (monotonically) nonincreasing if A\ D A2 Э ··· Dil» D · · ·, i.e., if 4Λ+ι is a subset of An for every positive integer я. A sequence (Л«) is called monotonic if it is either non- decreasing or nonincreasing. (Hereafter, the adjective monotonically may be deleted.) In example (1) above, the sequence is monotonically nondecreasing. In (2), the sequence is not monotonic, which shows that there are other sequences, besides the monotonic ones, whose limits exist. Example (3) shows that there are sequences of sets whose limits do not exist. Another example is the following: 1. If An в [0,1/я], я « 1, 2, · · ·, the sequence Au A% · · ·, An> · · · is nonincreasing, and lim An exists and consists of the single point 0. proposition 3. If Au A2, · · ·, Ani · · · is nondecreasing, then lim An exists and lim An — u Л*. n-l 00 во Proof. We first show that u An « lim An- Suppose * < u An- Then there is n-i n-l 10 Sets and Operations an η such that χ e Л*. But since the sequence Au Л2, · · ·, Л*, " · is non- «ft «ft 0» decreasing, x€An+u x€An+2 ,···. Hence x<n Л* so that χ € u η Л* fc—η η—1 ft—η » lim An- Therefore, sj An С Km An. n-l «ft «0 Suppose χ € lim Лп. Then xe и л Л* and there is an η such that n-l fc-n «» «» «» X€ л Ль Hence x€An, so that *€ \J Л*. Therefore, lim Ая С \j An. fc-n n-l n-l «ft The two inclusions show that u Ля = lim Л». n-l «» ^^ «· We now show that u Лл β fim Л*. Suppose χ € ν; Л*. Then there is n-l n-l an m such that χ € Лт. But, since Αχ, Лг, * * * , Лл, · · · is nondecreasing, «» χ с Л<»+1, χ € Лw+2, · · ·. It follows that χ € и Л* for every л. Hence ft-η χ € л и Л* β lim Л». Therefore, и ^»C lim An. n-l fc-n n-l «» ·» «» Suppose χ с Em An- Then χ € a и Л* so that χ € sj Ak. Therefore, n-l *-n fc-1 fim An С u Лл. The two inclusions show that u An — IEn Ля. n-l n-l We have thus shown that the limit of Ль Лг, · · ·, Л*, · · · exists and is the «· set \J A». n-l proposition 4. If Ль Лг, * · ·, Л», · · · is monotonically nonincreasing, then «0 lim An exists and lim Л» = л Л*. n-l The proof, which is very similar to that of "Proposition 3, is left to the student* Examples: 1. If An - [l/(« + 1), (л + l)/(« + 2)1» = 1, 2, ···, then lim Ля - (ο,ΐ). 2. If An * [0,1/iQ, я « 1, 2, · · ·, then lim Л* - [0], the set consisting of the single point 0. 3. If Лп - [0, (n + l)/nl η - 1, 2, ··· , then lim An - CO, 1]. 7. -4 connection with algebra A set 5 together with two operations for pairs of elements of 5 is called a ring if the operations satisfy certain conditions. In a ring» the opera- Equivalence Relations 11 tions called sum and product, and written а + ft and abi are such that for every a € Sy btS9 both a + ft and eft are elements of 5, and satisfy the following conditions: 1. a+(b+c) = (a + b)+c, 2. β 4- 6 « 6 «f л, 3. for every auS9b*$, there is an л с 5 such that j + * — ft, 4. j(ftr) = (fl^)tf, and 5. e(ft + c) «* eft + <#, (b + c)a ** ba + #j. Examples of rings are 1. The set of integers with the usual operations of addition and multiplication. 2. The set of square matrices of order я with real elements and the usual addition and multiplication of matrices as operations. We now return to the set of subsets of a given set U. One may ask whether this system forms a ring if the two basic operations union and intersection are taken as the sum and product operations. Let U be the set of subsets of I/. If A € U, В € U, then evidently A sj В and Л л В are both subsets of (/and so belong to U. Moreover, (\) A \j (B \j C) » (Л u B)sjC;(2)AkjB = B\j Л; (4) Л л (В л C) « (A n B) r\ C; and (5) А л (В ν С) - (Α η В) и (А л С), (В ν С) л А = (В л Л) и (С η Л). But it is not always true that (3) if A с U, В с U, then there is an X € U such that A\j X =* B. For there is no such ЛГ if Л — В is not empty. Hence the set of subsets of U does not form a ring if the operations are union and intersection. If, however, the addition operation is taken to be the operation (А и В) — (Л л В) and the multiplication is still А л В9 the system does form a ring. Notice that the addition operation results in the set of all elements which belong to Л or to В but not to both Л and B. The proof that the set of subsets of U now constitutes a ring is left to the student. In addition, this ring satisfies the condition: 6. aa s= a. Rings which satisfy this latter condition are called Boolean rings. 8. Equivalence relations One of the most important concepts in all of mathematics is that of equivalence relation. Since equivalence relations arise in a variety of places in real-variable theory, it is appropriate to discuss equivalence relations, in general, at the outset. An equivalence relation is simply a subdivision of a given set U into disjoint subsets, i.e., into a system ·£«, a € A, of subsets of U such that every

12 Sefs and Operations χ б U is in one and only one set S«. Two elements atU,btU, are said to be equivalent, a *-> b, if and only if they belong to the same £«. theorem 2. A relation "~" is an equivalence relation for a set U if and only if for every a*U9 b tU, с tU, 1. a ~ a, 2. if a ~ bt then b ~ a9 3 if a ~ b and b ~ c, then a ~ с Proof. Suppose the relation "~" is an equivalence relation. Then U is sub- divideo by the relation into disjoint subsets £«, a € A, such that every a e U is in Sa for one and only one a, and a ~ b if and only if a € Sa, b € Sa, for the same α € A. Now a ~ a. If я ~ b then <2 € &», be Sa, for the same a, so that * ^ a. If α ~ £ and b ~ с then a e Sa and 6 € Sa for the same α 6 A, while δ € £« and с eSa for the same α € A, whence a and £ belong to the same Sa, so that a ~ c. Conversely, a relation which satisfies (1), (2), and (3) above is an equivalence relation, but we leave the proof to the student. We give some examples of equivalence relations. 1. Congruence modulo m is an equivalence relation. Let U be the set of integers and m a fixed integer. Then, by definition, asj modulo m if and only if a — b is a multiple of w. This relation is an equivalence relation. For а ж a modulo т;\£ашЬ modulo m, then b ж a modulo m; and, if а ж Ь modulo m and Ьж с modulo m, then а ж с modulo m. 2. The extension from the system of positive integers to the system of all integers can be accomplished by denning a certain equivalence relation in the set U whose elements are the pairs of positive integers (a, b). The relation we have in mind is that (a,6)~M) if and only i£a-\-d**b + c. This is an equivalence relation; for conditions (1) and (2) are obviously satisfied. To show that (3) is satisfied, suppose (a, b) ~ (c9 d) and (*, d) ~ (*,/). Then а -f d « b + с and с +/ = d + e. It follows that a + d + c+f=b+c + d+eso that a +f » b -f e and (β, b) ~ (*,/). Equivalence classes of pairs of positive integers may be considered as elements. If addition, muldplication, etc., are defined appropriately for these elements, and the elements are then designated in the conventional manner, the system obtained is the system of all integers.

Exercises 13 Exercises 1.1 (a) Give five examples of sets of numbers, at least four of which are infinite, different from those given in the text, (b) Give five examples of other kinds of mathematical sets. 2.1 Show that if А С В and В С С then Л С С. 2.2 If А С В and С С В, what can be said about A in relation to C? Illustrate. 3.1 Show that every finite set of positive integers has a largest element. 3.2 Prove by the principle of finite induction that for every positive integer я, 14.04. | «fr + D 1 +2 + ··* + Л =« Г * 3.3 Prove the binomial theorem by using the principle of finite induction. 4.1 If Л is the set of all rational numbers which can be written as fractions whose denominator is 6 and В is the set of all rational numbers which can be written as fractions whose denominator is 8, what is the set Α ν Β? Α η Β? 5.1 If Л«, cc € A, is a set of sets, show that л Л« С υ Л«. or«A α«Α «о 5.2 If Ait Аг, · · · is a sequence of subsets of i/, show that r\ An = «0 C( u С (An)), where complementation is relative to £/. n-l 5.3 If В С A and Aa is given for every α € A, show that u Aa С и Л« «А <мА and η Л« С л Ла. а«А смВ 5.4 ИАа С B«foreverya« Ashowthat υ А„ С υ B«andn Л« С л В«. ««А в«А «мА вмА 5.5 If A and 2? are subsets of ί/, express A — В in terms of their union, intersection, and complements relative to U. 5.6 Show that if Л is the complement of В relative to U then В is the complement of Л relative to U. 5.7 Show that Л and В are disjoint if and only it Л - В = Л and В — Л «В. «о · 6.1 If Ль Лг, ··· is a sequence of sets, А» и Л* and С* « η Л*, show that Bb ife, · · * is nonincreasing and Cj, C2, · · · is nondecreasing. «0 40 CO «0 6.2 Show that и An =* υ Β* and л Л* — π Сл, where the Вя and n«l n»I n»l n«l Cn are defined as in Exercise 6.1. 6.3 If for every ppsitive integer я, Ая is the set of real numbers greater than я, find lim Ля. η-*·

14 Sets and Operations 6.4 Show, for a sequence of disjoint sets, that the limit exists and is empty. 6.5 Show that i£ Au A2, · · · is a nonincreasing sequence of sets, then lim An » lim A» » lim An· 6.6 Prove the second part of Proposition 2. 6.7 Show that if A\, Л2, · · · is any sequence of sets then lim A * С Em An- 7.1 Show that if A — В is not empty there is no set ATsuch that A sj X « B. 7.2 If for the set of all subsets of a given set U the operations of addition and multiplication of sets A and В are taken to be (A u B) — (А л В) and Л л 5, respectively, show that the system obtained is a Boolean ring. 7.3 Consider all sets which are unions of finite numbers of open intervals, together with the empty set. Is this system a ring? Take operations as in Exercise 7.2. 7.4 Consider all finite sets together with the empty set. Is this system a ring? Take operations as in Exercise 7.2; as union and intersection. 8.1 Show that a relation which satisfies conditions (1), (2), and (3) of Theorem 2 is an equivalence relation. 8.2 Which of the following relations are equivalence relations? (a) The set is all triangles in the plane and the relation is congruence. (b) The set is all triangles in the plane and the relation is similarity. (c) The set is all positive integers and the relation is being relatively prime. (d) The set is all lines in the plane and the relation is parallelism. (e) The set is all lines in the plane and the relation is perpendicularity. 8.3 Give three further equivalence relations, and two relations which are not equivalence relations. 8.4 If we have the equivalence relation (л, b) ~ (c, d) if a+d^b+c for pairs of positive integers, show that if addition for pairs is defined by (j,*) + (*,<0« (a+c,b+d), then (л, b) ~ (а', У) and (<?, d) *>·> (*', c') implies (j, b) + (c,d) ~ (a'\$ bf) + (0', c') so that we have, in effect, defined addition between equivalence classes. 8.5 Define multiplication between the equivalence classes of Exercise 8.4 so that the definition is appropriate for the interpretation of equivalence classes of pairs of positive integers as integers. 8.6 Show that the system given by Exercises 8.4 and 8.5 is a ring, the elements being equivalence classes of pairs of positive integers.

References 15 References Birkhoff and S. MacLane, A Sumy of Modern Algebra, New York, 1941. H. Hahn, Reelle Funktionen, New York, 1948, Chap. 1. P. R. Halmos, Measure Theory, New York, 1950, Chap. 1. F. Hausdorff, Mengenhhre, New York, 1944, Chap. 1. E. Landau, Grundlagen der Analysis, New York, 1946. N. H. McCoy, Rings and Ideals, New York, 1948, Chap. 1.

2 Equivalence of Sets. Cardinal Numbers 1. Equivalence between sets Two sets are called equivalent if their elements may be put into one-to-one correspondence with each other. «Thus the set of fingers and the set of toes of a normal human are equivalent sets. The concept of equivalence for sets is formulated in terms of mappings, which we discuss in some detail. A set Л is mapped into a set В by a mapping f if there is associated with each a € A a single element /(e) in B. Such a mapping/ has the property: 1. If a € Л, there is one and only one b в/(я) е В. For example, the set Л of all integers is mapped into the set В of non- negative integers by the mapping/(«) « я2. A set A is mapped onto a set В if A is mapped into В by a mapping/such that each b t В is the associate of some at A, Such a mapping has property (1) and also property 2. lib eBt there is an a t A such that/(a) = b. In the above example, A is mapped onto the set С of all squares of nonnegative integers. A set Л is in one-to-one correspondence with a set В if A is mapped onto В by a mapping/ such that every b € A is the associate of a single at A, Such a mapping has properties (1), (2), and property 3. If αχ, J2 e A and αχ τ* лг, then b\ =/(<*i), Ьг «/(лг) e B> and δι *■* &2· We remark that if Л is in one-to-one correspondence with B, then В is in one-to-one-correspondence with Л. Let/(л) « δ be a mapping of Л onto В with properties (1), (2), (3). We define the (inverse) mapping J"1 by 16

Examples of Equivalent Sets 17 f~4b) β * whenever /(e) » b. Then property (1) for f"1 follows from properties (2), (3), for/, and properties. (2), (3), for/"1, follow from property (1) for/. definition 1. If A is in one-to-one correspondence with B, we shall write A ~ В and say that A is equivalent to B. In order to avoid possible misunderstanding, we mention that it is only necessary to obtain one example of a mapping of A onto В with properties (1), (2), (3), in order to establish the equivalence of A and B. An example of a mapping which lacks one or more of these properties does not show that A and В are not equivalent. In order to have a proof that A and В are not equivalent, we must show that no mapping of A into В has properties (1), (2), (3). This point will be illustrated, but first we should like to show that the equivalence concept of Definition 1 has the general properties of an equivalence relation as given by Theorem 2, Chapter 1. theorem i. If A is any class whose members are sets A,B, · · ·, the relation "/^/" is an equivalence relation for the sets in A. Proof. If Л € A, the mapping a —/(e) of A into itself has properties (1), (2), (3), so that A ~ A. If Л, В с A and the mapping/ of A into В has properties (1), (2), (3), then by the above remark, the mapping/"1 of В into A also has properties (1), (2), (3), so that A ~ В implies В ~ Л. If Л, B9 С € A, and the mappings/ and g of A into В and В into C, respectively, have properties (1), (2), (3), then the mapping Λ, where k(a) =* g(f(a)), of A into С also has properties (1), (2), (3). Hence A ~ В and В ~С imply A~C. Accordingly, the relation "~" is by Theorem 2, Chapter 1, an equivalence relation and it decomposes A into disjoint classes of sets. 2. Examples of equivalent sets The nontrivial nature of the above ideas is indicated by the following examples. 1. Let Л be the set of even positive integers and В the set of all positive integers. The mapping.? — χ of A into В does not have property (2), for the number 1 in В has not been given a mate by this particular mapping. However, this does not prove that A and В are not equivalent. Indeed, consider the mappings = x/2 of Л into B. This mapping has properties (1), (2), (3), as the student may readily verify. In accordance with our previous remarks, it follows that A and В are equivalent. Incidentally, we have here an example of a set which is equivalent to one of its proper subsets. A set S is finite if it is either empty or is equivalent to a finite set of positive integers; otherwise it is infinite. We shall show later that every infinite set is equivalent to one of its proper

18 Equivalence of Sets. Cardinal Numbers subsets. We leave it as an exercise for the student to show that no finite set has this property. 2. Let Л be the set of positive integers and В the set of all integers. The mappings « C*/2](—l)x, where [*/2] denotes the largest integer not greater than x/2, establishes a one-to-one correspondence between A and B. Hence the set of positive integers and the set of aU integers are equivalent sets. 3. Let A be the set of real numbers in the closed interval [дь аг] and В the set of real numbers in the closed interval [fix, fa]· The mapping fa — b\ fa — δι , , - · χ — αχ h 01 U2 — U\ U2 — й\ 'establishes a one-to-one correspondence between A and B. 4. The same mapping function as in (3) establishes a one-to-one correspondence between the open intervals (βχ, вг) and (blt fa). Thus all closed intervals belong to the same equivalence class and all open {intervals belong to the same equivalence class. We ask now whether these equivalence classes are the same class. 5. Let A be the open interval (βχ, аг) and В the closed interval [j>x, fa]. Since A coi tains a closed subinterval, say [αχ + $$аг — βχ), аг — }(д2 — tfi)j and В contains an open subinterval, say (fa, fa), it follows from examples (3) and (4) that A is equivalent to a subset of В and В is equivalent to a subset of A. It is not an immediate consequence of these observations that A is equivalent to B. However, it woul J follow that A is equivalent to В if we knew in general that whenever each of two sets is equivalent to a subset of the other then the two sets are equivalent. This is a valid result which is known as Bernstein's Theorem. The brunt of the proof of Bernstein's Theorem, which is quite subtle, is absorbed by the following lemma. lemma 1. If Λ is a set, Αχ a subset of A9 and Аг a subset of Αχί and if A /-W Аг> then A ~ Αχ. fr*9. Since A ~ A2 there is a one-to-one correspondence between A and Аг* This correspondence establishes a one-to-one correspondence between the subset Αχ of A and a subset A% of Л2, which in turn establishes a one-to-one correspondence between the subset Аг of Αχ and a subset 44 of Аг, and so on. We obtain in this way by finite induction a monotonically nonincreasing sequence of sets АО AxD A2D ··· D АпЭ ··· such that Л ~ Аг ^ · · · ~ Аг* *■*/··· Denumerable Sets 19 and Αχ ~ As ~ · · · *·* i42n+i ^ * * *. Moreover, Л = (A-Ax)sj (A1~A2)sj ··· u (Л.-х-ЛлЭи ··· υ (Л л i4i л Αι л ···), and Αχ « Ui - Л2) и (Л2 - Лз) и · · · и (Лп-ι - Л«) и · · · υ (Αχ л Лг л · ··). For if χ с A and л < (Л — Αχ)9 χ 4 (Αχ — Лг), · · · , then χ с Αχ, χ € Лг, · · · , so that * с Л л Лх л Аъ η · · · ; and if* с Л1 and χ 4 (Αχ — Л*), л * (Лг — Лз), • · · , then * f Л2, * € Лз, · · · , so that χ с Αχ λ Α2 λ Now the one-to-one correspondence between Л and Л2 which establishes a one-to-one correspondence between Αχ and Л3 also establishes a one-to- one correspondence between Л — Л1 and Л2 — Л3. Hence (A — Αχ) ~ (Л2 - Лз). Similarly (A2 - Л3) ~ (Л4 - Л5), (Л4 - Л5) ~ (Л« - Л7), • · · . Also, obviously (Αχ — Лг) ~ (Л1 — Лг), (Лз — Л4) ~ (Лз — Л4), • · · , and Л л Л1 η · · · л An π · · · ~ Αχ л Лг π · · · п^яп The one-to-one correspondences between the disjoint subsets of Л of the above decomposition and the corresponding disjoint subsets of Αχ determine a one-to-one correspondence between Л and Αχ. Hence Л ^ Л1. theorem 2. (Bernstein's Theorem). If Л ^ Βχ, Βχ С В, and В ~ Ль Л! С Л,.then Л ~ В. f-~/" - "г 'г '' ' Proof. Since A ~ B\ there is a one-to-one correspondence between A and Βχ. This establishes a one-to-one correspondence between the subset Αχ С Л and a subset B2 С Βχ. Now, since В ^ Л1 and B2 ~ Лх, it follows that B~B2. But В Э Βχ D B2 so that, by Lemma 1, Β~Βχ. But Л ~Βχ. Therefore A~B. Bernstein's Theorem may now4 be used to complete the proof that open intervals and closed intervals are equivalent. We shall find numerous other applications of this theorem^ 3. Denumerable sets The above considerations make it possible to distinguish between denumerable and nondenumerable infinite sets. definition 2. A set S is denumerable if it is equivalent to the set of all positive integers. An infinite set S which is not denumerable will be called nondenumerable. Thus far, we have not shown the existence of nondenumerable sets; this will be done later in the chapter. 20 Equivalence of Sets. Cardinal Numbers A denumerable set St in view of its one-to-one correspondence with the positive integers, can be arranged as a sequence a\, аг» * * * , <2n, · · · , where an is the mate of the positive integer η as gwen by'the correspondence. proposition i. The union of two denumerable sets is a denumerable set. Proof. * We first prove the theorem for disjoint sets. Let A and В be disjoint denumerable sets. We have shown that the set of even positive integers is denumerable. It is similarly true that the set of odd positive integers is denumerable. There exist, accordingly, one-to-one correspondences between A and the set of even positive integers and between В and the set of odd positive integers. If the element of A, mated with the even integer 2n, is labeled 02 n, and the element of B, mated with the odd integer 2n — 1, is labeled tf2n-i> the set A u В has as its elements al9 <*2, * * * , an> ·»· This establishes a one-to-one correspondence between Α υ Β and the set of positive integers. Hence A u В is denumerable. The case where A and В are not disjoint follows by Bernstein's Theorem. corollary 1. The union of a denumerable set and a finite set is a denumerable set. The proof is left to the student» The following stronger proposition holds. proposition 2. If each set of a denumerable set of sets is denumerable, their union is a denumerable set. Proof. We again prove the theorem for disjoint sets. The case for which sets are not disjoint then follows by Bernstein's Theorem. The set of sets may be set up in one-to-one correspondence with the positive integers; then the sets are labeled A\9 A%> · · · , 4», · · · . For every я, the j»et An may be set up in one-to-one correspondence with the positive integers so that its elements are labeled a%\9 fln2, * * · , βη»», · · · .To prove the theorem we have only to establish a one-to-one correspondence between the double sequence of elements anm> я, m = 1, 2, · · · , and the positive integers. This may be done in the following way: for every pair я, m, the number of distinct pairs of positive integers £, / for which к + I <n + m\s given by *«(« + « — 2)(w + η - 1), and the number of distinct pairs of positive integers k> I for which к + / » η + m and к ^ η is я. The mapping к = i(m + n — 2)(m + я — 1) -Ь я accordingly establishes a one-to-one correspondence between the elements β™, η, m ~ 1, 2, · · · , and the positive integers u. We now prove two propositions of a different sort pertaining to denumerable sets. Nondenumerable Sets 21 proposition 3. Every infinite set S has a· denumerable subset. Proof. Let ax € S. Suppose we have, for any positive integer я, distinct elements au u2, · · · , an in S. Since S is infinite there is an element βΛ+ι € S distinct frjta all the elements дь Д2> * * * , Дя. By the principle of finite induction, S has a subset {аь Я25 * · · , ля, * * *) in one-to-one correspondence with the set of positive integers. Accordingly, S has a denumerable subset. proposition 4. Every infinite set is equivalent to one of its proper subsets Proof. First, every denumerable set D = [au a^ · · ·) is equivalent to a proper subset. For the set Ε « {β2, Дз, · · *} is a proper subset of D. The function βη+ι — /fan) establishes a one-to-one correspondence between D and £. Now let A be any infinite set. Then, by Proposition 3, A has a denumerable subset. Let A — С υ {<ц, аг, * * *}, where С and {ль аг, * * *} are disjoint sets and where С could possibly be empty. Then 5 = Cu {*2, вз, · · ·} is a proper subset of A. But Л and В are evidendy equivalent. 4. Nondenumerable sets Not all irfinite sets are denume dble. For example, the set of real numbers in the closed interval / = [0,1] is nondenumerable. A full development of the real-number system will be given in Chapter 3. We take the liberty at this time to borrow a property of the real numbers. It is the following property: If Д Э h Э · · · D /» Э · · · is a monotonically nonincreasing sequence to of closed intervals then the set л In is a nonempty set. n-l We now show that the closed interval [0,1] of real numbers is nondenumerable. Let there be a mapping of the positive integers into [0,1]. Label the mates of the positive integers 1, 2, · · · , я, · · · by this mapping as aX) a2> * * * , Дп, · · · . We shall show that there is a point in [0,1] which has not been labeled so that the mapping does not have property (2). Let I\ С I be a closed interval such that αχ4ΐ\. This is possible since a\ is just a single point. Then let h С I\ be a closed interval such that ач, 4 &. If we have closed intervals I\ D h D * * * D h such that ax 4 Ib azih, * * * , an 4 In> then there is a closed interval /Λ+ι С In such that an+\ 4 /*+ι· By finite induction we have a nonincreasing sequence of closed intervals /i Э /2 Э · · · Э /n Э · · · «0 such that αχ 4l\, лг ik, · · · , an i/«, · · · . Since η /» is nonempty, there n»l is an а б л /«. Now a € /. But a ^ βΛ, for every positive integer я, since n-l 22 Equivalence of Sets. Cardinal Numbers a* In but an 4 h* Hence our mapping does not have property (2) and, since it was arbitrary, the set of positive integers and the closed interval [0,1] are not equivalent, so that [0,1] is nondenumerable. 5. Cardinal number Of course, two finite sets which are equivalent have the "same number of elements," and conversely, if they have the same number of elements there is a one-to-one correspondence between them. This leads us to give the following definition pertaining to cardinal number for arbitrary sets. definition 3. Two sets have the same cardinal number if and only if they are equivalent. The denumerable sets all have the same cardinal number, that of the set of positive integers. The symbol Ко (aleph zero, aleph being the first letter of the Hebrew alphabet) designates this cardinal number. We have already shown that the set of all integers has cardinal number Ко* The rational numbers form a set of cardinal number Ко also. The proof of this, which follows from Proposition 2, is left to the student. The closed interval [0,1] of real numbers does not have cardinal Ко* The letter с is often used to designate this cardinal number. Since, as we have shown, all intervals, open or closed, are equivalent, it follows that every interval has cardinal number с It is desirable to order the cardinal numbers according to "size." The following remarks are pertinent in this connection. For any two sets A and B> there are four possibilities: 1. Л is equivalent to a subset of В and В is equivalent to a subset of A, 2. Л is equivalent to a subset of В but В is not equivalent to any subset of 4, 3. В is equivalent to a subset of Л but A is not equivalent to any subset of B, 4. Ah not equivalent to any subset of В and В is not equivalent to any subset of A. Case (4) can never arise. We shall assume this without proof for the present. The proof is a rather delicate matter which depends on the theory of ordinal numbers. It will be given in Chapter 10. proposition 5. If A' ~ A and Bf ~ B, the same relation [(1), (2), or (3)] holds between A' and B' as between A and B. The proof is left to the student. If (1) holds for two sets A and В then, by Bernstein's Theorem, A~ B. Anfhmetk of Cardinal Numbers 23 Now, if a is the cardinal number of Л and b is the cardinal number of B, we may, in view of Proposition 5, say without fear of confusion: a = b9 if (1) holds, a < b, if (2) holds, a>b, if (3) holds. This orders the cardinal numbers. As a consequence of Proposition 3, the cardinal number Ко is the smallest infinite cardinal number. For suppose A has cardinal number Ко and В is any infinite set. By Proposition 3, В has a denumerable subset so that A is equivalent to a subset of B. Hence the sets A and В satisfy either condition (1) or (2). Accordingly, if b is the cardinal number of В then Ко = b. 6. Arithmetic of cardinal numbers The usual operations between finite cardinal numbers may be defined for arbitrary cardinal numbers. In these definition*, the cue is taken from the finite case. The problem is to find that characterization of the operation in the finite case which does not depend on "finiteness" but is meaningful in the general case. definition 4o. If a and b are any cardinal numbers, the sum, a -h b, is the cardinal number of A u B, where A and В are any two disjoint sets whose cardinal numbers are, respectively, a and b. One must show, in order for the definition to have meaning, that the sum a -f b does not depend upon the choice of the sets A and B. To show this let A and A' be any two sew» whose cardinal number is a, and В and B' be any two sets whose cardinal number is b, such that A and В are disjoint and A ' and B' are disjoint. Then since A ~ A' and В ~ B', it follows that Av B~ A' \j B* so that they have the same cardinal number. More generally, the sum may be defined for any set of cardinal numbers. definition 4b. The sum of a set of cardinal numbers e«, a с A, is the cardinal number of a set u Л«, where the sets Aa are any mutually disjoint sets omA (i.e., if a 9* ft then Aa л Αβ β φ) such that, for every α € A, the cardinal number of the set Aa is aa. Some of the properties of finite cardinal numbers are not valid for infinite cardinal numbers. A first example pertains to the sum of cardinal numbers. For every finite cardinal number a (a different from the cardinal number of the empty set) we have β + u^e. But, for every infinite cardinal number β, we have a -f a = a. We shall prove this for the two infinite cardinal numbers we have met, Ко and c. 24 Equivalence of Seis. Cardinal Numbers. 1. Ко + Ко — Ко· For let A be the set of even positive integers and В the set of odd positive integers. Since all of these sets are of cardinal number Ко it follows, by Definition 4a, that Ко + Ко — Ко· 2. с + с « с. For.* is the cardinal of any interval, open, closed, or closed on one end. The sets (1/2,1], [0,1/2], and [0,1] are all of cardinal number с But (1/2, 1] υ [0,1/2] - [0,1]. We now give another property of the cardinal number c. 3. * + *+···+*+···**. For the sets (1/2,1], (1/3,1/2], · · · and (0,1] all have cardinal number с A consequence of· (3) is that the set of ail real numbers has cardinal number с. То show this consider the intervals *',/■-- - *"- [0,1), [1, 2), [-1, 0), [2, 3), [-2, -1), · · · Each of these sets has cardinal number с and their union is the set of all real numbers. It follows from this that the set of real numbers has cardinal number c. definition s. The product ab of two cardinal numbers a and b is the cardinal number of the set of all pairs (*,^), χ « At у с В, where A is a set whose cardinabnumber is a and В is a set whose cardinal number is b. Again, this definition is independent of the choice of sets A and В whose cardinal numbers are a and b respectively. For if A1 and B' are two such sets then A ~ A' and В ~ B' so that there are mappings x' = /(*) andy « g(y) of A into A' and В into B\ respectively, with properties (1), (2), (3), it is clear that (*',У) β (f(*), £(?)) is'a mapping of the set of all pairs (x,y) into the set of all paiir (*',У) with properties (1), (2), (3), so that the two sets of pairs have the same cardinal number. The product may also be defined for any set of cardinal numbers. We leave this definition for the student to make. As with the sum operation, multiplication of infinite cardinal numbers is different from that of finite cardinal numbers. For example: 1. Ko'Ko — Ко· Let A and В each be the set of'positive integers. The set of all pairs (яг, п) of positive integers has cardinal number Ко*Ко- But this set is mapped onto the set of positive integers by и = \(m +n - 2)(m -f η - 1) -f и, as was shown in the proof of Proposition 2. Hence its cardinal number is Ко, so that Ко-Ко = Ко- 2. c-c «= с. Let A and В each be the set of real numbers in the half-open interval 7beSet2A 25 (0,1]. Then the cardinal number of the set of points in the square О < χ й 1, 0<^^liscf. We have only to show that there is a one-to-one correspondence between this souare and the interval (0,1]. Write every [x,y) as a pair of nonterminating decimals (»fliu2 · · · an · · · , .Ь\Ъг · · · An · · ·)· This may be done in one and only one way. Mate with this point the number a\bx αύ>2 * * * anbn ·' · . This is the desired mapping. The proof that this mapping establishes a'one-to-one correspondence between the square and a subset of the interval (0,1] is left to the student. Bernstein's Theorem mav then be invoked to complete the proof that c-c = с definition 6. If a and b are two cardinal numbers, я* is the cardinal number of all functions on В with values in Л, where A and В have cardinal numbers a and b, respectively. For example, consider 23. Here Л is a set of cardinal number 2 and В a set of cardinal number 3. Let В = {a, b, c\ and A « {0,1}. The set of functions on В with values in A has the following members: {0,0,0}, {0,0,1), {0,1,0}, {1,0, OK {0,1,1},.{1, 0,1}, {1,1,0}, {1,1,1}. Hence 23 = 8. 7. The set 2A We associate with every set A the set 2A of all functions on A with values 0 or 1. If a is the cardinal number of Л, then, by Definition 6, the cardinal number of 2A is 2e. A,one-to-one correspondence between the set 2A and the set of subsets of Л, including the empty set φ, is obtained by mating with each function on A with values 0 or 1 the subset of A on which the function is 1. This remark is helpful in proving the following theorem. Indeed, we shall also refer to the set of subsets of Л as 2A. theorem 3. For every cardinal number β, 2β > α. Pre©f. Let Л be a set whose cardinal number is a. Let there be ainapping Aa =/(α) of Л into a class of subsets^«j-of A. AV«how~that-this--ts- not-a ■one-to.one correspondence between A and the class 2A of all- subsets-of-Л •Per w^snow that there is a subset В of Л such that В з* Л« for every a € A. For every α € Л we decide whether a € В as follows: if а с Л«, then a 4 B, and if α * Aay then a € B. This set В has the asserted property since for every α с Л, Л« И В, since α is in exactly one of the sets Л« and 5. It follows that there is no one-to-one correspondence between Л and 2A. It is easy, however, to establish a one-to-one correspondence between A and a subset of 2A. Such a correspondence is given by the mapping which associates with every a € Л the function which is 1 at a and 0 everywhere else. Therefore, 2a > a. In particular, 2^> > Ко- We show that 2K» — c, the cardinal number of the 26 Equivalence of Sets. Cardinal Numbers interval [0, lj In order to show this we use the binary expansions of the numbers in [[0,1J Every ν € [0,1] may be expressed m the form Σ- .Ti2- where an β 0 or 1 for every я. The numbei may accordingly be represented by the expression м&ъ · · · an · * *, which is called its binary expavsion The set of binary expansions is in obvious one-to-one correspondence with the set of all functions on the positive integers with values 0 or 1. Hence the set of binary expansions is of cardinal number 2Ko. But every number in [0,1 ] has either one or two binary expansions. For those numbers which have two binary expansions, one of them is terminating. It is easy to show that the terminating expansions are Ко in number. All of this means that с + Ко = 2K». But, since с + с » с, by Bernstein's Theorem, с + Ко ж с. We have thus shown that с - 2Ko. Since 2Ko > Ко, we have a second prcf that с > Ко. Exercises 1.1 Prove that a finite set is not equivalent to any of its proper subsets. 2.1 Show that the set of points interior to any ellipse and the set of points interior to any circle are equivalent. 2.2 Show that if the sets Aa> « € A, are disjoint and the sets Ba, α € A, are disjoint and if A« ~ B* for every α € A, then ewA oeA 2.3 Prove, without using Bernstein's Theorem, that an open interval and a closed interval are equivalent. 2.4 Prove, without using Bernstein's Theorem, that the interior of a circle is equivalent to the union of the interior and the boundary. 2.5 Give a proof that the set of all real numbers and an open interval are equivalent by establishing a one-to-one correspondence between the two sets. 2.6 If у =/(*) « 0/2](-l)* on the positive integers, what isf"l(y)? 3.1 Complete the proof of Proposition 1, 3.2 Prove, without using Bernstein's Theorem, that the union of a finite set and a denumerable set is denumerable. 3.3 Give details of the proof that the diophantine equation _* = Нт-г-я-2)(т-М- 1) + я has a unique solution in positive integers m and η for every positive integer A. References 2? 3.4 Generalize Exercise 3.3. 3.5 Show that the set of all algebraic numbers is denumerable. 3.6 Show that the union of a finite or denumerable aumber of denumera э!е sets is denumerable whether the sets are disjoint or not. 3.7 Show that every infinite set can be decomposed into a denumerable number of disjoint sets each oi which is infinite. 4.1 Show that the set of irrational numbers is nondenumerable. 5.1 Show that с -f Ко = с9 without using Bernstein's Theorem. 5.2 Prove Proposition 5. 6.1 Complete the proof started in the text that cc — c. 6.2 For any cardinal numbers a> b, and c, show that <rV » a*"1**. 6.3 Show that (ab)e « A 6.4 Generalize the definition of product of cardinal numbers to an arbitrary set aa, α € A, of cardinal numbers. 6.5 Show that the definition oi power is a special case of the one for product given in Exercise 6.4. 6.6 Using the definition of Exercise 6.4, find Ко! « 1-2-3 ···«··· . 6.7 Given cardinal numbers bu <a9icte A, show that Σ ba < Ua*. oeA oeA This is known as Konig's Theorem. 7.1 Show that Л = с. 7.2 Show that the terminating binary expansions are Ко in number. References H. Hahn; Reelle Funktionen, New York, 1948, Chap. 2. F. Hausdorff, Mengenlehre, New York, 1944, Chap. 2. E. W. Hobson, Tfo Theory of Functions of a Red Variable, Cambridge, 1927, Chap. 4. E. Kamke, Theory of Sets, translated by F. Bagemihl, New York, 1950. W. SiERPmsKi, Uyms sur les nombres trans/mis, Paris, 1928. 3 The Real Numbers 1. The rational numbers New number systems are introduced in mathematics mainly in order to remedy certain defects in existing systems. For example, a defect in the system of positive integers is that subtraction is sometimes impossible; the removal of this defect yields the system of all integers. On the other hand, the system ot all integers is defective when one wishes to divide, and the removal of this defect leads to the rational numbers. We shall assume that the rational number system R is given at the start. We list some properties of R: (a) R is afield. That is, R is a set such that for every a*R,b eR, there are two unique elements a + b and ab in R, called respectively the sum and product of a and b, such that 1. a+(b+c)± (a + b)+c, 2. a + b - b + tf, 3. for every a and b in R there is an χ € R such that a + χ = b9 4. a (be) β (ab)c, 5. a(b + c) = ab + β*, (δ + *}β = fo + л*· These, as we have noted in Chapter 1, are the axioms for a ring. It is an easy matter to show that every ring has a unique element в such that, for ever/ a in the ring, σ + θ == θ + a = α. θ is called the identity with respect to addition. It is then easy to shew that a · θ — θ for every β. In the rational number system R, the number 0 is the identity with respect to addition. That the ring R of radonal numbers is a field is expressed by the following additional conditions which it satisfies: 6. if a * R, a ^ 0, and b e R> there is an χ € R such that ax =* b, 28 The Rational Numbers 29 7. ab =» ba, 8. Л has at least two elements. Every field Fhas a unique element e such that for every ae Ftae = ea = a. The proof of this is left for the student. The element e is called the identity (in F) with respect to multiplication For the rational numbers R the identity with respect to multiplication is the number 1. (b) R is an order. This means that there is a relation called ''greater than" and written ">" in R such that 1. for every a*Rt b *R, a ?* b, either a > b or b > a. 2. if a > bt then it is not true that b > a, 3. if a > b and b > c, then a > с (c) R is an ordered field. This connects the field operations with the order relations as follows: 1. If a > 0 and b > 0, then a + b > 0 and ab > 0. (d) R has no proper ordered subfield. Two ordered fields F\ and F2 are said to be isomorphic if there is a one-to-one correspondence X2 =/(^1) between them such that 1. for every xx e F^yx € F\9 Л*1+Л)=/(*1)+/Ы>ап<* Μη) -/(ί)/0ί). 2. for every *i cFb^cF}, if *i >л then/(*i) >Λ>Ί). We show that the above properties which the rational numbers have are categorical. This means that if F\ and F2 are both ordered fields having no proper ordered subfields, then F\ and F2 are isomorphic. In other words, the rational numbers are effectively the only system with the above properties. We first prove the following proposition. (Since our main interest is in the real-number system, certain routine portions of the proof will be left for the student to complete.) proposition ϊ. Every ordered field F contains a subfield isomorphic with the ordered field of rational numbers. Proof. F has identities θ and e with respect to addition and multiplication, respectively. Since F has at least two elements, there is an χ t F such that χ^θ. But xe - χ and χθ = θ so that e 9* Θ. We note without proof that for every χ 9* θ, χχ > θ. In particular, ее > θ. But ее s= e. Hence e > Θ. For every positive integer n, we use the notation ne 10 designate that member of F obtained by adding e to itself η times, and by —ne the solution to the equation ne + * β θ. Then, for all integers n, m, m ψ* 0, we designate 30 The Real Numbers by ne/me the solution to the equation (me)x « re. The following facts aie easy to еъ^ u>hsh: 1. For ary mtegcrs k> щ w vhere к ¥· 0 and m ^ 0, кпе/кте « tte/„ie. 2. IT/»i, /??2, »i, «2 are integers, vhere /hj 5* 0 and mz 7* 0, then 1 a- ana = · m\e ffize ηΐχϊΐΐϊβ m\e τη& τήχτηκε 3. If tf»i »i2, я;, яг are integers, where mi *& 0 and ягг 5^ 0, then П\е П& . , Я! П2 — > — if ana only if — > — · m\e τη& mx Ш2 It is clear that the mapping f(m/n) = me/ne produces an isomorphism between the rationale R and a subfield of the ordered field F. proposition 2. If an ordered field F has no ordered proper subfield it is isomorphic with the field of rational numbers. Proof. By Proposition 1, F has a subfield Ff which is isomorphic with the rational numbers. Since F has no proper subfield of this kind, F' =» F. One further property of the rational numbers that we wish to mention at this time is that they form an archimedean system. By this we mean that if a > 0 and b > 0 there is a positive integer η such that na > b and nb > a. We proceed now to a discussion of the main defect in the rational number system. The defect is that there are infinite sequences of rational numbers which converge but which have no limits. Another defect which the rational number system has is that not all quadratic equations with rational coefficients have solutions, but this is not within our present line of interest The decimal expansions of the rational numbers between 0 and 1 have the following property: after a finite number of terms, the expansion becomes a repeating one, i.e., it has the form .a · · anb\ · · · bmbi · · · bj>x ···£«···. For example: ^-.21 142857 142857 142857 142857 ···, where the sequence 142857 repeats ad infinitum. This property of the rational numbers permits us to give an example of a sequence of rational numbers which converges but whose limit does not exist among the rational numbers. Let rx ш .01 Г2 - .01001 r* - .01001 · · · 010 · · · 01 Equivalence Classes of Sequences 31 This sequence {r«} converges, since for every € > 0, if Λ" is large enough so that (.1)" < €, then for every я, τη > JV, we have \rn - rm\ < e. We notice, however, that there is no rational number r such that r ■« lim r». n—*· For if r is a rational number, the decimal expansion of r has the form r = .ay · · · anbx · · · bmbi · · · bm6i · · · bm · · · . But it is obvious that no such decimal expansion can be the limit of the sequence {г*}. For the theory of real functions, this is a serious defect which we shall remedy by making an appropriate extension of the rational number system. It is convenient to consider a new system whose elements are themselves certain classes of sequences of rational numbers. With addition, multiplication, and order defined for the new entities, they will be shown to be an ordered field which does not have the above-mentioned defect. They are the real numbers. 2. Equivalence classes of fundamental sequences of rational numbers Our treatment will be essentially self-sufficient. We define fundamental sequences of rational numbers and prove all the facts regarding them which are needed. We also say a few words about convergent series. A sequence \rn) of rational numbers is a mapping which mates a rational number гл with every positive integer я. The terms in the sequence need not all be different from each other. For example, {0, 0, · · · , 0, · · ·} and {1, 1/2, · · · , 1/я, · · ·} are sequences. A sequence is not a set but every sequence has associated with it a set, the set of values assumed by its terms. In the above examples, the value sets are {0} and {1, 1/2, · · · , 1/я, · · ·}, respectively. The main idea in our discussion is that of fundamental sequence of rational numbers. definition i. For every rational number r, \r\ = max [r, — r]. \r\ is called the absolute value of r. proposition 3. For every pair rb r% of rational numbers, \rx -f гг| < |ri| 4- Ы and |ιν*| - ЫМ- The proof is left for the student. definition 2. A sequence {rn\ of rational numbers is fundamental if, for every rational number € > 0, there is a positive integer JV such that if я, m > N then |гл — rm\ < €. A fundamental sequence is also called a convergent sequence, a regular sequence, or a Cauchy sequence. 32 The Real Numbers The theory of convergent series is closely related to that of convergent sequences. definition 3. A series rx + гг Η- · · · -f rn + · · · of rational numbers is said to converge if the associated sequence [sn] is fundamental, where Sk - rx + r2 + · · · + г*, к « 1, 2, · · · . For example, the series *♦©'+···+©'+ is convergent since the associated sequence {1 — l/2n} is fundamental. The series i+i+...+i + is not convergent. proposition 4. A series rx -f гг 4- · · · + rn 4- · * * of positive rational numbers converges if and only if, for every rational number € > 0, there is a positive integer η such that, for every positive integer p, Гп 4" Гя+: + * * * 4· Гп-гр < €. The proof is left for the student. definition 4. In the rational number system, τ is said to be a limit of a fundamental sequence [rn] if, for every rational number 6 > 0, there is an JV such that if η > JV then \r - rn\ < 6. If r is the limit of {гл}, we write r = lim гл. pnoposmoN 5. In the rational number system, every fundamental sequence {r«} has at most one limit. Proof. Suppose r is a limit of {гл}. Let s j* r. There is an N such that for every η > Ν, But then, for every η > JV, so that j is not a limit. We have already seen that there are fundamental sequences which have no limits in the set of rational numbers. Hence a fundamental sequence of rational numbers has either one limit or no limit. We shall define an equivalence relation for the set of fundamental sequences of rational numbers. Sum of Equivalence Classes 33 definition 5. Two fundamental sequences of rational numbers, [rn] and {ίΛ}, are equivalent if for every rational number € > 0 there is an integer JV such that if η > JV, then \r» - sn\ < «; i.e. if lim (r„ - sn) « 0. n—+«e We shall designate this relation by {r«} ~ {in}. For the case where a fundamental sequence {rn} has a limit, the following proposition holds. proposition 6. If [гя] is a fundamental sequence whose limit r exists and if (in} is equivalent to {гл}, then r is also the limit of [sn]. Proof. Let 6 > 0. There is an Mi such that if η > M\, then \r — гл| < с/2. There is an JVs such that if я > JV2, then \r» - sn\ < e/2. Let jV = max (Mi9 JV2) "For every η > My |r--rJS|r-rJ+|r.-i.|<|4|-·. Hence r is the limit of {ίΛ}. Proposition 6 is the motivation for the above definition of equivalence. We now prove: proposition 7. The equivalence relation of Definition 2 satisfies: 1. Ы ~ M. 2. If {r„} ~ {*„}, then {in} ~ M· 3. If {r„j ~ {in} and {in} ~ [U}> then {r„} ~ {*«}. The proof is left to the student. By Theorem 1, Chapter 1, it follows that our equivalence relation separates the set of all fundamental sequences into disjoint classes. We shall use Greek letters ρ, σ, τ, · · · to denote classes of equivalent fundamental sequences. ρ, σ, τ, · · · will be the elements which we consider. We define sum and product of these elements. 3. Sum of equivalence classes In order to be able to define the sum of two equivalence classes of fundamental sequences of rational numbers, we need the following two propositions. proposition 8. If {r*} and {sn} are fundamental sequences of rational numbers, then {rn + sn] is a fundamental sequence. Proof. Let € > 0. Since 1* „} is a fundamental sequence, there is an M\ such that if n, m > M% then \rn - rM\ < e/2. Likewise, there is an JV2 such that if и, m > N2 then \sn - sm\ < «/2. Let JV - max (A'b JV2), i.e., the larger 34 The Real Numbers of the two integers N1 and N2* If я, m > JV, then I (rn + s») - (r. + *«)| S |r. - rj + к - /J < 5 + 5 - ·. Hence, for every 6 > 0, there is an TV such that if и, m > jV then | (гл -f- sn) — (r« + sm)\ < €, so that {r* + in} is a fundamental sequence. proposition 9. If \rn) ~ {r«'} and [sn] ~ {s.')9 where {r„}, {гл'}, {i»}, {in'}, are fundamental sequences of rational numbers, then jr» + i„) ~{r/+*»'}. Proof. Let 6 > 0. Since {rn} ^ {гл'} there is an ΛΊ such that if я > ΛΊ then \rn — r%'\ < β/2. Likewise, there is an N2 such that if я > JVb then |ίΛ — ίΛ'| < €/2. Let JV - max (ΛΊ, JVj). If я > TV, then |(fn + in) - (#·' + OI ^ К - r.1 + I*· - *'| < \ + \ - «· Hence, for every € > 0, there is an JV such that if я > JV then |(r. + in) - (rn' + *.')l < 6, so that \rn + ^} - {rV + in'}. Propositions 8 ?nd 9 allow us to define the sum of two equivalence classes of fundamental sequences of rational numbers. definition 6. If ρ and σ are equivalence classes of fundamental sequences of rational numbers, ρ + σ is the equivalence class to which {rn + sn) belongs, where {rn} is any element of ρ and {sn\ is any element of <r. Proposition 8 assures that [rn + sn) is a fundamental sequence and Proposition 9 assures that all [rn + sn) for all permissible choices of \rn) and {in} belong to the same equivalence class. 4. Product of equivalence classes We again need certain simple facts about fundamental sequences. proposition ό. If {rn | is a fundamental sequence of rational numbers, there is a rational number Μ > 0 such that \гя\ < Μ for every positive integer n. Proof. Since {r„ J is a fundamental sequence, there is an JVsuch that for every n> N, \rN - г J < 1. Let Μ » max (\rx\, |r2|, · · · , Ы) -f 1. For every η й К Ы < Μ. Suppose n > N. Since \rN - γ·| < 1, W < \r»\ + 1 й Μ so that \rn\ < Μ in this case also. proposition ii. If {гя( and {sH} are fundamental sequences of rational numbers, then {гя*п} is a fundamental sequence. Proof. Let 6 > 0. By Proposition 10, there are rational numbers M\ > 0, А/г > 0, such that \гя\ < Mx and |in| < M2 for every positive integer я. Product of Equivalence Classes 35 Since {r*} is a fundamental sequence, there is an JVi such that if я, яг > JVi then |rn — rm| < */2Λί2. Similarly, there is an JV2 such that if n, яг > JV2 then |*n - J J < «/2Afi. Let JV = max (JVb JV2). If я, ж > JV, then |rwJ* - r*jm| - \rnsn - r„j„ -f rnsm - rmxj Hence for every € > 0 there is an JV such that if я, m > JV then |г* *« — rmsj < 6. Therefore, {r« *л} is a fundamental sequence. proposition 12. If {гл} ~ {г/} and {*.} ~ {*·'), where {rn}, {r.'b {*·}» (in'}, are fundamental sequences of rational numbers, then {rnsn) ~|f·'*.'}. Proof. Let € > 0. By Proposition 10, there are rational numbers M\ > 0, M2 > 0, Mb > 0, and MA > 0 such that |rm| < Μχ, |гл'| < M2, W < Αί3, and \sn\ < Μ4 for every positive integer я. Let Μ » max (Mi, Af2, А/з, М4)· Since {rn} ^ [fn\ and {*«} ~ {*/}, there are positive integers JVi and N2 such that if я > JVi then \rn — rn'\ < */2M and if я > JV2 then |*я - sn'\ <€/2M.LetJV β max (JVi, JV2). For every η > N9 \fn Sn — Гп Sn\ * |r« Sn — r„' 5„ -f fj Sn — Γ»' *»'| й wir. - г л + mix. - *ι <^·2^+Αί'άβ€· Hence for every € > 0 there is an JVsuch that if η > JV then \rn sn — r»' sn'\ < €. Therefore, {rnsn} ~ {r/x.'}. Propositions 11 and 12 allow us to define the product of equivalence classes of fundamental sequences of rational numbers. DtFMmoN 7. If ρ and <r are equivalence classes of fundamental sequences of rational numbers, ρσ is the equivalence class to which {rn sn] belongs, where {r*} is any element of ρ and [sn\ is any element of <r. Proposition 11 assures that {r* sn) is a fundamental sequence and Proposition 12 assures that all {rnSn} for all permissible choices of {r»} and {sn\ belong to the same equivalence class. We now come to an important proposition. proposition i*. The set of equivalence classes of fundamental sequences of rational numbers is a field. Proof. We have already shown that for every ρ and σ in the system there are elements ρ + σ and ρ · σ in the system. We now prove that the field properties are satisfied. That' 1.р + (* + т)-(р + <г) + т, 2. ρ + σ ш 9 + ρ, 36 The Real Numbers 3# ρ -|- £ = * has a solution for every ρ and σ, 4. ρ(στ) = (p<r)r, and 5. ρ(σ + τ) = p<r -f ρτ, (σ + r)p = σρ -f τρ, follows immediately from the fact that these rules hold for the rational numbers. For example, consider (3): a solution to ρ -f- f — σ is the equivalence class of fundamental sequences which has the sequence [sn — гл} as a member, where {r«} and [sn] are any sequences in ρ and σ, respectively. The identity θ with respect to addition is the class which has the sequence {0, 0, · · · ,0, · · ·} as a member. The field property 7. ρσ «* e-p is obviously satisfied so that it remains only to prove 6. pf * <r has a solution for every ρ and σ provided that ρ ^ Θ. Let {гл} « Ρ be such that rn v* 0 for every n, and let {;*}€*. The sequence {sn/rn} has the property {r*} · {;«/r»| = fan}· We must show that this sequence is a fundamental sequence. To show this, let € > 0. Since ρ ?* 0, (rn| does not have 0 as limit. Hence, there is а К > 0 and an ΛΊ such that for every η > ΛΊ, |гл| > A". There are rational numbers M\ > 0, Af2 > 0, such that |r«| < M\ and |i*| < Af2 for every positive integer и. Let Μ = max (Λ/ι, Μ2)* There are positive integers N2 and Nz such that for every n>m>N*, \rn - rm\ < Kh/2M -and for every щ m > JVa, \sn - fe| < K4J1M. Let JV « max (JVi, JV2, JV3). For every я, m > JV, If? _ id = l5* r*» ~" *» r»| _ 1J» r» ~ s* r» + J» r» ~ J» r»| |r« rm\ \ rnrm \ \ rnrm J ^I *· li uLi· li \^м к2*,м к2* Hence [sn/гп] is a fundamental sequence. We must still show that if {r/} €p and и' 7* 0 for every и, {*«'} «<г, {ί»'Λ»Ί ~ U*/rn}· The proof of this, which involves the technique we have been using repeatedly, is left to the student. This completes the proof that our system is a field. 5. Order between equivalence classes The real-number system is an order as well as a field. In this section, we define an order relation for equivalence classes of fundamental sequences of rational numbers. Order between Equivalence Cbsses 37 proposition 14. If [гя] and [s») are fundamental sequences of rational numbers, then either 1· Ы ~ W, or 2. there isa* > 0 and an N such that if я > JVthenr,» > sn 4- k, or 3. there is a к > 0 and an JVsuch that if η > N then jn > r» -f A:. Pro«f. Suppose that \rn) is not equivalent to \sn). Then there is a * > 0 such that for every positive integer μ there is a v > μ such that \rp - j„| > 2k. On the other hand, since \rn) and {*,»} are fundamental sequences, there is an N\ such that if я, ж > ΛΊ then |гя — rm\ < k/2 and there is an N2 such that ji nrm> N2 then |xn - sm\ < к/2. Let Ν ^ max (jVb JV2) be such that |r# — sjf\ > 2*. There are two possibilities: tn > sn or n > rir. Suppose nr > sN. Then r# > str + 2k. For every я > Ν, \гц — rn\ < к/2 and \sjf — 5*1 < к/2, so that rn > rir - - > sir -h 2k - - « sN + γ · But iy > i* - *A so that гя> sN + Ък/2 > sH + k. If sn > rs it follows, in the same way, that for every я > Ν, sn > гя -f- к. proposition is. If {гя) ~ {гя'} and {*«} ~ {*»,'] are fundamental sequences of rational numbers, the same relation in Proposition 14 holds between {rj} and \sn) as between {гя\ and {sn}* Proe». Suppose {fn} ~ {sn}- Then {rj} ~ {r*} ~ {sn} ~ {sn'} so that {rn'} - W). Suppose there is а к > 0 and an N1 such that if η > Νι then r* > j* + к. Since {rn} ^ {rn'} and {j*} ^ {*»'}> there are-integers N2 and JV3 such that if я > JV2 then |r„ - гл'| < к/Ъ and if я > JV3 then \sn - J»'| < */3. Hence, for every я > Лг, г*' > гя — к/3 and for every я > Лз, J* > *n' — к/Ъ. Let JV - max Oft, JV2, .ft). For every я > JV, Г»' > ГЯ - " > in + Л - - > *.' + к - - - J - in' + — Hence there is an / » Л/3 > 0 and an N such that for every η > JV, r„' > Sn' + /. The proof for the third case is the same as that for the second. Propositions 14 and 15 allow us to define an order relation between our elements. oifriition ·. If ρ and <r are equivalence classes of fundamental sequences of rational numbers, then ρ > σ if for {гя} cpand {«} €σ there is a λ > Oand an JV such that if я > N then r* > *я -4-:& We show that this relation has the properties of an order relation. 38 The Real Numbers proposition it. (1) For every ρ and σ, ρ 5* <r, either ρ > σ or σ > ρ; (2) if ρ > <r, then it is not true that <r > ρ; (3) if ρ > <r and σ > r, then P>r. The proof is left to the student. Moreover, we now have the following proposition. proposition 17. The equivalence classes of fundamental sequences of rational numbers form an ordered field. Pt*ef. Suppose ρ > θ and σ > θ. Let (г*} € ρ and {sn} € σ. There are numbers fa > 0 and fa > 0 and JV*i and Λ2 such that if я > JVi then r* > fa and if я > JV2 then x* > fa. Let JV = max (JVb JV2). For every я > JV, r* + ^n > fa + fa and гnSn > fafe. Hence ρ + σ > ^ and ρσ > 0. 6. Convergence and limit We now show that the number system developed in the last few sections is free from the defect which the rational numbers have. This requires definitions of fundamental sequence and of limit for this system. definitions A sequence {г1я}, {гг*}, ***, {гжя}, ··· of fundamental sequences is called a fundamental sequence of fundamental sequences if, for every с > 0, there is an JV such that if k, I > JV then there is a ?(fa /) such that if μ > v(k\ I) then This definition says that for every € > 0 there is an N such that for every k, I > JVthe sequences r*b r*2, · · ·, r**, · · · and гц, г», · · ·', n*, · · · are in the following relationship to each other: there is a positive integer ν (к, /), which may be different for other choices of k and /, such that for every H>F<fc/), proposition is. If [гщ] ~ fai*}, {г2п} ~ {*2*}, * * * are two sequences of fundamental sequences of rational numbers and if {ri*}, {гг*}, · · · is a fundamental sequence of fundamental sequences of rational numbers, then fain}, {s2n)t * * * is also a fundamental sequence of fundamental sequences of rational numbers. Pre·!. Let « > 0. There is an JV such that if k, I > JV then there is a v(k91) such that if μ > v(k, I) then |r*p — rj < «/3. But, for every m, there is a v(m) such that if μ > ?(m) then |γ«^ — т^\ < «/3. For every pair Λ, / of positive integers, let nft0-mefrft0,K*),K0). Convergence and Until 39 For every μ > /*(*, /), where k, I > „V, we have Ub - %| й \*к> - ημ\ + |г*„ - ημ\ -f \ημ - stA < α + 7 + α = €· Hence, for every « > 0, there is an N such that Ик,1> N then there is a /t(*, /) such that if μ > ц(к, I) then |j*„ - stlt\ < €, so that {*i„|, {j2«I, · · · is a fundamental sequence of fundamental sequences of rational numbers. This proposition allows us to define fundamental sequence of equivalence classes. definition io. The sequence pi, P2, · * * of equivalence classes of fundamental sequences of rational numbers is fundamental if every sequence {гщ}, {г2п}, • · · is fundamental, where \rmn} €pm, m = 1, 2, ···. We now introduce the definition of limit of a fundamental sequence for the entities we are considering. First, we define the limit of a fundamental sequence of fundamental sequences. definition п. If {г1л}> {г2п}, · · · is a fundamental sequence of fundamental sequences of rational numbers, a fundamental sequence [rn\ is its limit if, for every 6 > 0, there is an N such that for every k> N there is a v (к) such that if η > ν (к) then \rn — r*n| < €. We are now ready to prove the basic proposition. proposition 19. Every fundamental sequence of fundamental sequences of rational numbers has a limit. proof. Let {fi«}, ton), ***, frmn), ··· be a fundamental sequence of fundamental sequences of rational numbers. Let «i + «2 + · · * + «p + · · · be a convergent series of positive rational terms. There is an г^щ such that if η > «ι then {гщъ — rmi%\ < «i and if m > mi then there is a y(m, mi) such that if ν > v(m, ηΐχ) then \гЩ9 — rw| < 4. There is an rmint such that \fmtnt - r^oj < 2сь such that if η > n2 then {r^^ — гШг»\ < «2 and if m > ТП2 then there is a Km, »*2) such that if ρ > v(m, тг) then \rm* — »w| < «2· For every positive integer £, suppose we already have гЩП1> rni^ · · ·, Гщ,*,. There is an r·^·^ such that Ir^n, - '«^η^Ι < 2€*> such Ла* if я > n,+i then Ir.^j^j - r^J < <p+i, and if m > mp+l there is a v(m, тпр+χ) such that if ? > r(m, m,+i) then Ir»^, — rmK| < c»+i. We consider the sequence rmint, rmn, · · ·, r^n,, · · ·. We prove (1) that it is a fundamental sequence, and (2) that it is a limit of the sequence {г1л}, fan], ···, {гщ*}, ··· of fundamental sequences. /W f (*)· Let с > 0. There is a positive integer £ such that if к > p, I > 0 then c* + €*+i + l· €*+i-i < €/2. So for every A: > p and / > 0, 40 The Real Numbers we have "' + Ι^*+ι-ι«Λ+ί-ι - гтшПкН\ < 2ek + 2ίΑ+ι Η + 2€fc+i-i < €· This shows that {гтрЛр) is a fundamental sequence. Proof of (2). Let € > 0. There is a p such thai, for every / > 0, p-H € k-P 4 Let m > mp. Consider the fundamental sequence rmi, rm2, * · ·. There is an N such that if μ, ν > N then |r*»M — rmv\ < e/4. There is a μ > JV such that Vm* - гЛрПр| < 2f p. For every у > max (/>, JV), we have € 2€ , € This proves (.2). Proposition 19 is complemented by the following proposition. proposition 20. If {гщ}, [*2n), ··· is a fundamental sequence of fundamental sequences and (ru) ~ ta»}, {гг*} ~ {*2п}, ··· then {ίΐΛ}, {*2n} • · · is a fundamental sequence of fundamental sequences, and if {rn} and {sn) are respectively their limits, then {r»J ~ [sn}· The proof, which is a simple application of the definitions involved» is left as an exercise for the student. Propositions 19 and 20 allow us to make the following definition. definition 12. If pi, рг, * · ·, p*, · · · is a. fundamental sequence of equivalence classes, the limit is the equivalence class to which {rn} belongs, where {rn} is a limit of {ri»}, {гг*}, ··· and {ri*} шри {г2*} €рг, ···. We may now state the following proposition. proposition 21. Every fundamental sequence pi, p2, * * *, P», * * * of equivalence classes of fundamental sequences of rational numbers has a limit ρ and the limit is unique. 7. Real numbers Real numbers are equivalence classes of fundamental sequences of rational numbers. The following facts about real numbers are simply summary statements of results obtained in the preceding sections. theorem i. The real numbers are a field. This is a restatement of Proposition 13. The identity with respect to addition is the null class Θ, i.e., the class to The Dedekind Method 41 which the sequence {0,0, · · ·, 0, · · ·} belongs. The identity with respect to multiplication is the unit class e, i.e., the class to which the sequence (1,1, ···, 1, ···} belongs. theorem 2. The real numbers are an ordered field. theorem 3. The real-number system is archimedean, i.e., if ρ > σ > θ there is an я such that ncr > p. The proof is leg to the student. definition 13. A number system R is complete if every fundamental sequence in R has a limit in R. theorem 4. The real-number system is complete. This is a restatement of Proposition 21. Consider those equivalence classes which have members of the form [r> r> '··>*> *' * }· This subset of the set of real numbers is in one-to-one correspondence with the set of rational numbers in such a way that sums and products of corresponding elements are corresponding elements: i.e., it is isomorphic with the rationals. We shall simply call this subset of the reals the rationals. We state the further theorem. theorem 5. There is a rational number between every two real numbers The pVSfcf is left to the student. 8. The Dedekind method The method of obtaining the real numbers from the rational numbers which has been elaborated here is due to G. Cantor. Another method which proceeds along rather^cUfFerent lines but yields the same results is the method of R. Dedekind. The Dedekind method rests upon the so-called Dedekind cut in the rational number system. definition 14. A Dedekind cut (L,U) is a subdivision of the rational numbers into two nonempty disjoint sets L and U such that L has no largest element and if χ e L, у с U, then χ <y. Every rational number r determines a unique Dedekind cut (!,, U) if L consists of all * < r and U of all χ £ r. Conversely, if for a Dedekind cut (I, CO, the set U has a smallest element r, we shall say that the cut (L, U) determines the rational number r. The defect in the rational number system as envisaged by Dedekind is that there are cuts (£, U) which determine no rational number. For example: Let L consist of all the negative rational numbers and the positive rational numbers whose squares are less than 2 and U consist of the positive rational numbers whose squares are greater than or equal to 2. (£, U) is a Dedekind cut, but U has no smallest element. 42 The Red Numbers We shall use the following property of Dedekind cuts. proposition 22. If (L, U) is a Dedekind cut, for every 6 > 0 there is an χ e L and ayeU such that^ — χ < €. Proof. Sine e L and i/" are nonempty, there are rationale r с L and * с £Λ There is a positive integer я such that i-r<«6. Consider the rational numbers r, r -f - (* - r), r + - (s - r), · · ·, x. Among the numbers in this sequence which are in L there is a largest t + (j/n)(s - r). Then r + CO' + l)/«](* - r) « V. But я L я J я This proves the proposition. We define sum, product, and order for Dedekind cuts. definition is. If (Lb Ui) and (I2, £/2) are Dedekind cuts, their sum (L9 U) is the Dedekind cut for which L is the set of all χ +y where χ с Lj, у € L2, ard where £/" is the set of all other rational numbers. ***> proposition 23. The (£, I/) of Definition 15 is a Dedekind cut. Proof. Since L\ and Z* are not empty there are xtL\9y* L%. Thus χ + у с Ζ, so that £ is not empty. By Proposition 22, there are xtL\,y iL^ such that * + 1/2 6 ui, j + 1/2 € l/2. Now χ +y + 1 « U. For if * +j> + 1 eL9 then ж +y + 1 β £ + w where £ € Ζ,χ and a> € £2· But since * с JLi, * < χ + 1/2; and since w € Z#, o> < j> + 1/2; so that * + w < χ +jr + 1> *n contradiction. Hence U is not empty. Next we show that L has no largest element. Let z*L. Then ζ = χ +y where χ*L\,y€Z*. But £1 has no largest element, so that there isaiocli such that w > x. Now w + у € L and w +y > z* Finally we show that for every ζ « A w € £/, we have a> > £. If ζ > и> then z^ w + к where к > 0. Now since £€£,£ = χ + j> where χ € I*, jp € L2. But χ — ktL\ and a; = ζ — £ » * 4-jr — k β (ж — A:) +y9 so that и; € £, contrary to 10 с U and l/ л JL - φ. The product of Dedekind cuts may also be defined. In order to do this we must first distinguish between positive and negative cuts. A cut (L, U) is called positive if 0 с L. Since, by definition, 1 has no largest number, a cut is positive if and only if L contains positive numbers. A cut (£, U) is negative if U contains negative numbers. The addition inverse — (JL, U) of the cut (L, U) is the cut with the property (L, U) -f (— (Z, U)) = 0, where θ is the cut which determines the rational number 0. The proof that every Dedekind cut has an addition inverse is left for the student ТЫ DtdekM Method 43 definition ι*. If (JLb Ut) and (I2, £/2) arc positive Dcdekind cuts, their product (L, U) is the cut for which L is the set of all negative numbers and all products xy, where * *Luy €£2, and * ^ 0, j> £ 0. If (Li, ί/i) and (£2, #2) are both negative, their product is the product of their addition inverses. If (Z,b Ui) is positive and (I2, £/2) is negative, their product is the addition inverse of the product of (£1, U$$ and the addition inverse of (I2, £/2). If one or more of the factors (JLi, U\) and (L2, £/2) is θ the product is 0. proposition 24. The product, as given by Definition 15, is a Dedekind cut. Proof. It will suffice to prove the assertion for the case where (Ll9 U\) and (£2, U2) are both positive. Since L contains all negative rationals, it is not empty. By Proposition 22, there are χ e L\ and у«la such that χ > 0, у > 0 and * + 1 € U\ and ;> + 1 6 i/2. Now xy + χ +y + 1 e i/. For if *y + *+.7+ lcZ, then *y -h χ -+· у + 1 β #t>, where * > 0, w > 0, and * € £i, w € Z,2. But since * с £1, * < * Η- 1, and since ш € I2, w <y -Ь 1. Hence m <t xy -{· χ +y -\"\t in contradiction. Thus U is not empty. Next we show that L has no largest element. Let £ с £, * > 0. Then £ = xy where л € L\9 у e £2, and * > 0, _>> > 0. But £1 has no largest element, so there is a w € L\ such that w > x. Now toy « L and wy > z. Finally, for every Ζ € I, w 6 £/, we have ш > *. If ζ > w then s « wJfe where A >1. Now since zeLyZ ^ xy where χ с L\,y t L2. But *Д € L\ and 10 « ζ/к β *уД = х/к(у) so that α> € L, contrary to w € £/ and In U «= φ. We also define an order relation for the set of Dedekind cuts. definition 17. (Lj, £/1) > (1*2, U2) if there is an χ e L\ — £2. Our next step is to establish a one-to-one correspondence between the equivalence classes of fundamental sequence of rational numbers and the Dedekind cuts. Let η, Γ2, · · ·, гя, · · · be a fundamental sequence of rational numbers. Let L consist of all rational numbers which are less than all but a finite number of terms of the sequence {r»}, except for the largest such number if there should be one, and let U consist of all other rational numbers. (I, £/) is a Dedekind cut. For, by Proposition 10, there is an Μ > 0 such that \rn\ < Μ for every я. Then -M — 1 € L and Μ e U so that neither I nor £/is empty. Let xtLzndytU. If у < x then, since χ is less than all but a finite number of terms of {r«}, so is y. Thus yeL, contrary to у с U and L л U s* φ, so that j> > дг. Hence (I, £/) is a Dedekind cut. Suppose \r%} and {*«} are equivalent fundamental sequences of rational numbers and that (JL, U) and (Z/, £/') correspond, respectively, to \rn} and {χΛ} by the above association. We show that L « V. Let χ tL· Then there is aj>€ JL such that χ <y. There is an N1 such that for every η > Νι, r» > j. Since {r«} ~ {in}, there is an N2 such that for

44 The Real Numbers every η > JV2, sn> rn — (у — χ)/2. Let N = max (JVi, JV2). For every t у — χ ν — χ у — χ * + ~γ- - J» - "-J" < r· - — < '"· Hence there is a number greater than χ which is less than all but a finite number of terms of the sequence {sn\- Hence χ с V\ It follows that V С L. Similarly, L С L'. Accordingly, our correspondence may be considered as being between equivalence classes of fundamental sequences and Dedekind cuts. In order to complete the proof that the correspondence is one-to-one, we must show that fundamental sequences which are not equivalent are associated with different Dedekind cuts, and that every Dedekind cut is associated with some fundamental sequence. We leave the proof of these details to the student: We designate the above correspondence by (L,U)=f(p). We show that this mapping has the following properties: 1./(ρ + σ)=/(ρ)+/(σ), 2. f(j*r) -/j» /Or), and, 3. ifp><r,then/(p) >/(σ). Proof of (1). Let (IbtfO-Zfr), №2, U2) -/(*), and (JL, U) - f(j> + o). We must show that the Dedekind cut (Lt U) is the sum of the cuts (JLx, Ui) and (£2, #2). Let {г») б-ρ and {*л) « *. We show that if χ с L\ and j; € £2 then χ +yeLt and if s с L then there are ж с Ζ,χ, j> € Z,2 such that χ +y — s, thereby accomplishing the proof. First, suppose χ с Lu у с £2. There is an xf e Lx such #\at xr > x. Now x[ is less than all but a finite number of terms of {rn} and у is less than all but a finite number of terms of [sn). Hence χ +yis less than all but a finite number of terms of* {rn + sH] and is not the smallest such rational number. Thus χ + у « L. Next suppose ζ « L. There is a z! > ζ such that z* *L.zf β ж' + У where л' is less than all but a finite number of terms of {rn\ and У is less than all but a finite number of terms of {sn}· Let χ ш χ' - & ^ z)/2, y~y' ~(z' -z)/2. Then xeLu yeL2 and z - χ +y. It follows that the Dedekind cut/(p + <r) is the sum of the Dedekind cuts /<p) and/ίσ). Pro^f </ (2). The proof, which is very much like that· of (1), is left to the student. Proof of (3). Suppose ρ > <r. Let {r*} с p, and {Μ € <r, .and let (Z,b C/i) гв/(р)э №2, ί/г) e/(<r). After a certain finite number of terms, r» > *» for every ». Hence if л is less than all but a finite number of the sH it is less than all but a finite number of the r». This shows that L\ Э L»; hence

Exercises 45 /(ρ) ^ /(σ). We leave it for the student to show that there is an χ * Lx — L2 so that/GO >/(σ). What we have shown leads to the following proposition. proposition as. The Dedekind cuts form an ordered field which U isomorphic with the ordered field of equivalence classes of fundamental sequences of rational numbers. Fundamental sequences of Dedekind cuts and limits of such sequences will-now be defined. definition is. A sequence (Ll9 C/i), (JL2, #2), · · · of Dedekind cuts is called a fundamental sequence if there are two equivalent fundamental sequences [ln] and {un\ such that for every я, l» с Ln and un « Un* definition 19. If {(L*, Un)) is a fundamental sequence of Dedekind cuts the limit of the sequence is the Dedekind cut (£, U) =/(ρ) where ρ is the equivalence class ίο which [U\ belongs. We finally state the following proposition without proof. proposition 26. If pi, p2, * * *, Pn, · · · is a sequence of equivalence classes and (Ln, Un) — /(рл) for every positive integer и, then {pn} is a fundamental sequence if and only if {(Ln, Un)} is a fundamental sequence, and ρ is the limit of the sequence [pn] if and only if (L, U) — /(p) is the limit of This proposition shows the complete equivalence of the two approaches to real numbers. Exercises 1.1 (a) Show that every ring R has a unique element θ such that β + ί* θ -f a — a for every a ei?. . (b) Show that θ-α « α-θ * θ for every aeR. 1.2 Show that every field Fbzs a unique element e such that for every a € F9 ae = ea « a, 1.3 Show that there is a field which has exacdy five elements but that there is no field which has exacdy ten elements. 1.4 Show that for every positive integer η there is a ring which has exacdy η elements. 1.5 Show that in an ordered field Ffor every * 7* 0, xx > 0. 1.6 Give the details of the proof that if e is the identity element of an ordered field F then the mapping /m\ me produces an isomorphism between the rational field R and a subfield of F.

46 The Real Numbers 1.7 Show that there is no rational number whose square is 2 but that fot every rational number € > 0 there are two rational numbers r and 5 such that f2<2<s2<r2 + e. 2.1 Show that for every pair of rational numbers a and 6, \a + b\ ^ \a\ + \b\ and \ab\ = \a\\b\. 2.2 Show that if a sequence of rational numbers has a limit it is a fundamental sequence. 2.3 Show that every subsequence of a fundamental sequence of rational numbers is fundamental. 2.4 Show that the relation for fundamental sequences of rational numbers —\TA ~ Un) if for every € > 0 there is an JVsuch that if я > JV then \fn — sn\ < «—satisfies the conditions for an equivalence relation given in Theorem 1, Chapter 1. 2.5 Show, for every rational number r, that \rn/n\) is a fundamental sequence. 3.1 Define the difference ρ — <r between equivalence classes of fundamental sequences of rational numbers. 4.1 Prove the distributive law ρ(σ + τ) « ρσ + ρτ for equivalence classes of fundamental sequences. 42 Show that if ρ and σ are equivalence classes of fundamental sequences of rational numbers such that ρ is not the null class and {rn} « p, {гп\ €p, where rn 9* 0, rn' 9* 0, for every «, and {s*} €<r, {*»'} *<r, then {sn/rn} ~ {*·/*% \· 4.3 Define p*. Assuming the laws of exponents for rational numbers, show that they hold for equivalence classes of fundamental sequences. 5.1 Prove that for every ρ and σ, if ρ 5* <r then (a) either ρ > <r or ρ < <r, (b) if ρ > <r then it is not true that σ > ρ, and (с) if ρ > σ and σ > τ then ρ > т. 6.1 Show that every equivalence class, ρ > & of fundamental sequences has a member of the form where *, Jfci, · · ·, *», · · · are integers and 0 й кп й 9 for every n. 6.2 Generalize Exercise 6.1. 6.3 If \ri»), {ran}, * * * is a fundamental sequence of fundamental sequences, let ттПт be such that if η > л* then \гШПя — r*»| < 1/m for every m. I* {'»*»} a fundamental sequence? Is it a limit of {гщ}, tan}, * *'? Either prove or give a counterexample. 6*4 If {ri»), {г2»}> * * * is a fundamental sequence of fundamental sequences and {rin} ~ {*inb {fa*) ~ ten}, · * * show that fai»), fa»), · · · is a fundamental sequence, and if {r*} and {sn} are the respective sequence limits then {r«} ~ {*»].

References 47 7.1 Show that a real number is rational if and only if its decimal expansion is repeating. .2 Find out what a continued fraction is and show that a continued fraction is terminating if and only if the real number it represents is rational. 7.3 Show that the real-number system is archimedean. 7.4 Show that the rational numbers are dense in the real numbers: i.e., if ρ and σ are real numbers and ρ < σ, then there is a rational number r such that ρ <τ <ir. 8.1 For every fundamental sequence of rational numbers \rn} let L consist of all rational numbers which are less than all but a finite number of terms of (r«J with the exception of the largest such number if there is one, and let U be all other rational numbers. This associates a Dedekind cut with every fundamental sequence. Show that if {r»} and {sn} are not equivalent then the corresponding Dedekind cuts are different. 8.2 Show that every Dedekind cut corresponds to some fundamental sequence in the correspondence given in Exercise 8.1. 8.3 Prove that every Dedekind cut in the rationals has an addition inverse. 8.4 Prove that every Dedekind cut in the rationals, except the cut which determines 0, has a multiplication inverse. 8.5 Prove that/Gwr) —/(ρ)/(σ) where/(ρ) is the mapping of equivalence classes onto Dedekind cuts denned in §8. 8.6 Prove Proposition 26. 8.7 Show that in the