Everand Logo

Welcome to Everand!

  • Discover millions of ebooks, audiobooks, and so much more with a free trial

    From $11.99/month after trial. Cancel anytime.

    Analysis in Euclidean Space
    Analysis in Euclidean Space
    Analysis in Euclidean Space
    Ebook859 pages18 hours

    Analysis in Euclidean Space

    By Kenneth Hoffman

    ()

    Read preview

    About this ebook

    Developed for an introductory course in mathematical analysis at MIT, this text focuses on concepts, principles, and methods. Its introductions to real and complex analysis are closely formulated, and they constitute a natural introduction to complex function theory.
    Starting with an overview of the real number system, the text presents results for subsets and functions related to Euclidean space of n dimensions. It offers a rigorous review of the fundamentals of calculus, emphasizing power series expansions and introducing the theory of complex-analytic functions. Subsequent chapters cover sequences of functions, normed linear spaces, and the Lebesgue interval. They discuss most of the basic properties of integral and measure, including a brief look at orthogonal expansions. A chapter on differentiable mappings concludes the text, addressing implicit and inverse function theorems and the change of variable theorem. Exercises appear throughout the book, and extensive supplementary material includes a bibliography, list of symbols, index, and appendix with background in elementary set theory
    LanguageEnglish
    PublisherDover Publications
    Release dateJan 16, 2013
    ISBN9780486135847
    Analysis in Euclidean Space

    Related to Analysis in Euclidean Space

    Titles in the series (100)

    View More
    View More

    Related ebooks

    Mathematics For You

    View More
    View More

    Related podcast episodes

    Related articles

    • UNLIMITED

      Physicists Attack Math’s $1,000,000 Question

      Apr 4, 2017

      Physicists are attempting to map the distribution of the prime numbers to the energy levels of a particular quantum system.

      4 min read

    • UNLIMITED

      Euclid as Founding Father: We hold this mathematics to be self evident.

      Oct 13, 2016

      Despite the man’s awkward gestures, unkempt hair, and ill-fitting suit, it was one of the most extraordinary speeches that Reverend John Gulliver had ever heard. It was March 1860, and the venue was Norwich, Connecticut. The following morning Gullive

      8 min read

    • UNLIMITED

      Why Mathematicians Should Stop Naming Things After Each Other

      Sep 2, 2020

      Any student of modern math must know what it feels like to drown in a well of telescoping terminology. For a high-profile example, let’s take the Calabi-Yau manifold, made famous by string theory. A Calabi-Yau manifold is a compact, complex Kähler ma

      6 min read

    • UNLIMITED

      In Mathematics, Mistakes Aren’t What They Used To Be: Computers can’t invent, but they’re changing the field anyway.

      May 7, 2015

      Vladimir Voevodsky had no sooner sat himself down at the sparkling table, set for a dinner party at the illustrious Institute for Advanced Study in Princeton, New Jersey, than he overturned his empty wine glass, flipping bowl over stem and standing t

      9 min read

    • UNLIMITED

      The Math That Takes Newton Into the Quantum World

      Feb 28, 2019

      In my 50s, too old to become a real expert, I have finally fallen in love with algebraic geometry. As the name suggests, this is the study of geometry using algebra. Around 1637, René Descartes laid the groundwork for this subject by taking a plane,

      9 min read

    • UNLIMITED

      How I Rewired My Brain to Become Fluent in Math: Sorry, education reformers, it’s still memorization and repetition we need.

      Oct 2, 2014

      I was a wayward kid who grew up on the literary side of life, treating math and science as if they were pustules from the plague. So it’s a little strange how I’ve ended up now—someone who dances daily with triple integrals, Fourier transforms, and t

      11 min read

    • UNLIMITED

      How I Rewired My Brain to Become Fluent in Math: The building blocks of understanding are memorization and repetition.

      Sep 15, 2016

      I was a wayward kid who grew up on the literary side of life, treating math and science as if they were pustules from the plague. So it’s a little strange how I’ve ended up now—someone who dances daily with triple integrals, Fourier transforms, and t

      11 min read

    • UNLIMITED

      Do The Maths

      Sep 2, 2024

      by Robyn Arianrhod (UNSW Press, $49.99) Imagine an arrow on a computer screen –in PowerPoint, say. You can drag it around the screen and it’s the same arrow. You can make different arrows by stretching it or rotating it to point in a different direc

      2 min read

    • UNLIMITED

      This Type of Learning Pays Off for STEM Students

      May 21, 2017

      College students studying STEM subjects who can learn and understand larger concepts rather than focusing on the examples used to explain them may have more success in early classes. The findings, published in the Journal of Chemical Education, may h

      3 min read

    • UNLIMITED

      Fanning The Flame

      Sep 13, 2023

      Fanning The Flame
      3 min read

    • UNLIMITED

      Why Virtual Classes Can Be Better Than Real Ones: An engineering professor takes online-course critics to school.

      Oct 29, 2015

      I teach one of the world’s most popular MOOCs (massive online open courses), “Learning How to Learn,” with neuroscientist Terrence J. Sejnowski, the Francis Crick Professor at the Salk Institute for Biological Studies. The course draws on neuroscienc

      11 min read

    • UNLIMITED

      Forget The Fads

      Jan 15, 2024

      Forget The Fads
      1 min read

    • UNLIMITED

      Inspiration By Incident

      Jan 16, 2022

      Inspiration By Incident
      9 min read

    • UNLIMITED

      The Twin Prime Hero: Rags, riches, and fame in mathematics.

      Dec 29, 2016

      Yitang “Tom” Zhang spent the seven years following the completion of his Ph.D. in mathematics floating between Kentucky and Queens, working for a chain of Subway restaurants, and doing odd accounting work. Now he is on a lecture tour that includes st

      8 min read

    Related categories

    Reviews for Analysis in Euclidean Space

    0 ratings

    0 ratings0 reviews

    What did you think?

    Tap to rate

    Review must be at least 10 words

      Book preview

      Analysis in Euclidean Space - Kenneth Hoffman

      Copyright

      Copyright © 1975, 2003 by Kenneth Hoffman

      All rights reserved.

      Bibliographical Note

      This Dover edition, first published in 2007, is an unabridged republication of the work originally published by Prentice-Hall, Inc., Englewood Cliffs, N.J., in 1975.

      Library of Congress Cataloging-in-Publication Data

      Hoffman, Kenneth.

      Analysis in Euclidean space / Kenneth HofFman – Dover ed.

      p. cm.

      Originally published: Englewood Cliffs, N.J. : Prentice-Hall, 1975.

      Includes bibliographical references and index.

      9780486135847

      1. Mathematical analysis. 2. Algebraic spaces. I. Title.

      QA300.H63 2007

      515—dc22

      2006053462

      Manufactured in the United States of America

      Dover Publications, Inc., 31 East 2nd Street, Mineola, N.Y. 11501

      Table of Contents

      Title Page

      Copyright Page

      Dedication

      Preface

      Preface to the Student

      1, Numbers and Geometry

      2. Convergence and Compactness

      3. Continuity

      4. Calculus Revisited

      5. Sequences of Functions

      6. Normed Linear Spaces

      7. The Lebesgue Integral

      8. Differentiable Mappings

      Appendix. - Elementary Set Theory

      List of Symbols

      Bibliography

      Index

      To my parents

      Preface

      This textbook has been developed for use in the two-semester introductory course in mathematical analysis at the Massachusetts Institute of Technology. The aim of the course is to introduce the student to basic concepts, principles, and methods of mathematical analysis.

      The presumed mathematical background of the students is a solid calculus course covering one and (some of) several variables, plus (perhaps) elementary differential equations and linear algebra. The linear algebra background is not necessary until the second semester, since it enters the early chapters only through certain examples and exercises which utilize matrices. At M.I.T. the introductory calculus course is condensed into one year, after which the student has available a one-semester course in dif ferential equations and linear algebra. Thus, over half the students in the course are sophomores. Since many students enter M.I.T. having had a serious calculus course in high school, there are quite a few freshmen in the course. The remainder of the students tend to be juniors, seniors, or graduate students in fields such as physics or electrical engineering. Since very little prior experience with rigorous mathematical thought is assumed, it has been our custom to augment the lectures by structured tutorial sessions designed to help the students in learning to deal with precise mathematical definitions and proofs. It is to be expected that at many institutions the text would be suitable for a junior, senior, or graduate course in analysis, since it does assume a considerable technical facility with elementary mathematics as well as an affinity for mathematical thought.

      The presentation differs from that found in existing texts in two ways. First, a concerted effort is made to keep the introductions to real and complex analysis close together. These subjects have been separated in the curriculum for a number of years, thus tending to delay the introduction to complex function theory. Second, the generalizations beyond Rn are presented for subsets of normed linear spaces rather than for metric spaces. The pedagogical advantage of this is that the original material can be developed on the familiar terrain of Euclidean space and then simply observed to be largely valid for normed linear spaces, where the symbolism is just like that of Rn. The students are prepared for the generalization in much the same way that high-school algebra prepares one for manipulation in a commutative ring with identity.

      The first semester covers the bulk of the first five chapters. It emphasizes the Four C’s: completeness, convergence, compactness, and continuity. The basic results are presented for subsets of and functions on Euclidean space of n dimensions. This presentation includes (of course) a rigorous review of the intellectual skeleton of calculus, placing greater emphasis on power series expansions than one normally can in a calculus course. The discussion proceeds (in Chapter 5) into complex power series and an introduction to the theory of complex-analytic functions. The review of linear geometry in Section 1.6 is usually omitted from the formal structure of the first semester. The instructor who is pressed for time or who is predisposed to separate real and complex analysis may also omit all or part of Sections 5.5 – 5.10 on analytic functions and Fourier series without interrupting the flow of the remainder of the text.

      The second semester begins with Chapter 6. It reviews the main results of the first semester, the review being carried out in the context of (subsets of and functions on) normed linear spaces. The author has found that the student is readily able to absorb the fact that many of the arguments he or she has been exposed to are formal and are therefore valid in the more general context. It is then emphasized that two of the most crucial results from the first semester—the completeness of Rn and the Heine-Borel theorem—depend on finite-dimensionality. This leads naturally to a discussion of (i) complete (Banach) spaces, the Baire category theorem and fixed points of contractions, and (ii) compact subsets of various normed linear spaces, in particular, equicontinuity and Ascoli’s theorem. From there the course moves to the Lebesgue integral on Rn, which is developed by completing the space of continuous functions of compact support. Most of the basic properties of integral and measure are discussed, and a short presentation of orthogonal expansions (especially Fourier series) is included. The final chapter of the notes deals with differentiable maps on Rn, the implicit and inverse function theorems, and the change of variable theorem. This chapter may be presented earlier if the instructor finds it desirable, since the only dependence on Lebesgue integration is the proof of the change of variable theorem.

      A few final remarks. Some mathematicians will look at these notes and say, How can you teach an introductory course in analysis which never mentions partial differential equations or calculus of variations? Others will ask, How can you teach a basic course in analysis which devotes so little attention to applications, either to mathematics or to other fields of science? The answer is that there is no such thing as the introductory course in analysis. The subject is too large and too important to allow for that. The three most viable foci for organization of an introductory course seem to be (i) emphasis on general concepts and principles, (ii) emphasis on hard mathematical analysis (the source of the general ideas), and (iii) emphasis on applications to science and engineering. This text was developed for the first type of course. It can be very valuable for a certain category of students, principally the students going on to graduate school in mathematics, physics, or (abstract) electrical engineering, etc. It is not, and was not intended to be, right for all students who may need some advanced calculus or analysis beyond the elementary level.

      Thanks are due to many people who have contributed to the development of this text over the last eight years. Colleagues too numerous to mention used the classroom notes and pointed out errors or suggested improvements. Three must be singled out: Steven Minsker, David Ragozin, and Donald Wilken. Each of them assisted the author in improving the notes and managing the pedagogical affairs of the M.I.T. course. I am especially grateful to David Ragozin, who wrote an intermediate version of the chapter on Lebesgue integration. I am indebted to Mrs. Sophia Koulouras, who typed the original notes, and to Miss Viola Wiley, who typed the revision and the final manuscript. Finally, my thanks to Art Wester and the staff of Prentice-Hall, Inc.

      KENNETH HOFFMAN

      Preface to the Student

      This textbook will introduce you to many of the general principles of mathematical analysis. It assumes that you have a mathematical background which includes a solid course (at least one year) in the calculus of functions of one and several variables, as well as a short course in dif ferential equations. It would be helpful if you have been exposed to introductory linear algebra since many of the exercises and examples involve matrices. The material necessary for following these exercises and examples is summarized in Section 1.6, but a linear algebra background is not essential for reading the book since it does not enter into the logical development in the text until Chapter 6.

      You will meet a large number of concepts which are new to you, and you will be challenged to understand their precise definitions, some of their uses, and their general significance. In order to understand the meaning of this in quantitative terms, thumb through the Index and see how many of the terms listed there you can describe precisely. But it is the qualitative impact of the definitions which will loom largest in your experience with this book. You may find that you are having difficulty following the proofs presented in the book or even in understanding what a proof is. When this happens, look to the definitions because the chances are that your real difficulty lies in the fact that you have only a hazy understanding of the definitions of basic concepts or are suffering from a lack of familiarity with definitions which mean exactly what they say, nothing less and nothing more.

      You will also learn a lot of rich and beautiful mathematics. To make the learning task more manageable, the notes have been provided with supplementary material and mechanisms which you should utilize:

      Appendix: Note that the text proper is followed by an Appendix which discusses sets, functions, and a bit about cardinality (finite, infinite, countable, and uncountable sets). Read the first part on sets and functions and then refer to the remainder when it comes up in the notes.

      Bibliography: There is a short bibliography to which you might turn if you’re having trouble or want to go beyond the notes.

      List of Symbols: If a symbol occurs in the notes which you don’t recognize, try this list.

      Index: The Index is fairly extensive and can lead you to various places where a given concept or result is discussed.

      One last thing. Use the Exercises to test your understanding. Most of them come with specific instructions or questions, Find this, Prove that, True or false?. Occasionally an exercise will come without instructions and will be a simple declarative sentence, Every differentiable function is continuous. Such statements are to be proved. Their occurrence reflects nothing more than the author’s attempt to break the monotony of saying, Prove that ... over and over again. The exercises marked with an asterisk are (usually) extremely difficult. Don’t be discouraged if some of the ones without asterisks stump you. A few of them were significant mathematical discoveries not so long ago.

      KENNETH HOFFMAN

      1, Numbers and Geometry

      1.1. The Real Number System

      The basic prerequisite for reading this book is a familiarity with the real number system. That familiarity should include a facility both with the elementary algebra of real numbers and with a few inequalities derived from the natural ordering of those numbers. This section is designed to emphasize some properties of the number system which may be less familiar.

      The first thing we shall do is to list a few fundamental properties of algebra and order from which all of the properties of the real number system can be deduced. Let R be the set of real numbers.

      A. Field Axioms. On the set R there are two operations, as follows. The first operation, called addition, associates with each pair of elements x, y in R an element (x + y) in R. The second operation, called multiplication, associates with each pair of elements x, y in R an element xy in R. These two operations have the following properties.

      Addition is commutative,

      x + y = y + x

      for all x and y in R.

      Addition is associative,

      (x + y) + z = x + (y + z)

      for all x, y, and z in R.

      There is a unique element 0 (zero) in R such that x + 0 = x for all x in R.

      To each x in R there corresponds a unique element – x in R such that x + ( – x) = 0.

      Multiplication is commutative,

      xy = yx

      for all x and y in R.

      Multiplication is associative,

      (xy)z = x(yz)

      for all x, y, and z in R.

      There is a unique element 1 (one) in R such that x1 = x for all x in R.

      To each non-zero x in R there corresponds a unique element x– 1 (or 1/x) in R such that xx– 1 = 1.

      1 ≠ 0.

      Multiplication distributes over addition,

      x(y + z) = xy + xz

      for all x, y, and z in R.

      B. Order Axioms. There is on R a relation <, called less than, with these properties.

      If x and y are in R, one and only one of the following holds:

      x < y; x = y; y < x.

      x < y if and only if 0 < y – x.

      If 0 < x and 0 < y, then 0 < (x + y) and 0 < xy.

      C. Completeness Axiom. If S and T are non-empty subsets of R such that

      (i) R = S ∪ T;

      (ii) s < t for every s in S and every t in T;

      then either there exists a largest number in the set S or there exists a smallest number in the set T.

      These properties are usually summarized by saying that the set of real numbers, with its usual addition, multiplication, and ordering, is (A) a field, which (B) is ordered and which (C) is complete in that ordering. Briefly, the real number system is a complete ordered field.

      From the field axioms (A), we could deduce the various algebraic relations which we shall use; however, we shall not do that. We shall use without comment basic identities such as the binomial theorem

      or the telescoping property of a geometric series

      1 – xn+1 = (1 – x)(1 + x + x² + ··· + xn).

      We could define the set of positive integers

      Z+ = {1, 2, 3, ...}

      (from the axioms) as the set consisting of the numbers 1,1 1 + 1, 1 + 1 + 1, ... ; and then we could prove the principle of mathematical induction: If S is a subset of R such that

      1 ∈ S;

      if x ∈ S then (x + 1) ∈ S,

      then S contains every positive integer. Then we could define the set of integers

      Z = {..., -2,-1,0, , 2,...}

      and the set of rational numbers

      and diligently verify that

      and so on. Perhaps (logically) we should carry out those deductions; however, that would be time-consuming and it would be of virtually no help in understanding analysis.

      A similar comment is applicable to a few inequalities which can be deduced (easily) from the order axioms (B). If x < y, then x + z < y + z; if x < y and 0 < c, then cx < cy. We use x ≤ y to mean x < y or x = y. It is understood that y > x means the same thing as x < y. The absolute value of a number x is defined by

      and absolute value has these properties:

      These inequalities will be used with little or no comment.

      Now, one might reasonably ask this. If we are not going to deduce the various properties of the real number system from (A), (B), and (C), why do we bother to list just those particular properties and to assert that they determine the real number system? There are two principal reasons.

      First, analysis is based upon the concept of number, and so we are obligated to state clearly what the real number system is. One way do to that is to state that the system is characterized by two theorems: (i) There exists a complete ordered field. (ii) Any two such fields are isomorphic; that is, there exists a 1: 1 correspondence between their members which preserves addition, multiplication, and order. The second reason for listing (A), (B), and (C) is that it will help us understand the completeness property (C). A fair fraction of introductory analysis consists of learning the meaning of the completeness of the real number system and learning to use various reformulations of it.

      As we have suggested, we shall not prove here that the real number system exists or that it is unique. What we assume is a familiarity with calculations in an ordered field. The one aspect of the number system with which we do not assume much familiarity is the completeness. In the next two sections we begin to look at some implications of completeness. Right now, let us try to be clear about what it says.

      Intuitively, property (C) is intended to say that if one thinks of real numbers as corresponding to points on a line, then the line has no holes in it. How can one subdivide the line R into the union of two non-empty sets, S and T, such that every number in S is less than every number in T? The only way to do that is to cut the line at some point, to let S be everything on one side of the cut and to let T be everything on the other side of the cut. Of course, the point where we cut must be put either in S or in T, and it will accordingly be the largest number in S or the smallest number in T.

      Precisely, suppose we choose any real number c. From c we obtain two slightly different subdivisions as described in (C):

      or

      The completeness property states that there are no other examples, the first type being the one in which S has a largest member, the second type being the one in which T has a smallest member.

      EXAMPLE 1. Let us look at the rational number system, which consists of Q (the set of rational numbers), together with the addition, multiplication, and ordering inherited from R. Since sums, differences, products, and quotients of rational numbers are rational, we see that if we substitute Q for R in (A), the field axioms are satisfied. Similarly, Q satisfies the order axioms (B). Thus, the rational number system is an ordered field; however, it is not ought to be. More precisely, they proved that there is no rational number x such that xand show that Q is not complete, because the set

      S = [s ∈ Qhas no largest member and the set T = {t ∈ Q

      Suppose we define

      (1.1)

      Clearly S and T are non-empty subsets of Q and each number in S is less than each number in T. Here is the important point. Since there does not exist any x in Q with x² = 2, it follows that

      Q = S ∪ T.

      Does T have a smallest member? If t ∈ T, then t² > 2. If r is a very small positive rational number, then we shall have (t – r)² > 2 (as well as t – r > 0); i.e., we shall have (t – r) ∈ T. Hence T has no smallest member. By similar reasoning, S has no largest member. Thus, the rational number system does not have the completeness property (C).

      Why can’t we give the same example in the real number system? Of course, the completeness property says that we cannot. But, let’s try it and see exactly what goes wrong. We define sets S and T as in (1.1), but replace Q by R. Again, we conclude that S and T are non-empty and that every number in S is less than every number in T. Again, we can show that S has no largest member and that T has no smallest member. The completeness property (C) leaves us with only one possibility, namely, that R ≠ S ∪ T, i.e., that some real number belongs neither to S nor to T. It is very easy to see that if x is a real number such that x ∉ S and x ∉ T, then x² = 2. Thus, one of the things which completeness guarantees is that there exists in R a square root for the number 2.

      Exercises

      In Exercises 1-10, deduce the stated properties of real numbers from the basic properties (A), (B), and (C).

      If x < y and z < w, then x + z < y + w.

      If x < 0, then – x > 0.

      If x + y = x, then y = 0.

      For each x in R, x0 = 0.

      If x < y and y < z, then x < z.

      If xy = 0, then either x = 0 or y = 0.

      ( – x)y = – (xy). Hint: [x + ( – x)]y = ?

      ( – x)( – y) = xy.

      For each x in R, x² ≥ 0.

      If x < y,

      If x² + y² = 0, then x = y = 0.

      Use the completeness of the real number system to prove that each positive real number has a unique positive square root.

      The set of integers, with the addition, multiplication, and ordering inherited from R, is not a complete ordered field Precisely which of the conditions listed under the headings (A), (B), (C) are not satisfied?

      1.2. Consequences of Completeness

      We shall discuss a few applications of the completeness of the real number system. First, we need some basic terminology.

      Definition. Let A be a set of real numbers, i.e., a subset of R. We say that A is bounded above if there exists a number b ∈ R such that

      a < b, for all a ∈ A.

      Any such b is called an upper bound for the set A. We say that A is bounded below if there exists a number c ∈ R such that

      c ≤ a, for all a ∈ A.

      Any such c is called a lower bound for the set A. We say that A is bounded if A is bounded above and bounded below.

      There are various simple observations we should make. The set A is bounded below if and only if the set – A = { – x; x ∈ A} is bounded above. If b is an upper bound for – A, then – b is a lower bound for A. Such things are immediate from the fact that the condition x > y is equivalent to – x < – y. The set A is bounded if and only if the set |A| = {|x|; x ∈ A} is bounded above. If b is an upper bound for |A|, then

      – b ≤ x ≤ b, x ∈ A.

      On the other hand, if

      c < x < d, x ∈ A

      then the larger of |c| and |d| is an upper bound for the set |A|

      EXAMPLE 2. The set of positive real members

      (1.2)

      is an elementary example of a subset of R which is not bounded. It is bounded below; in fact, any x ≤ 0 is a lower bound for R+. But it is not bounded above. If b were an upper bound, we could deduce in order: b > 0, (b + 1) ∈ R+, b ≥ b + 1, ??

      EXAMPLE 3. The set of positive integers Z+ is bounded below. It is not bounded above. One of the properties of the real number system with which the reader is supposed to be familiar is the Archimedean ordering property, which states that if b ∈ R, then there is a positive integer greater than b. We shall call that a theorem (Theorem 2), and prove it as an exercise in the use of completeness.

      Theorem 1. Let A be a non-empty subset of R which is bounded above. Then A has a least (smallest) upper bound.

      Proof. Let T be the set of all upper bounds for A:

      T = {b ∈ R; x ≤ b for all x ∈ A}.

      Let S be the complement of T

      S = {x ∈ R; x ∉ T}.

      We can see easily that

      (i) R = S ∪ T;

      (ii) if s ∈ S and t ∈ T, then s < t.

      We defined S so that (i) would be true. What does (ii) say? It says, if s ∈ T and t ∈ T, then s < t; or, if t ∈ T and s ≥ t then s ∈ T. The last statement is clearly true. Look at the definition of T.

      The hypothesis that A is bounded above is precisely the statement that T is non-empty. The hypothesis that A is non-empty tells us that S is non-empty, as follows. Choose any x ∈ A. Then S contains every number y < x, because, if y < x then y is not an upper bound for A.

      The completeness condition now tells us that either S has a largest member or T has a smallest member. But S does not have a largest member. Let s ∈ S, that is, let s ∉ T. Then s is not an upper bound for A. Consequently there exists a number a ∈ A with a > s. The number d satisfies

      s < d < a.

      Since d < a, we have d ∈ S. Since s < d, we see that s is not the largest member of S.

      Therefore, T has a smallest (least) number in it. That is a number c such that

      (1.3)

      In other words, c is the least upper bound for A.

      Evidently, there is a companion result which asserts that, if a subset A of R is non-empty and bounded below, it has a greatest lower bound. That is a number c such that

      (1.4)

      Notation and Terminology. Let A be a non-empty subset of R. If A is bounded above, the least upper bound for A is also called the supremum of A and is denoted by

      sup A.

      If A is bounded below, the greatest lower bound for A is also called the infimum of A and is denoted by

      inf A.

      One might wonder why we introduce other names for least upper bound and greatest lower bound. One reason is that they occur so often that they must be abbreviated, and lub and glb leave a little to be desired.

      Theorem 1 is a reformulation of the completeness of the real number system. In Section 1.1, if one assumes Theorem 1 instead of property (C), then it is easy to prove (C) as a theorem. The two properties are only slightly different. Let’s use Theorem 1 to prove that the set of positive integers is not bounded.

      Theorem 2 (Archimedean Ordering Principle). If x is a real number, there exists a positive integer n such thatx

      Proof. Suppose Z+ is bounded above. Let c = sup Z+. Since c is the least upper bound for Z+, c – 1 is not an upper bound for Z+. Therefore, there exists a positive integer n such that c – 1 < n. So c < n + 1. But that says that c is not an upper bound for Z+. (?)

      Corollary. If x > 0, there exists a positive integer n such that 1/n < x.

      Corollary. If y – x ≥ 1, there is an integer n such that x ≤ n ≤ y.

      Proof According to Theorem 2, there exists an integer m such that x ≤ m. There are at most finitely many integers k such that x ≤ k ≤ m. (That follows from the principle of mathematical induction.) Let n be the least of those integers. It is a simple matter to verify that x < n ≤ y.

      Corollary. If A is a bounded set of integers, then sup A and inf A are integers.

      Corollary. If x < y, there exists a rational number r such that x < r < y.

      Proof. Choose a positive integer n such that n(y – x) > 1. Then find an integer m such that nx < m < ny. Let r = m/n.

      Theorem 3. Let x be a positive real number and let n be a positive integer. There is precisely one positive real number y such that yu = x.

      Proof. Let us make a simple basic observation. If s ≥ 0 and t ≥ 0, then t ≥ s if and only if tn ≥ sn. That follows from the fact that tn – sn (t – s)f(t, s) where f(t, s) = tn-1 + tn – ²s + ··· + tsn-2 + sn – ¹. Since f(t, s) > 0 unless s = t = 0, the numbers tn – sn and t – s have the same sign.

      Obviously (then) we cannot have two distinct positive nth roots. The only problem is to prove that there exists at least one.

      Let

      A={y ∈ R; y > 0 and yn > x}.

      Then A is bounded below. Furthermore A is non-empty. In case x < 1, we have 1n ≥ x so that 1 ∈ A; and, in case x > 1 we have

      xn – x = x(xn – ¹ – 1)

      ≥ 0

      so that x ∈ A. Let c = inf A. Certainly c ≥ 0, and the claim is that cn = x.

      First, we show that cn ≤ x. Suppose cn > x. Then we can find a small positive number r such that (c – r)n > x. (See following lemma.) By the definition of A, (c – r) ∈ A. But, c – r < c and c is a lower bound for A, a contradiction. It must be that cn ≤ x.

      The fact that no lower bound for A is greater than c will imply that cn ≥ x. Suppose cn < x. We can find a small positive number r such that (c + r)n < x. Thus (c + r)n < x < yn for all y ∈ A, which yields c + r < y for all y ∈ A. So, c + r is a lower bound for A. But c + r > c; hence, something is wrong. We conclude that cn > x.

      Lemma. Let n be a positive integer. Let a, b, and c be real numbers such that a < cn < b. There exists a number δ > 0 such that a < (c + r)n < b for every r which satisfies |r| < δ.

      Proof. We have

      tn – cn = (t – c)f(t, c)

      where

      f(t, c) = tn – ¹ + tn – ²c + ··· + tcn – ² + cn – ¹

      If we apply this with t = c + r, we obtain

      (c + r)n – cn, = rf(c + r, c)

      and hence

      |(c + r)n – cn| ≤ |r f(c + r, c)|.

      Now

      If |r| ≤ 1, then

      Therefore,

      Although it is not necessary, we shall rewrite this inequality in a more concrete form: Since (1 + |c|) – |c| = 1, the definition of f tells us that

      f(1 + |c|, |c|) = (1 + |c|)n – |c|n.

      So our inequality says

      (1.5)

      We are told that a < cn < b and we want to ensure that a < (c + r)n < b, provided |r| is small. Let s be the smaller of the two numbers cn – a and b – cn. Then, if

      |(c + r)n – cn| ≤ s

      we shall have a < (c + r)n < b. Define δ by

      δ[(1 + |c|)n – |c|n] = s.

      From (1.5) we then have

      a < (c + r)n < b, provided |r| < δ.

      The reader may already be familiar with the conclusion of the last lemma—the nth power function is continuous. One should look at the proof anyway, since one cannot have too much experience in handling inequalities.

      The unique y > 0 such that yn = x or x¹/n. Remember that x¹/n > 0. If n is even, there is another real number y such that yn = x, namely, y = – x¹/n. If n is odd, there is no other real nth root.

      Exercises

      1. Is the set of rational numbers bounded below?

      2. Give an example of a bounded set A such that sup A is in A but inf A is not in A.

      3. Find all non-empty bounded sets A such that sup A ≤ inf A.

      4. Is the empty set bounded above? Does it have a least upper bound?

      5. Every subset of a bounded set is bounded. Any set which contains an unbounded set is unbounded. (Unbounded means not bounded.)

      6. If A is bounded above and B is bounded below, then the intersection A ∩ B is bounded.

      7. Prove that, if x is any real number, then

      x = sup {r ∈ Q; r < x}.

      8. If x < y, there exists an irrational (not rational) number t such that x < t < y.

      9. Verify that every non-empty set of positive integers contains its infimum.

      10. Let A be a subset of R which has uncountably many points in it. Prove that there exists a non-empty set B ⊂ A such that sup B is not in B. (Uncountable is defined in the Appendix.)

      11. Prove the completeness property (C) from Theorem 1.

      12. Prove that, if a subset S of R (with the inherited addition, multiplication, and ordering) is a complete ordered field, then S = R.

      *13. Let R and S be complete ordered fields. Show that R and S are isomorphic, i.e., show that there is a 1:1 correspondence between the members of R and the members of S which preserves addition, multiplication, and order.

      1.3. Intervals and Decimals

      This is a short section, in which we shall discuss the decimal representations of real numbers. We shall not use these representations very much. The purpose of the section is twofold. It provides us with some concrete objects to which we can point and say, There, if you will, are the real numbers. More important, it will make us think about the relation of intervals to the completeness of the real number system.

      Definition. An interval is a set I ⊂ R such that

      (i) I contains at least two points;

      (ii) if x < t < yand if x, y ∈ I, then t ∈ I.

      There are four types of bounded intervals, to which we shall refer repeatedly:

      The open interval (a, b) = {x ∈ R; a < x < b}

      The closed interval [a, b] = {x ∈ R; a < x < b}

      The semi-closed interval (a, b] {x ∈ R; a < x ≤ b}

      The semi-closed interval [a, b) = {x ∈ R; a ≤ x < b}.

      It is understood that a, b are real numbers with a < b.

      There are five types of unbounded intervals, to which we shall refer occasionally:

      We have left for the exercises the proof that every interval is of one of the nine types listed. In the notations for unbounded intervals, there occur the symbols – ∞ and ∞. There are no objects – ∞ or ∞ in the real number system; indeed, we have assigned no meaning whatever to – ∞ and ∞. The far left and the far right have their uses, but we’ll talk about that later.

      The decimal representation of a real number simply locates the number in a nested sequence of intervals, the lengths of which go down by a factor of 10 each time. The Archimedean ordering property and mathematical induction locate each x ∈

      Enjoying the preview?
      Page 1 of 1
      pFad - Phonifier reborn

      Pfad - The Proxy pFad of © 2024 Garber Painting. All rights reserved.

      Note: This service is not intended for secure transactions such as banking, social media, email, or purchasing. Use at your own risk. We assume no liability whatsoever for broken pages.


      Alternative Proxies:

      Alternative Proxy

      pFad Proxy

      pFad v3 Proxy

      pFad v4 Proxy