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The Real Numbers and Real Analysis
Ethan D. BlochThe Real Numbersand Real Analysis
Ethan D. BlochMathematics DepartmentBard CollegeAnnandale-on-Hudson, NY 12504USAbloch@bard.eduISBN 978-0-387-72176-7e-ISBN 978-0-387-72177-4DOI 10.1007/978-0-387-72177-4Springer New York Dordrecht Heidelberg LondonLibrary of Congress Control Number: 2011928556Mathematics Subject Classification (2010): 26-01 Springer Science Business Media, LLC 2011All rights reserved. This work may not be translated or copied in whole or in part without the writtenpermission of the publisher (Springer Science Business Media, LLC, 233 Spring Street, New York,NY 10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use inconnection with any form of information storage and retrieval, electronic adaptation, computer software,or by similar or dissimilar methodology now known or hereafter developed is forbidden.The use in this publication of trade names, trademarks, service marks, and similar terms, even if theyare not identified as such, is not to be taken as an expression of opinion as to whether or not they aresubject to proprietary rights.Printed on acid-free paperSpringer is part of Springer Science Business Media (www.springer.com)
Dedicated to my two wonderful children Gil Nehemya and Ada Haviva,for whom my love has no upper bound
ContentsPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xiTo the Student . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xxiTo the Instructor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xxvii1Construction of the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.2 Entry 1: Axioms for the Natural Numbers . . . . . . . . . . . . . . . . . . . . . .1.3 Constructing the Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.4 Entry 2: Axioms for the Integers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.5 Constructing the Rational Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . .1.6 Dedekind Cuts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.7 Constructing the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1.8 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2Properties of the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 612.2 Entry 3: Axioms for the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . 622.3 Algebraic Properties of the Real Numbers . . . . . . . . . . . . . . . . . . . . . . 652.4 Finding the Natural Numbers, the Integers and the RationalNumbers in the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 752.5 Induction and Recursion in Practice . . . . . . . . . . . . . . . . . . . . . . . . . . . 832.6 The Least Upper Bound Property and Its Consequences . . . . . . . . . . 962.7 Uniqueness of the Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1072.8 Decimal Expansion of Real Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 1132.9 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1283Limits and Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1293.2 Limits of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129112111927334151
viiiContents3.33.43.53.6Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156Two Important Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1714Differentiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.2 The Derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1814.3 Computing Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1924.4 The Mean Value Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1984.5 Increasing and Decreasing Functions, Part I: Local and GlobalExtrema . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2074.6 Increasing and Decreasing Functions, Part II: Further Topics . . . . . . 2154.7 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2255Integration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2315.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2315.2 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2315.3 Elementary Properties of the Riemann Integral . . . . . . . . . . . . . . . . . . 2425.4 Upper Sums and Lower Sums . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2475.5 Further Properties of the Riemann Integral . . . . . . . . . . . . . . . . . . . . . 2585.6 Fundamental Theorem of Calculus . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2675.7 Computing Antiderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2775.8 Lebesgue’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2835.9 Area and Arc Length . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2935.10 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3126Limits to Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3216.2 Limits to Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3226.3 Computing Limits to Infinity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3316.4 Improper Integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3416.5 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3547Transcendental Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3577.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3577.2 Logarithmic and Exponential Functions . . . . . . . . . . . . . . . . . . . . . . . . 3587.3 Trigonometric Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3697.4 More about π . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3797.5 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3918Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.2 Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.3 Three Important Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .8.4 Applications of Sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .399399399412423
Contents8.5ixHistorical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4399Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4439.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4439.2 Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4439.3 Convergence Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4519.4 Absolute Convergence and Conditional Convergence . . . . . . . . . . . . . 4599.5 Power Series as Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4739.6 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48210Sequences and Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.2 Sequences of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.3 Series of Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.4 Functions as Power Series . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .10.5 A Continuous but Nowhere Differentiable Function . . . . . . . . . . . . . .10.6 Historical Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .489489489502509527534Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 539Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 545
PrefaceThe Origin of This BookThis text grew out of two types of real analysis courses taught by the author at BardCollege, one for undergraduate mathematics majors, and the other for students in themathematics section of Bard’s Masters of Arts in Teaching (M.A.T.) Program. Bard’sundergraduate real analysis course is a standard introductory course at the junior–senior level, but the M.A.T. real analysis course, as explained below, is somewhat lessstandard. The author was therefore unable to find an existing real analysis textbookthat exactly met the needs of the students in the M.A.T. course, and so this text waswritten to fill the gap. To make this text more broadly useful, however, it has beenwritten in a way that makes it sufficiently flexible to meet the needs of a standardundergraduate real analysis course as well, though with a few distinguishing features.One of the principles on which Bard’s M.A.T. Program was founded is thatsecondary school teachers need, in addition to sufficient training in pedagogy, a substantial background in their subject areas. In the Bard M.A.T. Program in Mathematics,not only are all students required to have completed the equivalent of a B.A. in mathematics to enroll in the program, but they are required to take four mathematics coursesin the M.A.T. Program, one of which is in real analysis. The M.A.T. mathematicscourses are different from standard first-year mathematics graduate courses, in thatrather than directing the students toward more advanced mathematical topics, the emphasis is on giving the students an advanced look at the material taught in secondaryschool mathematics courses. For example, it is important for prospective teachers ofcalculus to have a good understanding of the properties of the real numbers (includingdecimal expansion), and a detailed look at logarithmic, exponential and trigonometricfunctions, none of which is usually treated in detail in standard undergraduate realanalysis courses. Of course, a prospective teacher of calculus must also have a goodgrasp of limits, differentiation and integration, as found in any real analysis course. Bycontrast, it is not as important for prospective secondary teachers to spend valuablecourse time on some standard introductory real analysis topics such as sequences andseries of functions. Hence, the focus of a real analysis course for M.A.T. students issomewhat different from a standard undergraduate real analysis course.
xiiPrefaceThis text contains all the material needed for both a standard introductory coursein real analysis and for variants of such a course aimed at prospective teachers. It isthe hope of this author that for each intended audience, this text will offer a clear,accessible and interesting exposition of this beautiful material.AudienceThis text is aimed at three target audiences:1. Mathematics majors taking a standard introductory real analysis course;2. Prospective secondary school mathematics teachers taking an introductory realanalysis course;3. Prospective secondary school mathematics teachers taking a second real analysiscourse.For undergraduate mathematics majors taking an introductory real analysis course,this text covers all the standard topics that are typically treated in an introductorysingle-variable real analysis book. The order of the material is slightly different thanusual (with sequences being treated after derivatives and integrals), and as a result afew of the proofs are different, but all the standard topics are present, as well as a fewextras.For prospective secondary school mathematics teachers taking an introductory realanalysis course, this text has, in addition to the standard topics one would encounterin any
undergraduate real analysis course is a standard introductory course at the junior– senior level, but the M.A.T. real analysis course, as explained below, is somewhat less standard. The author was therefore unable to find an existing real analysis textbook that exactly met the needs of the students in the M.A.T. course, and so this text was
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May 02, 2018 · D. Program Evaluation ͟The organization has provided a description of the framework for how each program will be evaluated. The framework should include all the elements below: ͟The evaluation methods are cost-effective for the organization ͟Quantitative and qualitative data is being collected (at Basics tier, data collection must have begun)
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system of numbers is expanded to include imaginary numbers. The real numbers and imaginary numbers compose the set of complex numbers. Complex Numbers Real Numbers Imaginary Numbers Rational Numbers Irrational Numbers Integers Whole Numbers Natural Numbers The imaginary unit i is defi
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