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Problem Books in Mathematics
Edited by P. Winkler
For other titles in this series, go to
http://www.springer.com/series/714Asuman G. Aksoy
Mohamed A. Khamsi
A Problem Book in Real Analysis
123Asuman G. Aksoy
Department of Mathematics
Claremont McKenna College
Claremont, CA 91711
USAaaksoy@cmc.eduMohamed A. KhamsiDepartment of Mathematical SciencesUniversity of Texas at El PasoEl Paso, TX 79968USAmohamed@utep.edu
Series Editor:
Peter Winkler
Department of Mathematics
Dartmouth College
Hanover, NH 03755
USA peter.winkler@dartmouth.eduISSN 0941-3502
ISBN 978-1-4419-1295-4 e-ISBN 978-1-4419-1296-1
DOI 10.1007/978-1-4419-1296-1
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2009939759
Mathematics Subject ClassiÞcation (2000): 00A07 cSpringer Science+Business Media, LLC 2010
All rights reserved. This work may not be translated or copied in whole or in part without the written permission 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 in connection 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 they are not identiÞed as
such, is not to be taken as an expression of opinion as to whether or not they are subject to proprietary rights.
Printed on acid-free paper
Springer is part of Springer Science+
Business Media (www.springer.com)
Dedicated to Erc¨ument G. Aksoy and Anny Morrobel-SosaContents
Prefaceix
1 Elementary Logic and Set Theory1
Solutions............................................. 92 Real Numbers21
Solutions............................................. 273 Sequences41
Solutions............................................. 474 Limits of Functions63
Solutions............................................. 685 Continuity77
Solutions............................................. 846 Dierentiability97
7 Integration127
8 Series159
9 Metric Spaces181
10 Fundamentals of Topology197
11 Sequences and Series of Functions223
viiBibliography249
Index251
Preface
Education is an admirable thing, but it is well to remember from time to time that nothing worth knowing can be taught.Oscar Wilde, The Critic as Artist, 1890.
Analysis is a profound subject; it is neither easy to understand nor summarize. However, Real Analysis can be discovered by solving problems. This book aims to give independent students the opportunity to discover Real Analysis by themselves through problem solving. The depth and complexity of the theory of Analysis can be appreciated by taking a glimpse at its developmental history. Although Analysis was conceived in the 17th century during the ScienticRevolution, it has taken nearly two hundred years to establish its theoretical basis. Kepler, Galileo,
Descartes, Fermat, Newton and Leibniz were among those who contributed to its genesis. Deep conceptual changes in Analysis were brought about in the 19th century by Cauchy and Weierstrass. Furthermore, modern concepts such as open and closed sets were introduced in the 1900s. Today nearly every undergraduate mathematics program requires at least one semester of RealAnalysis. Often, students consider this course to be the most challenging or even intimidating of all
their mathematics major requirements. The primary goal of this book is to alleviate those concerns by systematically solving the problems related to the core concepts of most analysis courses. In doing so, we hope that learning analysis becomes less taxing and thereby more satisfying. The wide variety of exercises presented in this book range from the computational to the more conceptual and vary in diπculty. They cover the following subjects: Set Theory, Real Numbers, Sequences, Limits of Functions, Continuity, Diεerentiability, Integration, Series, Metric Spaces, Sequences and Series of Functions and Fundamentals of Topology. Prerequisites for accessing this book are a robust understanding of Calculus and Linear Algebra. While we dene the concepts and cite theorems used in each chapter, it is best to use this book alongside standard analysis books such as:Principles of Mathematical Analysisby W. Rudin,Understanding Analysisby S. Abbott,Elementary Classical Analysisby J. E. Marsden and M. J. Hoεman, andElements of Real Analysisby D. A. Sprecher. A list of analysis texts is provided at the end of the book. AlthoughA Problem Book in Real Analysisis intended mainly for undergraduate mathematics students, it can also be used by teachers to enhance their lectures or as an aid in preparing exams. The proper way to use this book is for students to rst attempt to solve its problems without looking at solutions. Furthermore, students should try to produce solutions which are diεerent from those presented in this book. It is through the search for a solution that one learns most mathematics. Knowledge accumulated from many analysis books we have studied in the past has surelyin"uenced the solutions we have given here. Giving proper credit to all the contributors is a diπcult
ix xPREFACE task that we have not undertaken; however, they are all appreciated. We also thank Claremont students Aaron J. Arvey, Vincent E. Selhorst-Jones and Martijn van Schaardenburg for their help with LaTeX. The source for the photographs and quotes given at the beginning of each chapter in this book are from the archive at http://www-history.mcs.st-andrews.ac.uk/ Perhaps Oscar Wilde is correct in saying "nothing worth knowing can be taught." Regardless, teachers can show that there are paths to knowledge. This book is intended to reveal such a path to understanding Real Analysis.A Problem Book in Real Analysisis not simply a collection of problems; it intends to stimulate its readers to independent thought in discovering Analysis.Asuman G¨uven Aksoy
Mohamed Amine Khamsi
May 2009
Chapter 1
Elementary Logic and Set Theory
Reserve your right to think, for even to think wrongly is better than not to think at all.Hypatia of Alexandria (370...415
Ifxbelongs to a classA, we writexΓAand read as xis an element ofA. Otherwise, we writexΓA. IfAandBare sets, thenAB(Ais a subset ofBorAis contained inB) means that each element ofAis also an element ofB. Sometimes we writeBA(B contains A) instead ofAB. We say two setsAandBareequal, writtenA=B,ifABandBA. Any statementShas anegationS(notS) dened bySis true ifSis false andSis false ifSis true.
LetP(x) denote apropertyPof the objectx. We writefor the quantier there exists.The expression
xΓX:P(x) means that there exists (at least) one objectxin the classXwhich has the propertyP.The symbolis called theexistential quantier.
1 A.G. Aksoy, M.A. Khamsi, A Problem Book in Real Analysis, Problem Books in Mathematics, DOI 10.1007/978-1-4419-1296-1_1, © Springer Science+Business Media, LLC 20102CHAPTER 1. ELEMENTARY LOGIC AND SET THEORY
We use the symbolfor the quantifier "for all." The expression xX:P(x) has the meaning "for each objectxin the classX,xhas propertyP." The symbolis called theuniversal quantier(or sometimes thegeneral quantier). We use the symbol := to mean "is dened by." We takex:=yto mean that the object or symbolxis defined by the expressiony. Note that for negation of a statement we have: (iA:=(A)=A (ii(AandB)=(A)or(B) (iii(AorB)=(A) and (B) (iv(xX:P(x))=(xX:P(x)) (v(xX:P(x))=(xX:P(x)). LetAandBbe statements.AimpliesBwill be denoted byAB.IfAimpliesB, we take this to mean that if we wish to proveB, it suffices to proveA(Ais a sufficient condition for B). The equivalenceAB("AandBare equivalent" or "Aif and only ifB," often writtenA iffB) of the statementsAandBis defined byAB):=(AB) and (BA).
Ais a necessary and sufficient condition forB, or vice versa. The statementBAis called thecontrapositiveof the statementAB. In standard logic practices, any statement is considered equivalent to its contrapositive. It is often easier to prove a statement"s contrapositive instead of directly proving the statement itself. To proveABby contradiction, one supposesBis false (thatBis true). Then, also assuming thatAis true, one reaches a conclusionCwhich is already known to be false. This contradiction shows that ifAis trueBcannot be true, and henceBis true ifAis true.Given two setsAandB, we defineA
B("the union ofAwithB") as the set
AB:={x:xAorxBor both}.
When speaking about unions, if we sayxAorxBit also includes the possibility thatx is in bothAandB. We defineAB("the intersection ofAwithB") as the setAB:={x:xAandxB}.
3LetAandBbe subsets ofX. Then
A\B:={xX:xAandx/B}
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