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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/714

Asuman G. Aksoy

Mohamed A. Khamsi

A Problem Book in Real Analysis

123

Asuman G. Aksoy

Department of Mathematics

Claremont McKenna College

Claremont, CA 91711

USA

aaksoy@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.edu

ISSN 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 c

Springer 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-Sosa

Contents

Prefaceix

1 Elementary Logic and Set Theory1

Solutions............................................. 9

2 Real Numbers21

Solutions............................................. 27

3 Sequences41

Solutions............................................. 47

4 Limits of Functions63

Solutions............................................. 68

5 Continuity77

Solutions............................................. 84

6 Dierentiability97

7 Integration127

8 Series159

9 Metric Spaces181

10 Fundamentals of Topology197

11 Sequences and Series of Functions223

vii

Bibliography249

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,Ž 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 Scienti“c

Revolution, 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 Real

Analysis. 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 de“ne 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 surely

in"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.Ž Otherwise, we writexΓA. €IfAandBare sets, thenAB(Ais a subset ofBŽorAis contained inBŽ) means that each element ofAis also an element ofB. Sometimes we writeBA(B contains AŽ) instead ofAB. €We say two setsAandBareequal, writtenA=B,ifABandBA. €Any statementShas anegationS(notSŽ) de“ned by

Sis true ifSis false andSis false ifSis true.

€LetP(x) denote apropertyPof the objectx. We writefor the quanti“er there exists.Ž

The expression

xΓX:P(x) means that there exists (at least) one objectxin the classXwhich has the propertyP.Ž

The symbolis called theexistential quanti“er.

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 2010

2CHAPTER 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 quanti“er(or sometimes thegeneral quanti“er). €We use the symbol := to mean "is de“ned 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 by

AB):=(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

A

B:={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 set

AB:={x:xAandxB}.

3

€LetAandBbe subsets ofX. Then

A\B:={xX:xAandx/B}

is therelative complementofBinA. When the setXis clear from the context we write also A c :=X\A and callA c thecomplementofA. €IfXis a set, then so is itspower setP(X). The elements ofP(X) are the subsets ofX.

Sometimes the power set is written 2

X for a reason which is made clear in Problem 2.8.

€Letf:XYbe a function, then

im(f):={yY;xX:y=f(x)} is called theimage of f.Wesayfissurjective(or onto) ifim(f)=Y,injective(or one-to-one) iff(x)=f(y) impliesx=yfor allx,yX, andfisbijectiveiffis both injective and surjective. €IfXandYare sets, theCartesian productX×YofXandYis the set of all ordered pairs x,y) withxXandyY.

€LetXbe a set andA={A

i :iI}be a family of sets andIis an index set.Intersection and union of this familyare given by i I A i ={xX;iI:xA i and i I A i ={xX;iI:xA i

€Letf:XYbe a function, andA

XandB

Yare subsets.Image of A under f,f(A)

defined as f(A)={f(x)Y:xA}. €Inverse image of B under f(or pre-image ofB),f 1

B) defined as

f 1

B)={xX:f(x)B}.

Note that we can formf

1

B) for a setB

Yeven thoughfmight not be one-to-one or

onto. €We will use standard notation,Nfor the set natural numbers,Zfor the set of integers ,Qfor the set rational numbers , andRfor the set real numbers . We have the natural containments: N Z Q R. €Two setsAandBhave the samecardinalityif there is a bijection fromAtoB. In this case we writeAB.WesayAiscountableifNA. An infinite set that is not countable is called an uncountable set.CHAPTER 1. ELEMENTARY LOGIC AND SET THEORY

4CHAPTER 1. ELEMENTARY LOGIC AND SET THEORY

€Schr¨oder...Bernstein Theorem: Assume that there exists one-to-one functionf:AB and another one-to-one functiong:BA. Then there exists a one-to-one, onto function h:ABand henceAB.

Problem 1.1Consider the four statements

a )xRyRx+y>0; b )xRyRx+y>0; c )xRyRx+y>0; d )xRyRy 2 >x.

1. Are the statementsa,b,c,dtrue or false?

2. Find their negations.

Problem 1.2Letf:RR. Find the negations of the following statements:

1. For anyxRf(x)1.

2. The functionfis increasing.

3. The functionfis increasing and positive.

4. There existsxR

such thatf(x)0.

5. There existsxRsuch that for anyyR,ifxf(y).

Problem 1.3Replace...by the appropriate quantifier:,,or. 1.xRx 2 =4...... x=2;

2.zCz=

z ...... zR;

3.xRx=Γ ...... e

2 ix =1. Problem 1.4Find the negation of: "Anyone living in Los Angeles who has blue eyes will win the Lottery and will take their retirement before the age of 50." 5 Problem 1.5Find the negation of the following statements:

1. Any rectangular triangle has a right angle.

2. In all the stables, the horses are black.

3. For any integerxπZ, there exists an integeryπZsuch that, for anyzπZ, the inequality

zProblem 1.6Show thatδ>0βNπNsuch that n?Nθ2Š<2n+1 n+2<2+). Problem 1.7Letf,gbe two functions defined fromRintoR. Translate using quantifiers the following statements:

1.fis bounded above;

2.fis bounded;

3.fis even;

4.fis odd;

5.fis never equal to 0;

6.fis periodic;

7.fis increasing;

8.fis strictly increasing;

9.fis not the 0 function;

10.fdoes not have the same value at two different points;

11.fis less thang;

12.fis not less thang.

Problem 1.8For two setsAandBshow that the following statements are equivalent: a)AαB b)AκB=B c)A∂B=A

CHAPTER 1. ELEMENTARY LOGIC AND SET THEORY

6CHAPTER 1. ELEMENTARY LOGIC AND SET THEORY

Problem 1.9Establish the following set theoretic relations: a)A B=B

A, AB=BA(Commutativity)

b)A (B C)=(A B)

C, A(BC)=(AB)C(Associativity)

c)A (BC)=(A B)(A

C) andA(B

C)=(AB)

(AC) (Distributivity) d)ABB c A c e)A\B=AB c f) (A B) c =A c B c and (AB) c =A c B c (De Morgan"s laws) Problem 1.10Suppose the collectionBis given byB=1,1+ 1 n :nΓN. Find BffB B and BffBquotesdbs_dbs7.pdfusesText_13

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