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MathematicalAnalysis

Mathem i

Elias Zakon

Copyright Notice

Mathematical Analysis I

c �1975 Elias Zakon c �2004 Bradley J. Lucier and Tamara Zakon Distributed under a Creative Commons Attribution 3.0 Unported (CC BY 3.0) license made possible by funding from The Saylor Foundation's Open Textbook Challenge in order to be incorporated into Saylor.org's collection of open courses available athttp://www.saylor.org. Full license terms may be viewed at:http://creativecommons.org/licenses/by/3.0/. First published by The Trillia Group,http://www.trillia.com, as the second volume of The Zakon

Series on Mathematical Analysis.

First published: May 20, 2004. This version released: July 11, 2011. Technical Typist: Betty Gick. Copy Editor: John Spiegelman.

Contents

t Pr efaceix

About the Authorxi

Chapter fi- Set Theoryfi

1{3. Sets and Operations on Sets. Quantiers:::::::::::::::::::::::::::1

Problems in Set Theory:::::::::::::::::::::::::::::::::::::::::6

4{7. Relations. Mappings:::::::::::::::::::::::::::::::::::::::::::::::8

Problems on Relations and Mappings:::::::::::::::::::::::::::14

8. Sequences::::::::::::::::::::::::::::::::::::::::::::::::::::::::15

9. Some Theorems on Countable Sets::::::::::::::::::::::::::::::::18

Problems on Countable and Uncountable Sets::::::::::::::::::21

Chapter 2- Real Numbers- Fields23

1{4. Axioms and Basic Denitions:::::::::::::::::::::::::::::::::::::23

5{6. Natural Numbers. Induction::::::::::::::::::::::::::::::::::::::27

Problems on Natural Numbers and Induction:::::::::::::::::::32

7. Integers and Rationals::::::::::::::::::::::::::::::::::::::::::::34

8{9. Upper and Lower Bounds. Completeness::::::::::::::::::::::::::36

Problems on Upper and Lower Bounds:::::::::::::::::::::::::40

10. Some Consequences of the Completeness Axiom:::::::::::::::::::43

11{12. Powers With Arbitrary Real Exponents. Irrationals:::::::::::::::46

Problems on Roots, Powers, and Irrationals:::::::::::::::::::::50

13. The Innities. Upper and Lower Limits of Sequences::::::::::::::53

Problems on Upper and Lower Limits of Sequences inE:::::::60

Chapter 3- Vector Spaces- Metric Spaces63

1{3. The Euclideann-space,En:::::::::::::::::::::::::::::::::::::::63

Problems on Vectors inEn:::::::::::::::::::::::::::::::::::::69

4{6. Lines and Planes inEn:::::::::::::::::::::::::::::::::::::::::::71

Problems on Lines and Planes inEn::::::::::::::::::::::::::::75 \Starred" sections may be omitted by beginners. viContents

7- Intervals inEn:::::::::::::::::::::::::::::::::::::::::::::::::::76

Problems on Intervals inEn::::::::::::::::::::::::::::::::::::79

8- Complex Numbers::::::::::::::::::::::::::::::::::::::::::::::::8Q

Problems on Complex Numbers::::::::::::::::::::::::::::::::83

9-Vector Spaces- The SpaceCn- Euclidean Spaces::::::::::::::::::85

Problems on Linear Spaces:::::::::::::::::::::::::::::::::::::89 fiQ- Normed Linear Spaces::::::::::::::::::::::::::::::::::::::::::::9Q Problems on Normed Linear Spaces:::::::::::::::::::::::::::::93 fifi- Metric Spaces::::::::::::::::::::::::::::::::::::::::::::::::::::95 Problems on Metric Spaces:::::::::::::::::::::::::::::::::::::98 fi2- Open and Closed Sets- Neighborhoods:::::::::::::::::::::::::::fiQfi Problems on Neighborhoodsff Open and Closed Sets::::::::::::fiQ6 fi3- Bounded Sets- Diameters::::::::::::::::::::::::::::::::::::::::fiQ8 Problems on Boundedness and Diameters::::::::::::::::::::::fifi2 fi4- Cluster Points- Convergent Sequences:::::::::::::::::::::::::::fifi4 Problems on Cluster Points and Convergence::::::::::::::::::fifi8 fi5- Operations on Convergent Sequences::::::::::::::::::::::::::::fi2Q Problems on Limits of Sequences::::::::::::::::::::::::::::::fi23 fi6- More on Cluster Points and Closed Sets- Density::::::::::::::::fi35 Problems on Cluster Pointsff Closed Setsff and Density::::::::::fi39 fi7- Cauchy Sequences- Completeness::::::::::::::::::::::::::::::::fi4fi Problems on Cauchy Sequences::::::::::::::::::::::::::::::::fi44

Chapter 4. Function Limits and Continuity149

fi- Basic Deynitions::::::::::::::::::::::::::::::::::::::::::::::::fi49 Problems on Limits and Continuity::::::::::::::::::::::::::::fi57

2- Some General Theorems on Limits and Continuity:::::::::::::::fi6fi

More Problems on Limits and Continuity::::::::::::::::::::::fi66

3- Operations on Limits- Rational Functions:::::::::::::::::::::::fi7Q

Problems on Continuity of VectorIValued Functions::::::::::::fi74

4- Inynite Limits- Operations inE::::::::::::::::::::::::::::::::fi77

Problems on Limits and Operations inE:::::::::::::::::::::fi8Q

5- Monotone Functions::::::::::::::::::::::::::::::::::::::::::::fi8fi

Problems on Monotone Functions:::::::::::::::::::::::::::::fi85

6- Compact Sets:::::::::::::::::::::::::::::::::::::::::::::::::::fi86

Problems on Compact Sets::::::::::::::::::::::::::::::::::::fi89

7-More on Compactness:::::::::::::::::::::::::::::::::::::::::::fi92

Contentsvii

"S Continuity on Compact SetsS Uniform Continuity::::::::::::::::OP: Problems on Uniform Continuityfi Continuity on Compact Sets:x'' PS The Intermediate Value Property::::::::::::::::::::::::::::::::x'w Problems on the Darboux Property and Related Topics::::::::x'P O'S Arcs and CurvesS Connected Sets::::::::::::::::::::::::::::::::xOO Problems on Arcsf Curvesf and Connected Sets::::::::::::::::xOG pOOS Product SpacesS Double and Iterated Limits:::::::::::::::::::::xO" pProblems on Double Limits and Product Spaces::::::::::::::xx: OxS Sequences and Series of Functions:::::::::::::::::::::::::::::::xx" Problems on Sequences and Series of Functions::::::::::::::::xww OwS Absolutely Convergent SeriesS Power Series::::::::::::::::::::::xw" More Problems on Series of Functions:::::::::::::::::::::::::x:G Chapter 5. Diαerentiation and Antidiαerentiation251 OS Derivatives of Functions of One Real Variable::::::::::::::::::::xGO Problems on Derived Functions in One Variable:::::::::::::::xG" xS Derivatives of ExtendedgReal Functions::::::::::::::::::::::::::xGP Problems on Derivatives of ExtendedgReal Functions::::::::::x,G wS LUH^opitalUs Rule:::::::::::::::::::::::::::::::::::::::::::::::::x,, Problems on LUH^opitalUs Rule:::::::::::::::::::::::::::::::::x,P :S Complex and VectorgValued Functions onE1::::::::::::::::::::x"O Problems on Complex and VectorgValued Functions onE1:::::x"G GS Antiderivatives pPrimitivesf Integrals(::::::::::::::::::::::::::::x"" Problems on Antiderivatives:::::::::::::::::::::::::::::::::::x"G ,S DinerentialsS TaylorUs Theorem and TaylorUs Series:::::::::::::::x"" Problems on TaylorUs Theorem::::::::::::::::::::::::::::::::xP, "S The Total Variation pLength( of a Functionf6E1!E ::::::::::w'' Problems on Total Variation and Graph Length:::::::::::::::w', "S Rectiyable ArcsS Absolute Continuity::::::::::::::::::::::::::::w'" Problems on Absolute Continuity and Rectiyable Arcs:::::::::wO: PS Convergence Theorems in Dinerentiation and Integration::::::::wO: Problems on Convergence in Dinerentiation and Integration::::wxO O'S SuIcient Condition of IntegrabilityS Regulated Functions::::::::wxx Problems on Regulated Functions:::::::::::::::::::::::::::::wxP OOS Integral Deynitions of Some Functions:::::::::::::::::::::::::::wwO Problems on Exponential and Trigonometric Functions::::::::ww"

Index341

Preface

Th is text is an outgrowth of lectures given at the University of Windsorf CanadaS One of our main objectives isupdatingthe undergraduate analysis as a rigorous postcalculus courseS While such excellent books as DieudonnEeUs Foundations of Modern Analysisare addressed mainly to graduate studentsf we try to simplify the modern Bourbaki approach to make it accessible to suIciently advanced undergraduatesS pSeef for examplefx:of Chapter GS( On the other handf we endeavor not to lose contact with classical textsf still widely in useS Thusf unlike DieudonnEef we retain the classical notion of a derivative as anumberpor vector(f not a linear transformationS Linear maps are reserved for later pVolume II( to give a modern version ofdierentialsS Nor do we downgrade the classical meangvalue theorems psee Chapter Gfxx( or Ri emann{Stieltjes integrationf but we treat the latter rigorously in Volume IIf inside Lebesgue theoryS Firstf howeverf we present the modern Bourbaki theory ofantidierentiationpChapter GfxGnS(f adapted to an undergraduate courseS Me tric spaces pChapter wfxOOn

S (a rei ntroducedc autiouslyfa ftert heng

sp aceEnf with simple diagrams inE2prather thanE3(f and many \advanced calculusDgtype exercisesf along with only a few topological ideasS With some adjustmentsf the instructor may even limit all toEnorE2pbut not just to the real linefE1(f postponing metric theory to Volume IISWe do not hesitate to deviate from tradition if this simplies cumbersome formulationsf unpalatable to undergraduatesS Thus we found useful someconsistentfthough not very usualfconventionspsee Chapter GfxOand the end of Chapter :fx:(f and ane arly use of quantierspChapter OfxO{w( f e veni nf ormulatingt heoremsS Co ntrary to some existing prejudicesf quantiyers are easily grasped by students after some exercisef and help clarify all essentialsS Several yearsU class testing led us to the following conclusions6 pO( Volume I can be pandwas( taught even to sophomoresf though they only gradually learn toreadandstaterigorous argumentsS A sophomore often does not even know how tostarta proofS The main stumbling block remains theε, δgprocedureS As a remedyf we provide most exercises with explicit hintsf sometimes with almost complete solutionsf leaving only tiny \whysD to be answeredS px( Motivations are good if they are brief and avoid terms not yet knownS Diagrams are good if they aresimpleand appeal to intuitionS xPreface L39 Flexibility is a must- One must adapt the course to the level of the class- \StarredF sections are best deferred- LContinuity is not anected-9 L49 \ColloquialF language fails here- We try to keep the exposition rigorous andincreasingly conciseff but readable- L59 It is advisable to make the studentsprereadeach topic and prepare quesI tions in advanceff to be answeredin the contextof the next lecture- L69 Some topological ideas Lsuch as compactness in terms of open coverings9 are hard on the students- Trial and error led us to emphasize the seI quential approach instead LChapter 4ffx69- \CoveringsF are treated in Ch apter 4ffx7L\starredF9- L7

9 To students unfamiliar with elements of set theory we recommend our

Basic Concepts of Mathematicsfor supplementary reading- LAt Windsorff this text was used for a preparatory yrstIyear oneIsemester course-9 The yrst two chapters and the yrst ten sections ofChapter 3of the present te xt are actually summaries of the corresponding topics of the author4s Basic Concepts of Mathematicsff to which we also relegate such topics as the construction of the real number systemff etc- For many valuable suggestions and corrections we are indebted to H- AtkinI sonff F- Lemireff and T- Traynor- ThanksR

Publisher's Notes

Chapters fiand2andxxfi{fiQof Chapter 3 in the present work are sumI ma ries and extracts from the author4sBasic Concepts of Mathematicsff also published by the Trillia Group- These sections are numbered according to their appearance in the yrst book- Several annotations are used throughout this book: This symbol marks material that can be omitted at yrst reading- )This symbol marks exercises that are of particular importance-

About the Author

El ias Zakon was born in Russia under the czar in OP'"f and he was swept along in the turbulence of the great events of twentiethgcentury EuropeS Zakon studied mathematics and law in Germany and Polandf and later he joined his fatherUs law practice in PolandS Fleeing the approach of the German Army in OP:Of he took his family to Barnaulf Siberiaf wheref with the rest of the populacef they endured yve years of hardshipS The Leningrad Institute of Technology was also evacuated to Barnaul upon the siege of Leningradf and there he met the mathematician IS PS Natansonfi with NatansonUs encourageg mentf Zakon again took up his studies and research in mathematicsS Zakon and his family spent the years from OP:, to OP:P in a refugee camp in Salzburgf Austriaf where he taught himself Hebrewf one of the six or seven languages in which he became suentS In OP:Pf he took his family to the newly created state of Israel and he taught at the Technion in Haifa until OPG,S In Israel he published his yrst research papers in logic and analysisS Throughout his lifef Zakon maintained a love of musicf artf politicsf historyf lawf and especially chessfi it was in Israel that he achieved the rank of chess masterS In OPG,f Zakon moved to CanadaS As a research fellow at the University of Torontof he worked with Abraham RobinsonS In OPG"f he joined the mathematg ics faculty at the University of Windsorf where the yrst degrees in the newly established Honours program in Mathematics were awarded in OP,'S While at Windsorf he continued publishing his research results in logic and analysisS In this postgMcCarthy eraf he often had as his housegguest the proliyc and eccentric mathematician Paul Erd}osf who was then banned from the United States for his political viewsS Erd}os would speak at the University of Windsorf where mathematicians from the University of Michigan and other American universities would gather to hear him and to discuss mathematicsS While at Windsorf Zakon developed three volumes on mathematical analysisf which were bound and distributed to studentsS His goal was to introduce rigorous material as early as possiblefi later courses could then rely on this materialS We are publishing here the latest complete version of the second of these volumesf which was used in a twogsemester class required of all secondg year Honours Mathematics students at WindsorS

Chapter fi

Se t Theory xxfi{3- Sets and Operations on Sets- Quantiyers Asetis a collection of objects of any specied kind. Sets are usually denoted by capitals. The objects belonging to a set are called itselementsormembers. We writex2Aifxis a member ofA, andx62Aif it is not. A=fa, b, c, ...gmeans thatAconsists of the elementsa, b, c, .... In particular,A=fa, bgconsists ofaandb;A=fpgconsists ofpalone. The emptyorvoidset,,, hasnoelements. Equality (=) meanslogical identity. If all members ofAare also inB, we callAasubsetofB(andBasuperset ofA), and writeABorBA. It is an axiom thatthe setsAandBare equal(A=B)if they have the same members, i.e.,

ABandBA.

If, however,ABbutB6A(i.e.,Bhas some elementsnotinA), we callA apropersubset ofBand writeABorBA. \" is called theinclusion relation. Set equality is not aected by theorderin which elements appear. Thus fa, bg=fb, ag. Not so forordered pairs(a, b).1For such pairs, (a, b) = (x, y) i2a=xandb=y, but not ifa=yandb=x. Similarly, fororderedn-tuples, (a1, a2, ..., an) = (x1, x2, ..., xn) iak=xk, k= 1,2, ..., n. We writefxjP(x)gfor \the set of allxsatisfying the conditionP(x)." Similarly,f(x, y)jP(x, y)gis the set of allordered pairsfor whichP(x, y) holds;fx2AjP(x)gis the set of thosexinAfor whichP(x) is true.1

See Problem 6 for a denition.

2Short for?f a?d ??ly ?f; also written().

2Chapter OS Set Theory

For any setsAandBff we deyne theirunionA[BffintersectionA\Bff dilerenceABff andCartesian productLorcross product9ABff as follows: A[Bis the set of all members ofAandBtakentogether: fxjx2Aorx2Bg:3

A\Bis the set of allcommonelements ofAandB:

fx2Ajx2Bg:

ABconsists of thosex2Athat arenotinB:

fx2Ajx62Bg: ABis the set of allordered pairsLx; y9ff withx2Aandy2B: fLx; y9jx2A; y2Bg: SimilarlyffA1A2A?is the set of allorderedn-tuplesLx1; :::; x?9 such thatxk2Akffk= fi;2; :::; n- We writeA?forAA ALnfactors9- AandBare said to bedisjointinA\B=,Lno common elements9- Otherwiseff we say thatAmeetsBLA\B6=,9- Usually all sets involved are subsets of a \master setFSff called thespace- Then we writeXforSXff and callXthecomplementofXLinS9- Various other notations are likewise in use-

Examples.

LetA=ffi;2;3gffB=f2;4g- Then

A[B=ffi;2;3;4gffA\B=f2gffAB=ffi;3gff

IfNis the set of allnaturalsLpositive integers9ff we could also write

A=fx2Njx <4g:

Theorem ∞.

La9A[A=A;A\A=A;

Lb9A[B=B[AffA\B=B\A;

Lc9 LA[B9[C=A[LB[C9; LA\B9\C=A\LB\C9;

Ld9 LA[B9\C= LA\C9[LB\C9;

Le9 LA\B9[C= LA[C9\LB[C9-3

Theword \orD is used in theinclusivesense6 \?orQD means \?orQorbothSD xxO{wS Sets and Operations on SetsS Quantiyers3 T he proof of (d) is sketched in Problem 1. The rest is left to the reader. Because of (c), we may omit brackets inA[B[CandA\B\C; similarly for four or more sets. More generally, we may consider wholefamiliesof sets, i.e., collections of many (possibly innitely many) sets. IfMis such a family, we dene itsunion,?M, to be the set of all elementsx, each belonging toat least oneset of the family. The intersection ofM, denoted?M, consists of thosexthat belong to all sets of the familysimultaneously. Instead, we also write?fXjX2 Mgand?fXjX2 Mg, respectively.

Often we cannumberthe sets of a given family:

A ∞; A?; :::; An; :::: More generally, we may denote all sets of a familyMby some letter (say,X) with indicesiattached to it (the indices may, butneed not, be numbers). The familyMthen is denoted byfXigorfXiji2Ig, whereiis a variable index ranging over a suitable setIof indices (\index notation"). In this case, the union and intersection ofMare denoted by such symbols as ?fXiji2Ig=? iX i=?X i=? i2IX i; ?fXiji2Ig=? iX i=?X i=? i2IX i:

If the indices areintegers, we may write

m n=∞X n;1? n=∞X n;m? n=kX n;etc. ??e??em 2(De Morgan's duality laws).For any setsSandAi(i2I),the following are true: (i)S? iA i=? i(SAi); (ii)S? iA i=? i(SAi): (IfSis the entire space, we may writeAiforSAi,?AiforS?Ai, etc.) Before proving these laws, we introduce some useful notation. L?g?cal Qua?t??e??.From logic we borrow the following abbreviations. \(8x2A):::" means \For each memberxofA, it is true that:::." \(9x2A):::" means \There is at least onexinAsuch that:::." \(9!x2A):::" means \There is auniquexinAsuch that:::."

4Chapter OS Set Theory

The symbols \L8x2A9F and \L9x2A9F are called theuniversaland existential quantiersff respectively- If confusion is ruled outff we simply write \L8x9ffF \L9x9ffF and \L9Rx9F instead- For exampleff if we agree thatmffn denotenaturalsff then \L8n9 L9m9m > nF means \For each naturalnff there is a naturalmsuch thatm > n-F We give some more examples- LetM=fAiji2Igbe an indexed set family- By deynitionffx2?Ai means thatxis inat least oneof the setsAiffi2I- In other wordsffthere is at least one indexi2Isuch thatx2Ai; in symbolsff

L9i2I9x2Ai:

Thus we note that

x2? i?IA iin [L9i2I9x2Ai]:

Similarlyff

x2? iA iin [L8i2I9x2Ai]: Also note thatx =2?Aiinxis innoneof theAiff i-e-ff

L8i9x =2Ai:

Similarlyffx =2?Aiinxfails to be insomeAiff i-e-ff

L9i9x =2Ai:LWhy?9

We now use these remarks to prove Theorem 2Li9- We have to show that S?Aihas the same elements as?LSAi9ff i-e-ff thatx2S?Aiin x2?LSAi9- Butff by our deynitionsff we have x2S?A i()[x2S; x =2?A i] ()L8i9 [x2S; x62Ai] ()L8i9x2SAi ()x2?LSAi9; as required- One proves part Lii9 of Theorem 2 quite similarly- LExerciseR9 We shall now dwell on quantiyers more closely- Sometimes a formulaPLx9 holds not for allx2Aff but only for those with an additional propertyQLx9-

This will be written as

L8x2AjQLx99PLx9;

xxO{wS Sets and Operations on SetsS Quantiyers5 w here the vertical stroke stands for \such that." For example, ifNis again the naturals, then the formula (8x2Njx >3)x4(1) means \for eachx2Nsuch thatx >3, it is true thatx4." In other words, for naturals,x >3 =)x4 (the arrow stands for \implies"). Thus (1) can also be written as (8x2N)x >3 =)x4: In mathematics, we often have to form thenegationof a formula that starts with one or several quantiers. It is noteworthy, then, thateach universal quantier is replaced by an existential one(and vice versa), followed by the negation of the subsequent part of the formula. For example, in calculus, a real numberpis called thelimitof a sequencex1; x2; :::; xn; :::i the following is true: For every real" >0, there is a naturalk(depending on") such that, for all naturaln > k, we havejxnpj< ". If we agree that lower case letters (possibly with subscripts) denote real num- bers, and thatn,kdenote naturals (n; k2N), this sentence can be written as (8" >0) (9k) (8n > k)jxnpj< ":(2) Here the expressions \(8" >0)" and \(8n > k)" stand for \(8"j" >0)" and \(8njn > k)", respectively (such self-explanatory abbreviations will also be used in other similar cases). Now, since (2) states that \for all" >0" something (i.e., the rest of (2)) is true, the negation of (2) starts with \there is an" >0" (for which the rest of the formulafails). Thus we start with \(9" >0)", and form the negation of what follows, i.e., of (9k) (8n > k)jxnpj< ": This negation, in turn, starts with \(8k)", etc. Step by step, we nally arrive at (9" >0) (8k) (9n > k)jxnpj ": Note that herethe choice ofn > kmay depend onk. To stress it, we often writenkforn. Thus the negation of (2) nally emerges as (9" >0) (8k) (9nk> k)jxnkpj ":(3) Theorderin which the quantiers follow each other isessential. For exam- ple, the formula (8n2N) (9m2N)m > n

6Chapter OS Set Theory

L\eachn2Nis exceeded by somem2NF9 is trueff but

L9m2N9 L8n2N9m > n

is false- Howeverff twoconsecutiveuniversal quantiyers Lor twoconsecutive existential ones9 may be interchanged- We briesy write \L8x; y2A9F for \L8x2A9 L8y2A9ffF and \L9x; y2A9F for \L9x2A9 L9y2A9ffF etc- We conclude with an important remark- Theuniversalquantiyer in a forI mula

L8x2A9PLx9

does notimply the existence of anxfor whichPLx9 is true- It is only meant to imply thatthere is noxinAfor whichPLx9fails- The latter is true even ifA=;; we then say that \L8x2A9PLx9F is vacuously true- For exampleff the formula; vBff i-e-ff

L8x2 ;9x2B;

is always trueLvacuously9-

Problems in Set Theory

1.Prove Theorem fi Lshow thatxis in the leftIhand set in it is in the

rightIhand set9- For exampleff for Ld9ff x2LA[B9\C()[x2LA[B9 andx2C] ()[Lx2Aorx2B9;andx2C] ()[Lx2A; x2C9 or Lx2B; x2C9]:

2.Prove that

Li9LA9 =A;

Lii9AvBinBv A-

3.Prove that

AB=A\LB9 = LB9LA9 =[LA9[B]:

Alsoff give three expressions forA\BandA[Bff in terms of complements-

4.Prove the second duality law LTheorem 2Lii99-

xxO{wS Sets and Operations on SetsS Quantiyers7quotesdbs_dbs19.pdfusesText_25

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