Table Of Content

COMPANION NOTES A Working Excursion to Accompany Baby Rudin Evelyn M. Silvia1 April 1, 1999 1hc EvelynM.Silvia1999 ii Contents Preface vii 0.0.1 AbouttheOrganizationoftheMaterial . . . . . . . . . . . viii 0.0.2 AbouttheErrors . . . . . . . . . . . . . . . . . . . . . . . viii 1 TheFieldofRealsandBeyond 1 1.1 Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 OrderedFields . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2.1 Special Subsets ofanOrderedField . . . . . . . . . . . . . 15 1.2.2 BoundingProperties . . . . . . . . . . . . . . . . . . . . . 17 1.3 TheReal Field. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.3.1 DensityProperties oftheReals . . . . . . . . . . . . . . . . 27 1.3.2 Existence of nthRoots . . . . . . . . . . . . . . . . . . . . 29 1.3.3 TheExtendedReal NumberSystem . . . . . . . . . . . . . 33 1.4 TheComplexField . . . . . . . . . . . . . . . . . . . . . . . . . . 34 1.4.1 ThinkingComplex . . . . . . . . . . . . . . . . . . . . . . 39 1.5 ProblemSetA . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43 2 FromFinitetoUncountableSets 49 2.1 SomeReviewofFunctions . . . . . . . . . . . . . . . . . . . . . . 49 2.2 AReviewofCardinal Equivalence . . . . . . . . . . . . . . . . . . 54 2.2.1 DenumerableSets andSequences . . . . . . . . . . . . . . 61 2.3 ReviewofIndexedFamiliesofSets . . . . . . . . . . . . . . . . . 62 2.4 CardinalityofUnionsOverFamilies . . . . . . . . . . . . . . . . . 65 2.5 TheUncountableReals . . . . . . . . . . . . . . . . . . . . . . . . 69 2.6 ProblemSetB . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 3 METRICSPACESandSOME BASICTOPOLOGY 73 3.1 Euclideann-space . . . . . . . . . . . . . . . . . . . . . . . . . . . 73 iii iv CONTENTS 3.2 MetricSpaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 3.3 Point SetTopologyonMetric Spaces . . . . . . . . . . . . . . . . . 82 3.3.1 Complements andFamiliesofSubsetsofMetricSpaces . . 87 3.3.2 OpenRelativetoSubsets ofMetricSpaces . . . . . . . . . 96 3.3.3 Compact Sets . . . . . . . . . . . . . . . . . . . . . . . . . 98 3.3.4 Compactness inEuclideann-space . . . . . . . . . . . . . . 105 3.3.5 ConnectedSets . . . . . . . . . . . . . . . . . . . . . . . . 111 3.3.6 PerfectSets . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.4 ProblemSetC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118 4 Sequences andSeries–FirstView 123 4.1 Sequences andSubsequences inMetricSpaces . . . . . . . . . . . 124 4.2 CauchySequences inMetricSpaces . . . . . . . . . . . . . . . . . 135 4.3 Sequences inEuclideank-space . . . . . . . . . . . . . . . . . . . 139 4.3.1 UpperandLowerBounds . . . . . . . . . . . . . . . . . . 145 4.4 SomeSpecial Sequences . . . . . . . . . . . . . . . . . . . . . . . 149 4.5 Series ofComplexNumbers . . . . . . . . . . . . . . . . . . . . . 152 4.5.1 Some(Absolute)Convergence Tests . . . . . . . . . . . . . 156 4.5.2 AbsoluteConvergenceandCauchyProducts . . . . . . . . 169 4.5.3 Hadamard Products and Series with Positive and Negative Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173 4.5.4 DiscussingConvergence . . . . . . . . . . . . . . . . . . . 176 4.5.5 Rearrangements of Series . . . . . . . . . . . . . . . . . . 177 4.6 ProblemSetD . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178 5 Functions onMetricSpacesandContinuity 183 5.1 Limits ofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.2 Continuous Functions onMetricSpaces . . . . . . . . . . . . . . . 197 5.2.1 ACharacterization ofContinuity . . . . . . . . . . . . . . . 200 5.2.2 ContinuityandCompactness . . . . . . . . . . . . . . . . . 203 5.2.3 ContinuityandConnectedness . . . . . . . . . . . . . . . . 206 5.3 Uniform Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . 208 5.4 Discontinuities andMonotonicFunctions . . . . . . . . . . . . . . 211 5.4.1 Limits ofFunctionsintheExtendedRealNumberSystem . 221 5.5 ProblemSetE . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225 CONTENTS v 6 Differentiation: Our FirstView 229 6.1 TheDerivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229 6.1.1 FormulasforDifferentiation . . . . . . . . . . . . . . . . . 242 6.1.2 RevisitingAGeometricInterpretationforthe Derivative . . 243 6.2 TheDerivativeandFunctionBehavior . . . . . . . . . . . . . . . . 244 6.2.1 Continuity(orDiscontinuity)ofDerivatives . . . . . . . . . 250 6.3 TheDerivativeandFindingLimits . . . . . . . . . . . . . . . . . . 251 6.4 InverseFunctions . . . . . . . . . . . . . . . . . . . . . . . . . . . 254 6.5 Derivatives ofHigherOrder . . . . . . . . . . . . . . . . . . . . . 258 6.6 DifferentiationofVector-ValuedFunctions . . . . . . . . . . . . . . 262 6.7 ProblemSetF . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 266 7 Riemann-StieltjesIntegration 275 7.1 RiemannSums andIntegrability . . . . . . . . . . . . . . . . . . . 277 7.1.1 PropertiesofRiemann-Stieltjes Integrals . . . . . . . . . . 295 7.2 RiemannIntegralsandDifferentiation . . . . . . . . . . . . . . . . 307 7.2.1 SomeMethods ofIntegration . . . . . . . . . . . . . . . . . 311 7.2.2 TheNatural LogarithmFunction . . . . . . . . . . . . . . . 314 7.3 IntegrationofVector-ValuedFunctions . . . . . . . . . . . . . . . . 315 7.3.1 Recti¿ableCurves . . . . . . . . . . . . . . . . . . . . . . 318 7.4 ProblemSetG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321 8 Sequences andSeries ofFunctions 325 8.1 Pointwise andUniformConvergence . . . . . . . . . . . . . . . . . 326 8.1.1 Sequences ofComplex-ValuedFunctionsonMetricSpaces . 334 8.2 Conditions forUniformConvergence . . . . . . . . . . . . . . . . 335 8.3 PropertyTransmissionandUniform Convergence . . . . . . . . . . 339 8.4 FamiliesofFunctions . . . . . . . . . . . . . . . . . . . . . . . . . 349 8.5 TheStone-WeierstrassTheorem . . . . . . . . . . . . . . . . . . . 363 8.6 ProblemSetH . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363 9 SomeSpecialFunctions 369 9.1 Power Series OvertheReals . . . . . . . . . . . . . . . . . . . . . 369 9.2 SomeGeneral ConvergenceProperties . . . . . . . . . . . . . . . . 379 9.3 DesignerSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . 388 9.3.1 AnotherVisitWiththeLogarithmFunction . . . . . . . . . 393 9.3.2 ASeries DevelopmentofTwoTrigonometricFunctions . . 394 9.4 Series from Taylor’s Theorem . . . . . . . . . . . . . . . . . . . . 397 vi CONTENTS 9.4.1 SomeSeries ToKnow& Love . . . . . . . . . . . . . . . . 399 9.4.2 Series From OtherSeries . . . . . . . . . . . . . . . . . . . 404 9.5 FourierSeries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 408 9.6 ProblemSetI . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416 Preface These notes have been prepared to assist students who are learning Advanced Cal- culus/Real Analysis for the ¿rst time in courses or self-study programs that are using the text Principles of Mathematical Analysis (3rd Edition) by Walter Rudin. References to page numbers or general location of results that mention “our text” are always referringtoRudin’s book. Thenotes are designedto (cid:14) encourageorengenderaninteractiveapproachtolearningthematerial, (cid:14) providemoreexamples at theintroductorylevel, (cid:14) offersomealternative viewsofsomeoftheconcepts,and (cid:14) draw a clearer connection to the mathematics that is prerequisite to under- standingthedevelopment of themathematical analysis. Onourcampus,theonlyprerequisitesontheAdvancedCalculuscourseinclude anintroductiontoabstractmathematics(MAT108)courseandelementarycalculus. Consequently, the terseness of Rudin can require quite an intellectual leap. One needs to pause and re(cid:192)ect on what is being presented(cid:30) stopping to do things like drawpictures,constructexamplesorcounterexamplesfortheconceptsthearebeing discussed, and learn the de¿nitions is an essential part of learning the material. These Companion Notes explicitly guide the reader/participant to engage in those activities. With more math experience or maturity such behaviors should become a natural part of learning mathematics. A math text is not a novel(cid:30) simply reading it from end to end is unlikely to give you more than a sense for the material. On the other hand, the level of interaction that is needed to successfully internalize an understanding of the material varies widely from person to person. For optimal bene¿t from the combined use of the text (Rudin) and the Companion Notes ¿rst read the section of interest as offered in Rudin, then work through the relevant vii viii PREFACE section or sections in the Companion Notes, and follow that by a more interactive reviewofthe sectionfromRudinwithwhichyoustarted. One thing that should be quite noticeable is the higher level of detail that is offered for many of the proofs. This was done largely in response to our campus prerequisite for the course. Because most students would have had only a brief exposure to some of the foundational material, a very deliberate attempt has been made to demonstrate how the prerequisite material that is usually learned in an introduction to abstract mathematics course is directly applied to the development of mathematical analysis. You always have the elegant, “no nonsense” approach available in the text. Learn to pick and choose the level of detail that you need accordingtoyourownpersonal mathematical needs. 0.0.1 About the Organization of the Material ThechaptersandsectionsoftheCompanionNotesarenotidenticallymatchedwith their counterparts in the text. For example, the material related to Rudin’s Chapter 1 can be found in Chapter 1, Chapter 2 and the beginning of Chapter 3 of the Companion Notes. There are also instances of topic coverage that haven’t made it into the Companion Notes(cid:30) the exclusions are due to course timing constraints and notstatements concerningimportanceofthetopics. 0.0.2 About the Errors Of course, there are errors! In spite of my efforts to correct typos and adjust errors astheyhavebeenreportedtomebymystudents,Iamsurethattherearemoreerrors to be found and I hope for the assistance of students who ¿nd things that look like errorsastheyworkthroughthenotes. Ifyouencountererrorsorthingsthatlooklike errors, please sent me a brief email indicatingthe nature of the problem. My email address is [email protected]. Thank you in advance for any comments, corrections,and/orinsights thatyoudecidetoshare. Chapter 1 The Field of Reals and Beyond Our goal with this section is to develop (review) the basic structure that character- izes the set of real numbers. Much of the material in the ¿rst section is a review of properties that were studied in MAT108(cid:30) however, there are a few slight differ- ences in the de¿nitions for some of the terms. Rather than prove that we can get fromthepresentationgivenbytheauthorofourMAT127Atextbooktotheprevious set of properties, with one exception, we will base our discussion and derivations on the new set. As a general rule the de¿nitions offered in this set of Compan- ion Notes will be stated in symbolic form(cid:30) this is done to reinforce the language of mathematics and to give the statements in a form that clari¿es how one might prove satisfaction or lack of satisfaction of the properties. YOUR GLOSSARIES ALWAYS SHOULD CONTAIN THE(INSYMBOLIC FORM) DEFINITION AS GIVEN IN OUR NOTES because that is the form that will be required for suc- cessful completion of literacy quizzes and exams where such statements may be requested. 1.1 Fields RecallthefollowingDEFINITIONS: (cid:14) TheCartesian product oftwosets A and B,denotedby A(cid:24) B,is (cid:10)(cid:16)a(cid:29)b(cid:17) : a + AFb + B(cid:12). 1 2 CHAPTER1. THEFIELDOFREALSANDBEYOND (cid:14) Afunctionh from A into B isa subset of A(cid:24) B suchthat (i)(cid:16)1a(cid:17)[a + A " (cid:16)2b(cid:17)(cid:16)b + B F(cid:16)a(cid:29)b(cid:17) + h(cid:17)](cid:30)i.e.,domh (cid:7) A,and (ii)(cid:16)1a(cid:17)(cid:16)1b(cid:17)(cid:16)1c(cid:17)[(cid:16)a(cid:29)b(cid:17) + h F(cid:16)a(cid:29)c(cid:17) + h " b (cid:7) c](cid:30)i.e.,h issingle-valued. (cid:14) Abinaryoperationonaset A is afunctionfrom A(cid:24) A into A. (cid:14) A ¿eld is an algebraic structure, denoted by (cid:16)I(cid:29)(cid:5)(cid:29)(cid:23)(cid:29)e(cid:29) f(cid:17), that includes a set of objects, I, and two binary operations, addition (cid:16)(cid:5)(cid:17) and multiplication (cid:16)(cid:23)(cid:17), that satisfy the Axioms of Addition, Axioms of Multiplication, and the DistributiveLawas describedinthefollowinglist. (A) AxiomsofAddition((cid:16)I(cid:29)(cid:5)(cid:29)e(cid:17)isacommutativegroupunderthebinary operationof addition(cid:16)(cid:5)(cid:17)withtheadditive identitydenotedbye)(cid:30) (A1) (cid:5) : I(cid:24)I (cid:26) I (A2) (cid:16)1x(cid:17)(cid:16)1y(cid:17)(cid:16)x(cid:29)y + I " (cid:16)x (cid:5) y (cid:7) y (cid:5)x(cid:17)(cid:17)(commutativewithrespect toaddition) b d ec (A3) (cid:16)1x(cid:17)(cid:16)1y(cid:17)(cid:16)1z(cid:17) x(cid:29)y(cid:29)z + I " (cid:16)x (cid:5) y(cid:17)(cid:5)z (cid:7) x (cid:5)(cid:16)y (cid:5)z(cid:17) (asso- ciativewithrespect toaddition) (A4) (cid:16)2e(cid:17)[e + IF(cid:16)1x(cid:17)(cid:16)x + I "x (cid:5)e (cid:7) e(cid:5)x (cid:7) x(cid:17)](additiveidentity property) (A5) (cid:16)1x(cid:17)(cid:16)x + I " (cid:16)2(cid:16)(cid:19)x(cid:17)(cid:17)[(cid:16)(cid:19)x(cid:17) + IF(cid:16)x (cid:5)(cid:16)(cid:19)x(cid:17) (cid:7) (cid:16)(cid:19)x(cid:17)(cid:5)x (cid:7) e(cid:17)](cid:17) (additiveinverseproperty) (M) Axioms of Multiplication ((cid:16)I(cid:29)(cid:23)(cid:29) f(cid:17) is a commutative group under the binary operation of multiplication (cid:16)(cid:23)(cid:17) with the multiplicative identity denotedby f)(cid:30) (M1) (cid:23) : I(cid:24)I (cid:26) I (M2) (cid:16)1x(cid:17)(cid:16)1y(cid:17)(cid:16)x(cid:29)y + I " (cid:16)x (cid:23) y (cid:7) y (cid:23)x(cid:17)(cid:17) (commutative with respect tomultiplication) b d ec (M3) (cid:16)1x(cid:17)(cid:16)1y(cid:17)(cid:16)1z(cid:17) x(cid:29)y(cid:29)z + I " (cid:16)x (cid:23) y(cid:17)(cid:23)z (cid:7) x (cid:23)(cid:16)y (cid:23)z(cid:17) (associative withrespect tomultiplication) d e (M4) (cid:16)2f(cid:17) f + IF f /(cid:7) eF(cid:16)1x(cid:17)(cid:16)x + I " x (cid:23) f (cid:7) f (cid:23)x (cid:7) x(cid:17) (mul- tiplicativeidentityproperty) (M5) (cid:16)1x(cid:17)(cid:16)x + I(cid:19)db(cid:10)eb(cid:12) "ccb ce 2 x(cid:19)1 x(cid:19)1 + IF(cid:16)x (cid:23)(cid:16)x(cid:19)1(cid:17) (cid:7) (cid:16)x(cid:19)1(cid:17)(cid:23)x (cid:7) f (cid:17) (multiplicative inverseproperty)