Table Of ContentCOMPANION 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)
