Table Of ContentUniversitext
Sergei Ovchinnikov
Real Analysis:
Foundations
Universitext
Universitext
SeriesEditors
SheldonAxler
SanFranciscoStateUniversity
CarlesCasacuberta
UniversitatdeBarcelona
JohnGreenlees
UniversityofWarwick
AngusMacIntyre
QueenMaryUniversityofLondon
KennethRibet
UniversityofCalifornia,Berkeley
ClaudeSabbah
ÉcolePolytechnique,CNRS,UniversitéParis-Saclay,Palaiseau
EndreSüli
UniversityofOxford
WojborA.Woyczyn´ski,
CaseWesternReserveUniversity
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Sergei Ovchinnikov
Real Analysis: Foundations
SergeiOvchinnikov
DepartmentofMathematics
SanFranciscoStateUniversity
SanFrancisco,CA,USA
ISSN0172-5939 ISSN2191-6675 (electronic)
Universitext
ISBN978-3-030-64700-1 ISBN978-3-030-64701-8 (eBook)
https://doi.org/10.1007/978-3-030-64701-8
MathematicsSubjectClassification:26A03,26A15,26A24,26A42,40A05
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Preface
This book is an introductory text on elementary Real Analysis, focusing on the
modernapproachtothefoundationsofthesubject.
AlldefinitionsandpropositionsofRealAnalysisareformulatedintermsofthe
fieldofrealnumbersR.However,mostofthemdonotuseallthepropertiesofthe
fieldR,butonlyoperationsandrelationsdefiningitasanorderedfield.Thus,one
canattemptto“develop”Analysisoverageneralorderedfield.Yet,thereisaserious
problem with this idea—the proofs of all main theorems of Real Analysis depend
onthecompletenesspropertyofthefieldR.Infact,thereisnowaytobypassthis
problem because, during the past two decades, it has been recognized that many
theoremsofRealAnalysisareintrinsicallyequivalenttothecompletenessproperty
of the underlying ordered field. This brings the foundations of Real Analysis into
the paradigm of the “reverse mathematics” (Stillwell 2019), where axioms are
deduced from theorems. From this perspective, the foundations of Real Analysis
aremultifaceted,consistingofcompletenessaxiomsforthefieldRandpropositions
equivalenttothem.Themainemphasisofthebookistoillustratethisuniquefeature
ofRealAnalysis.
The number systems that are fundamental in mathematics include natural
numbers, integers, and rational and real numbers. In order to keep the book self-
contained,anoverviewofnaturalnumbersandintegersisincludedinAppendixA.
The subject of the first chapter is the field of rational numbers. The first two
sectionsofthischapteraredevotedtoarigorousdevelopment ofthearithmeticof
rationalnumbers.ThenotionofanorderedfieldisintroducedinSection1.3,where
it is shown that the set of rational numbers Q can be endowed with the structure
of an ordered field and, as such, is contained in every ordered field. There are
deficiencies of the number system Q that motivate the need for a larger ordered
field. Algebraic and geometric examples are found in Section 1.4. In Section 1.5,
animportant“analytical”conceptofconvergenceofasequenceinanorderedfield
appearsforthefirsttimeinthebook.Cauchysequences(alsoknownasfundamental
sequences)withtermsinanorderedfieldareimportanttoolsinthedevelopmentof
realnumbersandalsomorebroadlyinanalysis.Thesesequencesareintroducedin
vii
viii Preface
thesamesection,where,inparticular,anexampleofadivergentCauchysequence
inQisgiven,exhibitinganotherlimitationofthefieldofrationalnumbers.
TherealnumbersareintroducedinChapter2viaCantor’smethodofequivalent
Cauchysequencesofrationalnumbers.Anotherconstruction—byDedekindcuts—
isoutlinedinAppendixB.Thechapteropenswithtwopropertiesoforderedfields,
theDedekindCompletenessPropertyandtheCauchyCompletenessProperty.The
central theorem of Section 2.1 establishes an important relation between these
properties: an ordered field is Dedekind complete if and only if it is Cauchy
complete and Archimedean. In the book, an ordered field is called “complete” if
it is Dedekind complete. A rather lengthy Section 2.2 is devoted to the Cauchy
completionofanorderedfieldF,thatis,theconstructionofaminimalorderedfield
containingthegivenfieldinwhicheveryCauchysequenceconverges.Itisproved
that such a field exist, is unique and the given field is a dense subfield of it. The
field R of real numbers is introduced in Section 2.3 as the Cauchy completion of
thefieldQofrationalnumbers.ItisshownthatRistheuniqueDedekindcomplete
field.BasicpropertiesofthefieldRareestablishedinthelastsectionofChapter2,
whereitisprovedthatthesepropertiesareequivalenttothecompletenessproperty.
Thenamesofsomeoftheseproperties—BoundedMonotoneConvergenceProperty,
Bolzano–WeierstrassProperty,andNestedIntervalsProperty—areknowntoevery
studentofRealAnalysis.
Continuousfunctionsonanorderedfieldandtheirpropertiesarethemaintopics
ofChapter3.ThechapteropenswithSection3.1,wherebasictopologicalstructures
onanorderedfieldareintroduced.Openandclosedsubsetsaredefinedandtheusual
properties of these subsets are established. It is proved that every nonempty open
subsetoftherealfieldRisanatmostcountableunionofpairwisedisjointintervals
andthatthispropertycharacterizesthefieldR.Then,theconnectednesspropertyis
introducedandshowntocharacterizethefieldRintheclassoforderedfields.Next,
the compact sets are introduced, and the Heine–Borel theorem of Real Analysis
is proved. It is shown that this theorem characterizes the field R—any incomplete
ordered field contains a non-compact set that is closed and bounded. Section 3.2
focusesonthreefundamentalpropertiesofcontinuousfunctionsonclosedbounded
intervals: the Intermediate Value, Extreme Value, and Fixed Point properties. It is
proved that these properties are satisfied only in the field of real numbers R. In
otherwords,eachofthesepropertiesisequivalenttothecompletenesspropertyof
anorderedfield.UniformlycontinuousfunctionsarediscussedinSection3.3.
ThetheoryofthederivativeforfunctionsonorderedfieldsisgiveninChapter4.
ThechapterstartswithSection4.1coveringthelimitsoffunctions.Thederivativeis
definedinSection4.2,whereitsbasicpropertiesarealsoestablished.InSection4.3,
theMeanValue,ZeroDerivative,Monotonicity,andDarbouxiespropertyofdiffer-
entiablefunctionsareeachshowntobeequivalenttothecompletenesspropertyof
theunderlyingfield.Thechapterendswithasectiononconvexfunctions.
Thetheoriesoftwointegrals—RiemannandDarboux—aretreatedinChapter5.
The concept of a partition of an interval is instrumental in the definitions of
the integrals. This notion and relevant properties of ordered fields are covered
in Section 5.1. The Riemann integral and Riemann integrability are defined in
Preface ix
Section5.2.ItisprovedinthissectionthattheRiemannintegrabilityofcontinuous
functions is equivalent to the completeness property. Some standard properties of
the Riemann integral are established in Section 5.3 for a general ordered field.
StepfunctionsareintroducedinSection5.4,whereitisshownthatthesefunctions
areRiemannintegrableandtheirRiemannintegralsareevaluated.Thesefunctions
are then used in the definitions of the Darboux integral and Darboux integrability
in Section 5.5. The connections between Darboux integrability and existence of
the Darboux integral are established in the same section. Properties of Darboux
integrablefunctionsareexhibitedinthelastsectionofthechapter.
Thelastchapterofthebook—Chapter6—isdevotedtoinfiniteserieswithterms
in an ordered field and covers standard topics adopted from Real Analysis. Basic
definitionsarefoundinSection6.1,whereitisshown,inparticular,thatconvergent
series with terms in “large” ordered fields terminate with zeros. In the rest of the
chapter, it is shown that the validity of standard propositions about series depend
stronglyonpropertiesoftheunderlyingorderedfield.AtheoremofRealAnalysis
claimsthataserieswithrealnon-negativetermsconvergesifandonlyiftheseriesof
partialsumsisbounded.ItisprovedinSection6.2thatthispropositionisequivalent
to the completeness property of the field. Convergence of an alternating series is
establishedinSection6.3forserieswithtermsinaCauchycompleteorderedfield.
Relationsbetweenconvergenceandabsoluteconvergenceofaseriesarestudiedin
Section6.4.ItisshowntherethatanArchimedeanorderedfieldiscompleteifand
onlyifabsoluteconvergenceofseriesimpliestheirconvergence.TheRatioTestis
a powerful tool in Real Analysis. It is shown in Section 6.5 that the assertion of
theRatioTesttheoremisequivalenttothecompletenesspropertyoftheunderlying
field.
There are three appendices in the book. Theories of number systems of natural
numbers and integers are outlined in Appendix A. I include these topics to
complementthetheoriesofrationalandrealnumbersystemspresentedinthebook.
Together, the four number systems form the foundation of Real Analysis. The
constructionofrealnumbersbymeansofDedekindcutsisanalternativetoCantor’s
method used in Chapter 2. This construction is outlined in Appendix B, where it
is also traced back to Euclid’s Elements. Some theorems in the book use special
propertiesoforderedfields.Itwasnotmygoaltocoverthetheoryoforderedfields
inthebook.Thesealgebraicstructuresareintroducedinthebookonthe“needonly”
basis.Inorder tohelp the reader toseethe “big picture,”I include inAppendix C
anoverviewoforderedfields.
The intended audience of this book includes upper-undergraduate and lower-
graduate students interested in the foundations of Real Analysis. Although the
proofs in the book are rather elementary, some mathematical maturity is required.
Inwritingthisbook,Iusedoriginalpublicationsfoundonlyinjournalsandonthe
Internet. There are many propositions in Real Analysis that are equivalent to the
completeness property. The paper (Deveau and Tiesmann 2013/2014) contains a
long list of these propositions together with pertinent references. To keep the size
of the book within reasonable limits and proofs accessible to the target audience,
x Preface
I included in this text only selected results that are relevant to some “traditional”
topicsofelementaryRealAnalysis.
I wish to thank Sheldon Axler for his support of this project and comments
on some parts of the original draft. My special thanks go to Eric Hayashi for
reading drafts of the manuscript carefully and suggesting numerous mathematical
and stylistic corrections. Last but not least, I want to thank my Springer editor
LorettaBartoliniforhercareandassistancethroughoutthepreparationofthisbook.
Berkeley,CA,USA SergeiOvchinnikov
October2020
