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AMS / MAA TEXTBOOKS VOL 70 Welcome to Real Analysis VOL Continuity and Calculus, Distance and Dynamics AMS / MAA TEXTBOOKS 70 Benjamin B. Kennedy Welcome to Real Analysis is designed for use in an introductory undergraduate course in real analysis. Much of the development is in the setting of the general metric space. The book makes substantial dy e use not only of the real line and n -dimensional Euclidean space, but nn e K also sequence and function spaces. Proving and extending results aul P from single-variable calculus provides motivation throughout. The esy of more abstract ideas come to life in meaningful and accessible appli- ourt W c o cations. For example, the contraction mapping principle is used to ot e h P l prove an existence and uniqueness theorem for solutions of ordinary c o differential equations and the existence of certain fractals; the continuity of the integration B m operator on the space of continuous functions on a compact interval paves the way for some e n results about power series. ja e m t The exposition is exceedingly clear and well-motivated. There are a wide variety of exercises i o n and many pedagogical innovations. For example, each chapter includes Reading Questions B R . so that students can check their understanding. In addition to the standard material in a ﬁ rst K e e a real analysis course, the book contains two concluding chapters on dynamical systems and n l fractals as an illustration of the power of the theory developed. ne A d n y a l y s i s For additional information and updates on this book, visit www.ams.org/bookpages/text-70 A M S / M A A P TEXT/70 R E S S 4-Color Process 374 pages on 50lb stock • Softcover • Spine 3/4” Welcome to Real Analysis: Continuity and Calculus, Distance and Dynamics AMS/MAA TEXTBOOKS VOL 70 Welcome to Real Analysis: Continuity and Calculus, Distance and Dynamics Benjamin B. Kennedy MAATextbooksEditorialBoard WilliamR.Green,Co-Editor SuzanneLynneLarson,Co-Editor PaulT.Allen MarkBollman SusanCrook HughN.Howards WilliamJohnston EmekKose MichaelJ.McAsey ThomasC.Ratliff PamelaRichardson RonTaylor RuthVanderpool ErikaWard ElizabethWilcox 2020MathematicsSubjectClassification.Primary26-01,37-01,39-01. Foradditionalinformationandupdatesonthisbook,visit www.ams.org/bookpages/text-70 LibraryofCongressCataloging-in-PublicationData Names:Kennedy,BenjaminB.,author. Title:Welcometorealanalysis:continuityandcalculus,distanceanddynamics/BenjaminB.Kennedy. Description: Providence,RhodeIsland: MAAPress,animprintoftheAmericanMathematicalSociety, [2021]|Series: AMS/MAAtextbooks,2577-1205;Volume70|Includesbibliographicalreferencesand index. Identifiers:LCCN2021036922|ISBN9781470464547(paperback)|978-1-4704-6847-7(ebook) Subjects:LCSH:Mathematicalanalysis.|Functionsofrealvariables.|AMS:Realfunctions–Instructional exposition(textbooks,tutorialpapers,etc.).|Dynamicalsystemsandergodictheory–Instructionalexpo- sition(textbooks,tutorialpapers,etc.).|Differenceandfunctionalequations–Instructionalexposition (textbooks,tutorialpapers,etc.). Classification:LCCQA300.K462021|DDC515–dc23 LCrecordavailableathttps://lccn.loc.gov/2021036922 Copyingandreprinting. 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VisittheAMShomepageathttps://www.ams.org/ 10987654321 272625242322 ForSpencerandPaul Contents Preface xi 0 WhereWe’reStartingandWhereWe’reGoing 1 1 EssentialTools 5 1.1 Setsandstatements 5 1.2 Functions 9 1.3 Countabilityanduncountability 18 1.4 Induction 23 1.5 Orderintherealline 26 1.6 Somevitalinequalities 32 1.7 Exercises 36 2 MetricSpaces 41 2.1 Thedefinitionofametricspace 41 2.2 Importantmetricsinℝ𝑛 45 2.3 Openballsandopensetsinmetricspaces 48 2.4 Closedsetsandlimitpoints 53 2.5 Interior,closure,andboundary 59 2.6 Densesubsets 62 2.7 Equivalentmetrics 63 2.8 Normedvectorspaces 68 2.9 Abriefnoteaboutconventions 71 2.10 Exercises 71 3 Sequences 77 3.1 Convergence 78 3.2 Discretedynamicalsystems 87 3.3 Sequencesandlimitpoints 92 3.4 Algebraictheoremsforsequences 95 3.5 Subsequences 101 3.6 Completeness 107 3.7 Thecontractionmappingprinciple 110 3.8 Setsofsequencesasmetricspaces 119 3.9 Exercises 125 4 Continuity 131 4.1 Thedefinitionofcontinuity 131 vii viii Contents 4.2 Equivalentformulationsofcontinuity 142 4.3 Continuity and limit theorems for scalar- valuedfunctions 146 4.4 Continuityandproductsofmetricspaces 150 4.5 Uniformcontinuity 154 4.6 Themetricspace𝐶([𝑎,𝑏],ℝ) 160 4.7 Anapplicationtofunctionalequations 167 4.8 Exercises 170 5 CompactnessandConnectedness 177 5.1 Basicdefinitionsandresultsoncompactness 177 5.2 Thenestedsetpropertyforcompactsets 181 5.3 Compactnessandcontinuity 183 5.4 Otherfactsaboutcompactness 185 5.5 Connectedness 191 5.6 Periodicpointsofmapsonintervals 195 5.7 Injective continuous functions defined on intervals 200 5.8 Exercises 202 6 TheDerivative 209 6.1 Thedefinitionofthederivative 209 6.2 Differentiationrules 218 6.3 Applicationsofthederivative 222 6.4 Exercises 229 7 TheRiemannIntegral 235 7.1 Partitionsandthedefinitionoftheintegral 235 7.2 Basicpropertiesoftheintegral 243 7.3 Thefundamentaltheoremofcalculus 247 7.4 Ordinarydifferentialequations 253 7.5 Exercises 257 8 SequencesofFunctions 261 8.1 Infiniteseries 261 8.2 Powerseries 267 8.3 HigherderivativesandTaylorpolynomials 275 8.4 Differentiationandintegrationofsequencesoffunctions 281 8.5 Theexponentialfunction 287 8.6 Compactsubsetsin𝐶[𝑎,𝑏] 290 8.7 Exercises 295 9 ChaosinDiscreteDynamicalSystems 301 9.1 Thedefinitionofchaos 302 9.2 Semiconjugacy 310 9.3 Subshiftsoffinitetype 314 9.4 Itinerariesandpiecewiseexpandingmaps 319 9.5 Adynamicalsystemwithadenseorbitbutnoperiodicpoints 331 9.6 Exercises 337