Writing proofs in analysis PDF
Preview Writing proofs in analysis
Jonathan M. Kane Writing Proofs in Analysis Writing Proofs in Analysis Jonathan M. Kane Writing Proofs in Analysis 123 JonathanM.Kane DepartmentofMathematics UniversityofWisconsin-Madison Madison,WI,USA ISBN978-3-319-30965-1 ISBN978-3-319-30967-5 (eBook) DOI10.1007/978-3-319-30967-5 LibraryofCongressControlNumber:2016936668 ©SpringerInternationalPublishingSwitzerland2016 Thisworkissubjecttocopyright.AllrightsarereservedbythePublisher,whetherthewholeorpartof thematerialisconcerned,specificallytherightsoftranslation,reprinting,reuseofillustrations,recitation, broadcasting,reproductiononmicrofilmsorinanyotherphysicalway,andtransmissionorinformation storageandretrieval,electronicadaptation,computersoftware,orbysimilarordissimilarmethodology nowknownorhereafterdeveloped. Theuseofgeneraldescriptivenames,registerednames,trademarks,servicemarks,etc.inthispublication doesnotimply,evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant protectivelawsandregulationsandthereforefreeforgeneraluse. Thepublisher,theauthorsandtheeditorsaresafetoassumethattheadviceandinformationinthisbook arebelievedtobetrueandaccurateatthedateofpublication.Neitherthepublishernortheauthorsor theeditorsgiveawarranty,expressorimplied,withrespecttothematerialcontainedhereinorforany errorsoromissionsthatmayhavebeenmade. Printedonacid-freepaper ThisSpringerimprintispublishedbySpringerNature TheregisteredcompanyisSpringerInternationalPublishingAGSwitzerland Tothe memoryofSylvanBurgstaller,Duane E.Anderson,andespeciallyJamesL.Nelson who,attheUniversityofMinnesotaDuluth, taughtmethefundamentals ofwritingproofs inanalysis. Acknowledgments I wish to thank Natalya St. Clair for her excellent work creating the illustrations appearinginthistextbook.Shetookmycrudesketchesandvagueideasandturned themintopleasingartworkandinstructivediagrams.IalsowishtothankDanielM. Kane, Alan Gluchoff, Thomas Drucker, and Walter Stromquist for their insightful commentsaboutthepresentation,content,andcorrectnessofthetext. vii Preface After learning to solve many types of problems such as those found in the first courses in Algebra, Geometry, Trigonometry, and Calculus, mathematics students areusuallyexposedtoa“transition”coursewheretheyareexpectedtowriteproofs ofvarioustheorems.Itaughtsuchacourseforadozenyearsandwasneversatisfied with the textbooks available for that course. Although such textbooks often teach thefundamentalsoflogic(conditionals,biconditionals,negations,truthtables)and givesomecommonproofstrategiessuchasmathematicalinduction,thetextbooks failedtoteachwhatastudentneedstobethinkingaboutwhentryingtoconstructa proof.Manyofthesebookspresentagreatnumberofwell-writtenproofsandthen askstudentstowriteproofsofsimilarstatementsinthehopethatthestudentswill beabletomimicwhattheyhaveseen.Someofthesebooksarealsodesignedtobe usedasanintroductorytextbookinAnalysis,AbstractAlgebra,Topology,Number Theory,orDiscreteMathematics,and,assuch,theyconcentratemoreonexplaining the fundamentals of those topic areas than on the fundamentals of writing good proofs. This Book Is Not Your Traditional Transition Textbook The goal of this book is to give the student precise training in the writing of proofs by explaining what elementsmakeupacorrectproof,byteachinghowtoconstructanacceptableproof, byexplainingwhatthestudentissupposedtobethinkingaboutwhentryingtowrite a proof, and by warning about pitfalls that result in incorrect proofs. In particular, thisbookwaswrittenwiththefollowingdirectives: • Unlike many transition books which do not give enough instruction about how towriteproofs,mostoftheproofspresentedinthistextareprecededbydetailed explanationsdescribingthethoughtprocessonegoesthroughwhenconstructing the proof. Then a good proof is given that incorporates the elements of that discussion. • Forproofsthatsharethesamegeneralstructuresuchastheproofoflimf.x/DL x!a forvariousfunctions,prooftemplatesareprovidedthatgiveagenericapproach towritingthattypeofproof. ix
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