A Story of Real Analysis

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1、HowWeGotfromTheretoHere:AStoryofRealAnalysisRobertRogersEugeneBomanSUNYFredoniaPennsylvaniaStateUniversityHarrisburgcampus2014RobertRogers&EugeneBomanISBN:978-1-312-343869-1(print)ISBN:978-0-9897226-6-7(ebook)ThisworkislicensedunderaCreativeCommonsAttribut

2、ion-NonCommercial-ShareAlike3.0UnportedLicense.PublishedbyOpenSUNYTextbooks,MilneLibrary(IITGPI)StateUniversityofNewYorkatGeneseo,Geneseo,NY14454AboutthisTextbookThetypicalintroductoryrealanalysistextstartswithananalysisoftherealnumbersystemandusesthistode

3、velopthedefinitionofalimit,whichisthenusedasafoundationforthedefinitionsencounteredthereafter.Whilethisiscertainlyareasonableapproachfromalogicalpointofview,itisnothowthesubjectevolved,norisitnecessarilythebestwaytointroducestudentstotherigorousbuthighlyno

4、n-intuitivedefinitionsandproofsfoundinanalysis.Thisbookproposesthataneffectivewaytomotivatethesedefinitionsistotelloneofthestories(therearemany)ofthehistoricaldevelopmentofthesubject,fromitsintuitivebeginningstomodernrigor.Thedefinitionsandtechniquesaremot

5、ivatedbytheactualdif-ficultiesencounteredbytheintuitiveapproachandarepresentedintheirhistoricalcontext.However,thisisnotahistoryofanalysisbook.Itisanintroductoryanalysistextbook,presentedthroughthelensofhistory.Assuch,itdoesnotsimplyinserthistoricalsnippet

6、stosupplementthematerial.Thehistoryisanintegralpartofthetopic,andstudentsareaskedtosolveproblemsthatoccurastheyariseintheirhistoricalcontext.Thisbookcoversthemajortopicstypicallyaddressedinanintroductoryundergraduatecourseinrealanalysisintheirhistoricalord

7、er.Writtenwiththestudentinmind,thebookprovidesguidancefortransforminganintuitiveunderstandingintorigorousmathematicalarguments.Forexample,inadditiontomoretraditionalproblems,majortheoremsareoftenstatedandaproofisoutlined.Thestudentisthenaskedtofillinthemis

8、singdetailsasahomeworkproblem.AbouttheAuthorsRobertRogersRobertRogersreceivedhisBSinMathematicswithCertificationinSecondaryEducationfromBuffaloStateCollegein1979.HeearnedhisMSinMathematicsfromSyracuseUniversi