UTM - Understanding Analysis - Stephen Abbott - Solutions Manual

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1、Author’snoteWhatbeganasadesiretosketchoutasimple“answerkey”fortheproblemsinUnderstandingAnalysishasinevitablyevolvedintosomethingabitmoreambitious.AsIwasgeneratingsolutionsfortheover350exercisesinthetext,Ifoundmyselfaddingregularcommentaryoncommonpitfallsandstrategiesthatfrequentlyarise.

2、Mysenseisthatthismanualshouldbeausefulsupplementtoinstructorsteachingacourseortoindividualsengagedinanindependentstudy.Aswiththetextbookitself,Itriedtowritewiththeintroductorystudentfirmlyinmind.Inmyteachingofanalysis,Ihavecometounderstandthestrongcorrelationbetweenhowstudentslearnanalysi

3、sandhowtheywriteit.AfinalgoalIhaveforthesenotesistoillustratebyexamplehowtheformandgrammarofawrittenargumentareintimatelyconnectedtotheclarityofaproofand,ultimately,toitsvalidity.IwouldliketothankformerstudentsCarrickDetweiller,KatherineOtt,YaredGurmu,andYuqiuJiangfortheirconsiderablehelp

4、withapreliminarydraft.IwouldalsoliketothankthereadersofUnderstandingAnalysisforthemanycommentsIhavereceivedaboutthetext.Especiallyappreciatedaretheconstructivesuggestionsaswellasthepointerstoerrorsoffact,andIwelcomemoreofthesame.Middlebury,VermontStephenAbbottMay2004vContentsAuthor’snote

5、v1TheRealNumbers1p1.1Discussion:TheIrrationalityof2.................11.2SomePreliminaries..........................11.3TheAxiomofCompleteness.....................61.4ConsequencesofCompleteness...................81.5Cantor’sTheorem..........................142SequencesandSeries192.1Discus

6、sion:RearrangementsofInfiniteSeries...........192.2TheLimitofaSequence.......................192.3TheAlgebraicandOrderLimitTheorems.............212.4TheMonotoneConvergenceTheoremandaFirstLookatInfiniteSeries.............................252.5SubsequencesandtheBolzano–WeierstrassTheorem.....

7、..292.6TheCauchyCriterion........................312.7PropertiesofInfiniteSeries.....................332.8DoubleSummationsandProductsofInfiniteSeries........393BasicTopologyofR453.1Discussion:TheCantorSet.....................453.2OpenandClosedSets........................453