Writing proofs in analysis

364 Pages·2016·2.76 MB·English

by  Kane

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
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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.
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