Pep TalkNotationLogicSetsFunctions and MapsComposition and InverseMathematical InductionInfinite SetsAxioms for the Real NumbersDistanceLimitsConvergence of SequencesContinuityOpen Sets and Closed SetsConnected SetsCompact SetsDerivativesThe IntegralTaylor's FormulaSeriesUniform ConvergencePower Series - IAnalytic FunctionsPower Series - IIThe Heat EquationFourier Series - IFourier Series - IIUniqueness of the Real NumbersAdditional ProblemsMath 521 – Spring 2010521May 12, 2010Contents1 Pep Talk 22 Notation 53 Logic 64 Sets 85 Functions and Maps 116 Composition and Inverse 137 Mathematical Induction 158 Infinite Sets 169 Axioms for the Real Numbers 1810 Distance 2211 Limits 2412 Convergence of Sequences 2613 Continuity 3214 Open Sets and Closed Sets 34115 Connected Sets 3716 Compact Sets 3917 Derivatives 4318 The Integral 4419 Taylor’s Formula 4920 Series 5021 Uniform Convergence 5622 Power Series - I 5923 Analytic Functions 6124 Power Series - II 6425 The Heat Equation 6626 Fourier Series - I 6827 Fourier Series - II 68A Uniqueness of the Real Numbers 68B Additional Problems 711 Pep TalkWhat does not kill me, makes me stronger.Friedrich Nietzsche (1844 - 1900)in The Twilight of the Idols (1899)It is generally acknowledged in the math department that the two hardestundergraduate courses we teach are 222 and 521. The former is hard because2for many students it is their first college math course, and although they didwell in high school, they are unprepared for what is expected in college. Thelatter is hard for many reasons. For many students it is the first course wherethey are expected to understand and write proofs. (See 3.4.) There are a lotof hard words like ‘if’, ’for all’, ’there exists’, (see Chapter 3) and the orderin which they appear is crucial (see 13.3). Inequalities (see 9.2) play a crucialrole. The language of set theory (see Chapter 4) is used heavily and somestudents have a hard time visualizing what set is. On top of all this thereis some strange lingo like ’connected’, ’compact, ’open cover’, etc. The firsttime you think you understand and discover that I disagree you may welldecide that it is easier to get hit by a truck.The subject is hard. Calculus was invented by Leibniz and Newton inthe seventeenth century, but the ideas presented in these notes were not fullydeveloped until the early twentieth century. Some of the best mathematicalminds in human history contributed to the subject. But you have one ad-vantage over these luminaries: a teacher who understands the subject andwants to help you understand.Given the task before you, why should you make the effort? The abovequote of Nietzsche may inspire you. The ability to reason carefully willserve you well in any future endeavor. You may be driven (like I am) tounderstand the patterns in mathematical reasoning. These patterns help ussee the similarities between apparently different problems and enable us touse our understanding of one problem to solve the others. Finally, you mayactually need to use the material someday. Many mathematical problemshave no explicit solution: we must prove that there is a solution and usethat proof both to guide us to an approximate solution and to estimate theaccuracy of that solution. If we get to the end of these notes in this courseyou will get a glimpse of how this works when we solve the heat equation inproblem 25.2.Here are some tips:1. Read. It is a bad idea to try to learn just from lectures. A really goodlecturer can make the material look easy, but just listening is rarelyenough to lead to true understanding.2. Focus on definitions. It is impossible to understand why a compactset is closed and bounded if you don’t know what the words mean.Use the index and cross references in these notes or the text you are3reading to recall definitions and previous theorems. If you are readingthese notes on a computer and have a sufficiently recent version ofAdobe Acrobat Reader or some other pdf viewer, you will notice thatthis document contains embedded hypertext links. Some viewers evenallow text searches and links to URLs. (If you are online try clickingon the link to the official description below.) This makes the task ofjumping around the text much easier.3. Be active not passive. As you read or listen, make up examples to testyour understanding. When you see a definition make up an example ofsomething which satisfies it and something which doesn’t.4. Ask questions. If you can formulate a question when you are confused,you may discover that you become unconfused and don’t need to askthe question. But don’t hesitate to ask me questions in class or officehours.5. Be concise when you write. Excess verbiage is hard to follow and mayconceal confusion.6. Don’t fall behind. If you postpone understanding till the night beforethe exam, you may find that you are still lost. For each lecture therewill be a short prequiz on Moodle (see below) designed to get you toread the material before I lecture on it. In addition there will be ashort post quiz almost every day (I hope).We will use an online course management system called Moodle. To useit go to https://www.math.wisc.edu/moodle and click on Math 521. Youwill be prompted to authenticate. Use your net id and net password the sameas if you were logging on the MyUW or WiscMail. If you are not enrolled inthe course (according to the registrar) you will not be allowed in.The text references in these notes are to the following texts:(Buck) R.Creighton Buck, Advanced Calculus, 3rd edition, Waveland Press.(Lang) Serge Lang, Undergraduate Analysis, 2nd edition, Springer Under-graduate Texts in Mathematics.(Morgan) Frank Morgan, Real Analysis, American Mathematical Society4(Rudin) Walter Rudin, Principles of Mathematical Analysis, McGraw Hill.There are three sections of 521 this semester and each has chosen one of thefirst three. The instructor I am replacing chose Buck. I used this book the lasttime I taught 521. The terminology, content, even the title is are somewhatdated. (The math department will soon change the name of 521 from Ad-vanced Calculus to Real Analysis I.) Lang is good but somewhat ponderous.It contains more material than can be comfortably treated in a two semestercourse. My favorite is from the first three is Morgan. It is student friendlyand contains almost exactly the material I hope to cover. (See the official de-scription of 521 at http://www.math.wisc.edu/521-advanced-calculus.)It is also