Basic Analysis: Introduction To Real Analysis

Basic AnalysisIntroduction to Real Analysisby Jiří LeblApril 22, 2013

2Typeset in LATEX.Copyright c 2009–2013 Jiří LeblThis work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike 3.0United States License. To view a copy of this license, visit s/ or send a letter to Creative Commons, 171 Second Street, Suite300, San Francisco, California, 94105, USA.You can use, print, duplicate, share these notes as much as you want. You can base your own noteson these and reuse parts if you keep the license the same. If you plan to use these commercially (sellthem for more than just duplicating cost), then you need to contact me and we will work somethingout. If you are printing a course pack for your students, then it is fine if the duplication service ischarging a fee for printing and selling the printed copy. I consider that duplicating cost.During the writing of these notes, the author was in part supported by NSF grant DMS-0900885.See http://www.jirka.org/ra/ for more information (including contact information).

ContentsIntroduction0.1 About this book . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.2 About analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .0.3 Basic set theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .1234Real Numbers1.1 Basic properties . . . . . . . . . .1.2 The set of real numbers . . . . . .1.3 Absolute value . . . . . . . . . .1.4 Intervals and the size of R . . . .1.5 Decimal representation of the reals5578.212125313538Sequences and Series2.1 Sequences and limits . . . . . . . . . . . . . . . . . .2.2 Facts about limits of sequences . . . . . . . . . . . . .2.3 Limit superior, limit inferior, and Bolzano-Weierstrass2.4 Cauchy sequences . . . . . . . . . . . . . . . . . . . .2.5 Series . . . . . . . . . . . . . . . . . . . . . . . . . .2.6 More on series . . . . . . . . . . . . . . . . . . . . . .43435161697283.9595103109115120124.Continuous Functions3.1 Limits of functions . . . . . . . . . . . .3.2 Continuous functions . . . . . . . . . . .3.3 Min-max and intermediate value theorems3.4 Uniform continuity . . . . . . . . . . . .3.5 Limits at infinity . . . . . . . . . . . . .3.6 Monotone functions and continuity . . . .The Derivative1294.1 The derivative . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1294.2 Mean value theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1353

4CONTENTS4.34.45Taylor’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141Inverse function theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144The Riemann Integral5.1 The Riemann integral . . . . . . .5.2 Properties of the integral . . . . .5.3 Fundamental theorem of calculus .5.4 The logarithm and the exponential5.5 Improper integrals . . . . . . . . .1471471551631691746Sequences of Functions1856.1 Pointwise and uniform convergence . . . . . . . . . . . . . . . . . . . . . . . . . 1856.2 Interchange of limits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1916.3 Picard’s theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1977Metric Spaces7.1 Metric spaces . . . . . . . . . . . . . . . . . .7.2 Open and closed sets . . . . . . . . . . . . . .7.3 Sequences and convergence . . . . . . . . . . .7.4 Completeness and compactness . . . . . . . . .7.5 Continuous functions . . . . . . . . . . . . . .7.6 Fixed point theorem and Picard’s theorem again.203203210217221226230Further Reading233Index235

Introduction0.1About this bookThis book is a one semester course in basic analysis. It started its life as my lecture notes forteaching Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in Fall semester 2009.Later I added the metric space chapter to teach Math 521 at University of Wisconsin–Madison(UW). A prerequisite for the course should be a basic proof course, for example using [H], [F], orthe unfortunately rather pricey [DW].It should be possible to use the book for both a basic course for students who do not necessarilywish to go to graduate school (such as UIUC 444), but also as a more advanced one-semester coursethat also covers topics such as metric spaces (such as UW 521). Here are my suggestions for whatto cover in a semester course. For a slower course such as UIUC 444:§0.3, §1.1–§1.4, §2.1–§2.5, §3.1–§3.4, §4.1–§4.2, §5.1–§5.3, §6.1–§6.3For a more rigorous course covering metric spaces that runs quite a bit faster (such as UW 521):§0.3, §1.1–§1.4, §2.1–§2.5, §3.1–§3.4, §4.1–§4.2, §5.1–§5.3, §6.1–§6.2, §7.1–§7.6It should also be possible to run a faster course without metric spaces covering all sections ofchapters 0 through 6. The approximate number of lectures given in the section notes through chapter6 are a very rough estimate and were designed for the slower course.The book normally used for the class at UIUC is Bartle and Sherbert, Introduction to RealAnalysis third edition [BS]. The structure of the beginning of the book somewhat follows thestandard syllabus of UIUC Math 444 and therefore has some similarities with [BS]. A majordifference is that we define the Riemann integral using Darboux sums and not tagged partitions.The Darboux approach is far more appropriate for a course of this level.Our approach allows us to fit within a semester and still spend some extra time on the interchangeof limits and end with Picard’s theorem on the existence and uniqueness of solutions of ordinarydifferential equations. This theorem is a wonderful example that uses many results proved in thebook. For more advanced students, material may be covered faster so that we arrive at metric spacesand prove Picard’s theorem using the fixed point theorem as is usual.Other excellent books exist. My favorite is Rudin’s excellent Principles of MathematicalAnalysis [R2] or as it is commonly and lovingly called baby Rudin (to distinguish it from his other5

6INTRODUCTIONgreat analysis textbook). I have taken a lot of inspiration and ideas from Rudin. However, Rudinis a bit more advanced and ambitious than this present course. For those that wish to continuemathematics, Rudin is a fine investment. An inexpensive and somewhat simpler alternative to Rudinis Rosenlicht’s Introduction to Analysis [R1]. There is also the freely downloadable Introduction toReal Analysis by William Trench [T].A note about the style of some of the proofs: Many proofs traditionally done by contradiction,I prefer to do by a direct proof or by contrapositive. While the book does include proofs bycontradiction, I only do so when the contrapositive statement seemed too awkward, or whencontradiction follows rather quickly. In my opinion, contradiction is more likely to get beginningstudents into trouble, as we are talking about objects that do not exist.I try to avoid unnecessary formalism where it is unhelpful. Furthermore, the proofs and thelanguage get slightly less formal as we progress through the book, as more and more details are leftout to avoid clutter.As a general rule, I use : instead of to define an object rather than to simply show equality.I use this symbol rather more liberally than is usual for emphasis. I use it even when the context is“local,” that is, I may simply define a function f (x) : x2 for a single exercise or example.Finally, I would like to acknowledge Jana Maříková, Glen Pugh, Paul Vojta, Frank Beatrous,and Sönmez Şahutoğlu for teaching with the book and giving me useful feedback. I would also liketo thank Dan Stoneham, Jeremy Sutter, Eliya Gwetta, Daniel Alarcon, Steve Hoerning, Yi Zhang,Nicole Caviris, an anonymous reader, and in general all the students in my classes for suggestionsand finding errors and typos.

0.2. ABOUT ANALYSIS0.27About analysisAnalysis is the branch of mathematics that deals with inequalities and limits. The present coursedeals with the most basic concepts in analysis. The goal of the course is to acquaint the reader withrigorous proofs in analysis and also to set a firm foundation for calculus of one variable.Calculus has prepared you, the student, for using mathematics without telling you why whatyou have learned is true. To use, or teach, mathematics effectively, you cannot simply know what istrue, you must know why it is true. This course shows you why calculus is true. It is here to giveyou a good understanding of the concept of a limit, the derivative, and the integral.Let us use an analogy. An auto mechanic that has learned to change the oil, fix brokenheadlights, and charge the battery, will only be able to do those simple tasks. He will be unable towork independently to diagnose and fix problems. A high school teacher that does not understandthe definition of the Riemann integral or the derivative may not be able to properly answer all thestudent’s questions. To this day I remember several nonsensical statements I heard from my calculusteacher in high school, who simply did not understand the concept of the limit, though he could “do”all problems in calculus.We start with a discussion of the real number system, most importantly its completeness property,which is the basis for all that comes after. We then discuss the simplest form of a limit, the limit ofa sequence. Afterwards, we study functions of one variable, continuity, and the derivative. Next, wedefine the Riemann integral and prove the fundamental theorem of calculus. We discuss sequencesof functions and the interchange of limits. Finally, we give an introduction to metric spaces.Let us give the most important difference between analysis and algebra. In algebra, we proveequalities directly; we prove that an object, a number perhaps, is equal to another object. In analysis,we usually prove inequalities. To illustrate the point, consider the following statement.Let x be a real number. If 0 x ε is true for all real numbers ε 0, then x 0.This statement is the general idea of what we do in analysis. If we wish to show that x 0, wewill show that 0 x ε for all positive ε.The term real analysis is a little bit of a misnomer. I prefer to use simply analysis. The othertype of analysis, complex analysis, really builds up on the present material, rather than being distinct.Furthermore, a more advanced course on real analysis would talk about complex numbers often. Isuspect the nomenclature is historical baggage.Let us get on with the show. . .