Basic Analysis: Introduction to Real Analysis

1 Basic AnalysisIntroduction to real Analysisby Ji r LeblApril 26, 20112 Typeset in 2009 2011 Ji r LeblThis work is licensed under the Creative Commons Attribution-Noncommercial-Share Alike States License. To view a copy of this license, send a letter to Creative Commons, 171 Second Street, Suite300, San Francisco, California, 94105, 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 the writing of these notes, the author was in part supported by NSF grant more information (including contact information).

2 About these notes .. Analysis .. set theory ..81 real properties .. set of real numbers .. value .. and the size ofR.. 352 Sequences and and limits .. about limits of sequences .. superior, limit inferior, and Bolzano-Weierstrass .. sequences .. 683 Continuous of functions .. functions .. and intermediate value theorems .. continuity .. 984 The derivative .. value theorem .. s theorem .. 11534 CONTENTS5 The Riemann Riemann integral .. of the integral .. theorem of calculus .. 1336 Sequences of and uniform convergence .. of limits .. s theorem .. 151 Further Notes about these notesThis book is a one semester course in Basic Analysis . These were my lecture notes for teaching Math444 at the University of Illinois at Urbana-Champaign (UIUC) in Fall semester 2009.

3 The course isa first course in mathematical Analysis aimed at students who do not necessarily wish to continue agraduate study in mathematics. A prerequisite for the course is a Basic proof course, for exampleone using the (unfortunately rather pricey) book [DW]. The course does not cover topics such asmetric spaces, which a more advanced course would. It should be possible to use these notes for abeginning of a more advanced course, but further material should be book normally used for the class at UIUC is Bartle and Sherbert, Introduction to RealAnalysisthird edition [BS]. The structure of the notes mostly follows the syllabus of UIUC Math 444and therefore has some similarities with [BS]. Some topics covered in [BS] are covered in slightlydifferent order, some topics differ substantially from [BS] and some topics are not covered at example, we will define the Riemann integral using Darboux sums and not tagged Darboux approach is far more appropriate for a course of this level.

4 In my view, [BS] seemsto be targeting a different audience than this course, and that is the reason for writing this presentbook. The generalized Riemann integral is not covered at the integral is treated more lightly, we can spend some extra time on the interchange of limitsand in particular on a section on Picard s theorem on the existence and uniqueness of solutions ofordinary differential equations if time allows. This theorem is a wonderful example that uses manyresults proved in the excellent books exist. My favorite is without doubt Rudin s excellentPrinciples ofMathematical Analysis [R2] or as it is commonly and lovingly calledbaby Rudin(to distinguishit from his other great Analysis textbook). I have taken a lot of inspiration and ideas from , Rudin is a bit more advanced and ambitious than this present course. For those thatwish to continue mathematics, Rudin is a fine investment.

5 An inexpensive alternative to Rudin isRosenlicht sIntroduction to Analysis [R1]. Rosenlicht may not be as dry as Rudin for those juststarting out in mathematics. There is also the freely downloadableIntroduction to real AnalysisbyWilliam Trench [T] for those that do not wish to invest much want to mention a note about the style of some of the proofs. Many proofs that are traditionallydone by contradiction, I prefer to do by a direct proof or at least by a contrapositive. While the56 Introduction book does include proofs by contradiction, I only do so when the contrapositive statement seemedtoo awkward, or when the contradiction follows rather quickly. In my opinion, contradiction ismore likely to get the beginning student into trouble. In a contradiction proof, we are arguing aboutobjects that do not exist. In a direct proof or a contrapositive proof one can be guided by intuition,but in a contradiction proof, intuition usually leads us also try to avoid unnecessary formalism where it is unhelpful.

6 Furthermore, the proofs and thelanguage get slightly less formal as we progress through the book, as more and more details are leftout to avoid a general rule, I will use:=instead of=to define an object rather than to simply showequality. I use this symbol rather more liberally than is usual. I may use it even when the context is local, that is, I may simply define a functionf(x):=x2for a single exercise or you are teaching (or being taught) with [BS], here is the correspondence of the sections. Thecorrespondences are only approximate, the material in these notes and in [BS] differs, as in [BS] and and of , , , and (and )SectionSection in [BS] ? , in [BS]It is possible to skip or skim some material in the book as it is not used later on.

7 The optionalmaterial is marked in the notes that appear below every section title. Section can be coveredlightly, or left as reading. The material within is considered prerequisite. The section on Taylor stheorem ( ) can safely be skipped as it is never used later. Uncountability ofRin can safelybe skipped. The alternative proof of Bolzano-Weierstrass in can safely be skipped. And ofcourse, the section on Picard s theorem can also be skipped if there is no time at the end of thecourse, though I have not marked the section I would like to acknowledge Jana Ma r kov and Glen Pugh for teaching with the notesand finding many typos and errors. I would also like to thank Dan Stoneham, Frank Beatrous, andan anonymous reader for suggestions and finding errors and ABOUT About analysisAnalysis is the branch of mathematics that deals with inequalities and limiting processes.

8 Thepresent course will deal with the most Basic concepts in Analysis . The goal of the course is toacquaint the reader with the Basic concepts of rigorous proof in Analysis , and also to set a firmfoundation for calculus of one 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 knowwhatistrue, you must knowwhyit is true. This course is to tell youwhycalculus is true. It is here to giveyou a good understanding of the concept of a limit, the derivative, and the us give an analogy to make the point. An auto mechanic that has learned to change the oil,fix broken headlights, and charge the battery, will only be able to do those simple tasks. He willnot be able to work independently to diagnose and fix problems. A high school teacher that doesnot understand the definition of the Riemann integral will not be able to properly answer all thestudent s questions that could come up.

9 To this day I remember several nonsensical statements Iheard from my calculus teacher in high school who simply did not understand the concept of thelimit, though he could do all problems in will start with discussion of the real number system, most importantly its completenessproperty, which is the basis for all that we will talk about. We will then discuss the simplest formof a limit, that is, the limit of a sequence. We will then move to study functions of one variable,continuity, and the derivative. Next, we will define the Riemann integral and prove the fundamentaltheorem of calculus. We will end with discussion of sequences of functions and the interchange me give perhaps the most important difference between Analysis and algebra. In algebra, weprove equalities directly. That is, we prove that an object (a number perhaps) is equal to anotherobject.

10 In Analysis , we generally prove inequalities. To illustrate the point, consider the x be a real number. If0 x< is true for all real numbers >0, then x= statement is the general idea of what we do in Analysis . If we wish to show thatx=0, wewill show that 0 x< for all positive .The term real Analysis is a little bit of a misnomer. I prefer to normally use just Analysis . The other type of Analysis , that is, complex Analysis really builds up on the present material,rather than being distinct. Furthermore, a more advanced course on real Analysis would talk aboutcomplex numbers often. I suspect the nomenclature is just historical us get on with the show.. Basic set theoryNote: 1 3 lectures (some material can be skipped or covered lightly)Before we can start talking about Analysis we need to fix some language. Modern analysisuses the language of sets, and therefore that s where we will start.