Basic Analysis I - jirka.org

1 Basic AnalysisIntroduction to Real Analysisby Ji r LeblFebruary 29, 2016(version )2 Typeset in 2009 2016 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.

2 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 grants DMS-0900885and date is the main identifier of version. The major version / edition number is raised only if therehave been substantial changes. Edition number started at 4, that is, version , as it was not kepttrack of more information (including contact information).

3 This book .. Analysis .. set theory ..81 Real properties .. set of real numbers .. value .. and the size ofR.. representation of the reals .. 392 Sequences and and limits .. about limits of sequences .. superior, limit inferior, and Bolzano-Weierstrass .. sequences .. on series .. 873 Continuous of functions .. functions .. and intermediate value theorems .. continuity .. at infinity .. functions and continuity .. 1314 The derivative.

4 Value theorem .. s theorem .. function theorem .. 1535 The Riemann Riemann integral .. of the integral .. theorem of calculus .. logarithm and the exponential .. integrals .. 1866 Sequences of and uniform convergence .. of limits .. s theorem .. 2127 Metric spaces .. and closed sets .. and convergence .. and compactness .. functions .. point theorem and Picard s theorem again .. 248 Further About this bookThis book is a one semester course in Basic Analysis .

5 It started its life as my lecture notes forteaching Math 444 at the University of Illinois at Urbana-Champaign (UIUC) in Fall semester I added the metric space chapter to teach Math 521 at University of Wisconsin Madison(UW). A prerequisite for this course is a Basic proof course, using for example [H], [F], or [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).

6 Here are my suggestions for whatto cover in a semester course. For a slower course such as UIUC 444: , , , , , , a more rigorous course covering metric spaces that runs quite a bit faster (such as UW 521): , , , , , , , 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.

7 The first few chapters of thebook can be used in an introductory proofs course as is for example done at Iowa State UniversityMath 201, where this book is used in conjunction with Hammack s Book of Proof [H].The book normally used for the class at UIUC is Bartle and Sherbert,Introduction to RealAnalysisthird 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 Darboux approach is far more appropriate for a course of this approach allows us to fit a course such as UIUC 444 within a semester and still spendsome extra time on the interchange of limits and end with Picard s theorem on the existence anduniqueness of solutions of ordinary differential equations.

8 This theorem is a wonderful examplethat uses many results proved in the book. For more advanced students, material may be coveredfaster so that we arrive at metric spaces and prove Picard s theorem using the fixed point theorem asis excellent books exist. My favorite is Rudin s excellentPrinciples of MathematicalAnalysis[R2] or as it is commonly and lovingly calledbaby Rudin(to distinguish it from hisother great Analysis textbook). I took a lot of inspiration and ideas from Rudin. However, Rudinis a bit more advanced and ambitious than this present course.

9 For those that wish to continuemathematics, Rudin is a fine investment. An inexpensive and somewhat simpler alternative to Rudinis Rosenlicht sIntroduction to Analysis [R1]. There is also the freely downloadableIntroduction toReal Analysisby 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.

10 In my opinion, contradiction is more likely to get beginningstudents into trouble, as we are talking about objects that do not 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 a general rule, I use:=instead of=to define an object rather than to simply show 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 functionf(x):=x2for a single exercise or , I would like to acknowledge Jana Ma r kov , Glen Pugh, Paul Vojta, Frank Beatrous,S nmez Sahuto glu, Jim Brandt, Kenji Kozai, and Arthur Busch, for teaching with the book andgiving me lots of useful feedback.