Basic Analysis I Introduction to Real Analysis, Volume I by Jiríˇ Lebl June 8, 2021 (version 5.4) 2 Typeset in LATEX. Copyright ©2009–2021 Jiríˇ Lebl This work is dual licensed under the Creative Commons Attribution-Noncommercial-Share Alike 4.0 International License and the Creative Commons Attribution-Share Alike 4.0 International License. To view a copy of these licenses, visit https://creativecommons.org/licenses/ by-nc-sa/4.0/ or https://creativecommons.org/licenses/by-sa/4.0/ or send a letter to Creative Commons PO Box 1866, Mountain View, CA 94042, USA. You can use, print, duplicate, share this book as much as you want. You can base your own notes on it and reuse parts if you keep the license the same. You can assume the license is either the CC-BY-NC-SA or CC-BY-SA, whichever is compatible with what you wish to do, your derivative works must use at least one of the licenses. Derivative works must be prominently marked as such. During the writing of this book, the author was in part supported by NSF grants DMS-0900885 and DMS-1362337. The date is the main identifier of version. The major version / edition number is raised only if there have been substantial changes. Each volume has its own version number. Edition number started at 4, that is, version 4.0, as it was not kept track of before. See https://www.jirka.org/ra/ for more information (including contact information, possible updates and errata). The LATEX source for the book is available for possible modification and customization at github: https://github.com/jirilebl/ra Contents Introduction 5 0.1 About this book ....................................5 0.2 About analysis ....................................7 0.3 Basic set theory ....................................8 1 Real Numbers 21 1.1 Basic properties .................................... 21 1.2 The set of real numbers ................................ 26 1.3 Absolute value and bounded functions ........................ 33 1.4 Intervals and the size of R .............................. 38 1.5 Decimal representation of the reals .......................... 41 2 Sequences and Series 47 2.1 Sequences and limits ................................. 47 2.2 Facts about limits of sequences ............................ 56 2.3 Limit superior, limit inferior, and Bolzano–Weierstrass ............... 67 2.4 Cauchy sequences ................................... 77 2.5 Series ......................................... 80 2.6 More on series ..................................... 92 3 Continuous Functions 103 3.1 Limits of functions .................................. 103 3.2 Continuous functions ................................. 111 3.3 Min-max and intermediate value theorems ...................... 118 3.4 Uniform continuity .................................. 125 3.5 Limits at infinity ................................... 131 3.6 Monotone functions and continuity .......................... 135 4 The Derivative 141 4.1 The derivative ..................................... 141 4.2 Mean value theorem .................................. 147 4.3 Taylor’s theorem ................................... 155 4.4 Inverse function theorem ............................... 159 4 CONTENTS 5 The Riemann Integral 163 5.1 The Riemann integral ................................. 163 5.2 Properties of the integral ............................... 172 5.3 Fundamental theorem of calculus ........................... 180 5.4 The logarithm and the exponential .......................... 186 5.5 Improper integrals ................................... 192 6 Sequences of Functions 205 6.1 Pointwise and uniform convergence ......................... 205 6.2 Interchange of limits ................................. 212 6.3 Picard’s theorem ................................... 223 7 Metric Spaces 229 7.1 Metric spaces ..................................... 229 7.2 Open and closed sets ................................. 237 7.3 Sequences and convergence .............................. 246 7.4 Completeness and compactness ............................ 251 7.5 Continuous functions ................................. 259 7.6 Fixed point theorem and Picard’s theorem again ................... 267 Further Reading 271 Index 273 List of Notation 279 Introduction 0.1 About this book This first volume is a one semester course in basic analysis. Together with the second volume it is a year-long course. It started its life as my lecture notes for teaching 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). Volume II was added to teach Math 4143/4153 at Oklahoma State University (OSU). A prerequisite for these courses is usually 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 necessarily wish to go to graduate school (such as UIUC 444), but also as a more advanced one-semester course that also covers topics such as metric spaces (such as UW 521). Here are my suggestions for what to 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–§6.3 For 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–§7.6 It should also be possible to run a faster course without metric spaces covering all sections of chapters 0 through 6. The approximate number of lectures given in the section notes through chapter 6 are a very rough estimate and were designed for the slower course. The first few chapters of the book can be used in an introductory proofs course as is done, for example, at Iowa State University Math 201, where this book is used in conjunction with Hammack’s Book of Proof [H ]. With volume II one can run a year-long course that also covers multivariable topics. It may make sense in this case to cover most of the first volume in the first semester while leaving metric spaces for the beginning of the second semester. The book normally used for the class at UIUC is Bartle and Sherbert, Introduction to Real Analysis third edition [BS ]. The structure of the beginning of the book somewhat follows the standard syllabus of UIUC Math 444 and therefore has some similarities with [BS ]. A major difference 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 a course such as UIUC 444 within a semester and still spend some time on the interchange of limits and end with Picard’s theorem on the existence and uniqueness of solutions of ordinary differential equations. This theorem is a wonderful example that uses many results proved in the book. For more advanced students, material may be covered faster so that we arrive at metric spaces and prove Picard’s theorem using the fixed point theorem as is usual. 6 INTRODUCTION Other excellent books exist. My favorite is Rudin’s excellent Principles of Mathematical Analysis [ R2] or, as it is commonly and lovingly called, baby Rudin (to distinguish it from his other great analysis textbook, big Rudin). I took a lot of inspiration and ideas from Rudin. However, Rudin is a bit more advanced and ambitious than this present course. For those that wish to continue mathematics, Rudin is a fine investment. An inexpensive and somewhat simpler alternative to Rudin is Rosenlicht’s Introduction to Analysis [R1 ]. There is also the freely downloadable Introduction to Real 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 by contradiction, I only do so when the contrapositive statement seemed too awkward, or when contradiction follows rather quickly. In my opinion, contradiction is more likely to get beginning students 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 the language get slightly less formal as we progress through the book, as more and more details are left out 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 Maríková,ˇ Glen Pugh, Paul Vojta, Frank Beatrous, Sönmez ¸Sahutoglu,˘ Jim Brandt, Kenji Kozai, Arthur Busch, Anton Petrunin, Mark Meilstrup, Harold P. Boas, Atilla Yılmaz, Thomas Mahoney, Scott Armstrong, and Paul Sacks, Matthias Weber, Manuele Santoprete, Robert Niemeyer, Amanullah Nabavi, for teaching with the book and giving me lots of useful feedback. Frank Beatrous wrote the University of Pittsburgh version extensions, which served as inspiration for many more recent additions. I would also like to thank Dan Stoneham, Jeremy Sutter, Eliya Gwetta, Daniel Pimentel-Alarcón, Steve Hoerning, Yi Zhang, Nicole Caviris, Kristopher Lee, Baoyue Bi, Hannah Lund, Trevor Mannella, Mitchel Meyer, Gregory Beauregard, Chase Meadors, Andreas Giannopoulos, Nick Nelsen, Ru Wang, Trevor Fancher, Brandon Tague, an anonymous reader or two, and in general all the students in my classes for suggestions and finding errors and typos. 0.2. ABOUT ANALYSIS 7 0.2 About analysis Analysis is the branch of mathematics that deals with inequalities and limits. The present course deals with the most basic concepts in analysis. The goal of the course is to acquaint the reader with rigorous proofs in analysis and also to set a firm foundation for calculus of one variable (and several variables if volume II is also considered). Calculus has prepared you, the student, for using mathematics without telling you why what you learned is true.