Introduction to Real Analysis MATH 5200-5210 Theodore Kilgore Date of most recent revision is September 11, 2019 Contents Preface ix 1 Some Basic Tools 1-1 1.1 Sets......................................................... 1-1 1.1.1 Operations upon sets............................................ 1-6 1.1.2 More concerning set operations...................................... 1-9 1.1.3 The Cartesian Product of two sets.................................... 1-13 i ii CONTENTS 1.1.4 Relations and Functions.......................................... 1-13 1.2 Logic......................................................... 1-17 1.2.1 Propositions, operations, and truth tables................................ 1-17 1.2.2 Quantifiers................................................. 1-22 1.2.3 Negations.................................................. 1-26 2 Integers and Rational Numbers 2-1 2.1 Introductory Remarks............................................... 2-2 2.2 Basic properties of the integers.......................................... 2-4 2.2.1 Algebraic properties of the integers.................................... 2-4 2.2.2 Order properties of the integers...................................... 2-7 2.2.3 The Well Ordering Principle and Mathematical Induction....................... 2-10 2.2.4 Decimal and other representations of integers.............................. 2-16 2.2.5 Other properties of the integers...................................... 2-18 2.3 A small excursion.................................................. 2-19 2.4 Construction of the rational numbers....................................... 2-23 2.5 Inadequacy of the rational numbers........................................ 2-28 CONTENTS iii 3 Building the real numbers 3-1 3.1 Introduction..................................................... 3-2 3.2 Sequences...................................................... 3-12 3.3 Sequences of rational numbers........................................... 3-14 3.4 The real numbers.................................................. 3-22 3.5 Completeness.................................................... 3-32 4 Series 4-1 4.1 Finite series and sigma notation.......................................... 4-2 4.2 Tools { the binomial theorem........................................... 4-7 4.3 Infinite series.................................................... 4-9 4.4 Tools { the geometric series............................................ 4-13 4.5 A small excursion { the definition of e ..................................... 4-14 4.6 Some discussion of the exponential function................................... 4-19 4.7 More on convergence of series........................................... 4-21 4.7.1 The Root Test and the Ratio Test.................................... 4-21 4.7.2 Conditional Convergence and the Alternating Series Test....................... 4-27 iv CONTENTS 5 Topological concepts 5-1 5.1 Basics........................................................ 5-1 5.2 A brief discussion of topology........................................... 5-4 5.2.1 Base for a topology............................................. 5-6 5.2.2 Functions and Continuity......................................... 5-7 5.2.3 Topological Properties........................................... 5-8 5.2.4 Relative Topologies............................................. 5-9 5.3 Connectedness................................................... 5-11 5.4 Compactness.................................................... 5-12 5.5 Metric Spaces.................................................... 5-14 5.6 Some general results................................................ 5-17 6 Functions, limits, and continuity 6-1 6.1 Functions...................................................... 6-2 6.2 Limits of functions................................................. 6-6 6.3 Continuity...................................................... 6-13 6.4 Uniform continuity................................................. 6-20 6.5 lim sup and lim inf................................................. 6-21 CONTENTS v 7 Cardinality 7-1 7.1 Finite and countable sets............................................. 7-1 7.2 Uncountable sets.................................................. 7-6 8 Representations of the real numbers 8-1 8.1 Introduction..................................................... 8-2 8.2 Decimal representation............................................... 8-3 8.3 Binary representation............................................... 8-5 8.4 Other representations............................................... 8-7 8.5 The Cantor Set................................................... 8-8 9 Conclusion of MATH 5200, MATH 5210 begins 9-1 10 The Derivative and the Riemann Integral 10-1 10.1 The Derivative................................................... 10-2 10.2 Integrals....................................................... 10-6 10.2.1 The integral of a bounded non-negative function on a bounded closed interval............ 10-7 10.2.2 The integral of a bounded function on a bounded closed interval................... 10-13 10.2.3 The effect of unnatural ordering on integration............................. 10-14 10.2.4 The Riemann integral........................................... 10-16 vi CONTENTS 10.2.5 The linearity of the integral........................................ 10-18 10.2.6 Shortcuts can go wrong, when defining the integral........................... 10-19 10.3 The Fundamental Theorem of Calculus...................................... 10-21 10.4 Improper integrals and the integral test..................................... 10-25 10.5 A problem with the Riemann integral...................................... 10-30 11 Vector and Function Spaces 11-1 11.1 Normed vector spaces............................................... 11-2 11.2 Inner Products................................................... 11-11 11.3 Norms for continuous functions.......................................... 11-17 11.4 Banach Spaces................................................... 11-21 11.5 Linear Transformations and Continuity..................................... 11-24 11.6 Weierstrass Approximation Theorem....................................... 11-28 11.7 Linear operators defined by integral kernels................................... 11-34 11.8 Periodic functions and the Fourier series..................................... 11-35 11.9 The F´ej´eroperator................................................. 11-40 11.10Weierstrass Theorem for periodic functions................................... 11-43 CONTENTS vii 12 Finite Taylor-Maclaurin expansions 12-1 12.1 Introductory remarks................................................ 12-1 12.2 Finite Taylor expansions of a function...................................... 12-2 13 Functions given as series 13-1 13.1 Introductory remarks................................................ 13-1 13.2 Functions defined as series............................................. 13-4 13.3 Convergence of a power series........................................... 13-5 13.4 Differentiation and integration of power series.................................. 13-10 13.5 Double sums.................................................... 13-15 13.6 The rest of the story................................................ 13-16 14 Integrals on Rectangles 14-1 14.1 Integrals defined on Rectangles.......................................... 14-2 14.2 Fubini's Theorem.................................................. 14-5 15 The Stieltjes Integral 15-1 15.1 A broader view of integration........................................... 15-1 15.2 The Riemann-Stieltjes Integral.......................................... 15-8 viii CONTENTS Preface This document was begun as a text for MATH 5200, Spring 2012 and for MATH 5210, Fall 2012. Since then, some of the topics in it have been expanded or have been subjected to stylistic revision. Continued attention has been devoted to the seemingly never-ending task of removing typographical errors. The document is presented to the students and to their future instructors so that everyone interested in the matter can know what has been done in the course. The current version is \published" at <www.auburn.edu/∼kilgota> for downloading. Just in case that future revisions may be desirable, the students are encouraged to keep the \book" in a binder. ix x PREFACE That ought to make it easy to add any new pages, and also make it easy to incorporate any portions which are revised or expanded. Since the text is distributed in the form of a PDF file, students can keep an electronic copy and print any part as needed. Or, if desired, a student can merely keep an electronic copy and bring it to class on a laptop, netbook, or other electronic device. 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