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An Introduction to Real Analysis John K. Hunter1

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An Introduction to Real Analysis John K. Hunter1 An Introduction to Real Analysis John K. Hunter1 Department of Mathematics, University of California at Davis 1The author was supported in part by the NSF. Thanks to Janko Gravner for a number of correc- tions and comments. Abstract. These are some notes on introductory real analysis. They cover the properties of the real numbers, sequences and series of real numbers, limits of functions, continuity, differentiability, sequences and series of functions, and Riemann integration. They don't include multi-variable calculus or contain any problem sets. Optional sections are starred. c John K. Hunter, 2014 Contents Chapter 1. Sets and Functions 1 1.1. Sets 1 1.2. Functions 5 1.3. Composition and inverses of functions 7 1.4. Indexed sets 8 1.5. Relations 11 1.6. Countable and uncountable sets 14 Chapter 2. Numbers 21 2.1. Integers 22 2.2. Rational numbers 23 2.3. Real numbers: algebraic properties 25 2.4. Real numbers: ordering properties 26 2.5. The supremum and infimum 27 2.6. Real numbers: completeness 29 2.7. Properties of the supremum and infimum 31 Chapter 3. Sequences 35 3.1. The absolute value 35 3.2. Sequences 36 3.3. Convergence and limits 39 3.4. Properties of limits 43 3.5. Monotone sequences 45 3.6. The lim sup and lim inf 48 3.7. Cauchy sequences 54 3.8. Subsequences 55 iii iv Contents 3.9. The Bolzano-Weierstrass theorem 57 Chapter 4. Series 59 4.1. Convergence of series 59 4.2. The Cauchy condition 62 4.3. Absolutely convergent series 64 4.4. The comparison test 66 4.5. * The Riemann ζ-function 68 4.6. The ratio and root tests 69 4.7. Alternating series 71 4.8. Rearrangements 73 4.9. The Cauchy product 77 4.10. * Double series 78 4.11. * The irrationality of e 86 Chapter 5. Topology of the Real Numbers 89 5.1. Open sets 89 5.2. Closed sets 92 5.3. Compact sets 95 5.4. Connected sets 102 5.5. * The Cantor set 104 Chapter 6. Limits of Functions 109 6.1. Limits 109 6.2. Left, right, and infinite limits 114 6.3. Properties of limits 117 Chapter 7. Continuous Functions 121 7.1. Continuity 121 7.2. Properties of continuous functions 125 7.3. Uniform continuity 127 7.4. Continuous functions and open sets 129 7.5. Continuous functions on compact sets 131 7.6. The intermediate value theorem 133 7.7. Monotonic functions 136 Chapter 8. Differentiable Functions 139 8.1. The derivative 139 8.2. Properties of the derivative 145 8.3. The chain rule 147 8.4. Extreme values 150 8.5. The mean value theorem 152 Contents v 8.6. Taylor's theorem 154 8.7. * The inverse function theorem 157 8.8. * L'H^ospital'srule 162 Chapter 9. Sequences and Series of Functions 167 9.1. Pointwise convergence 167 9.2. Uniform convergence 169 9.3. Cauchy condition for uniform convergence 170 9.4. Properties of uniform convergence 171 9.5. Series 175 Chapter 10. Power Series 181 10.1. Introduction 181 10.2. Radius of convergence 182 10.3. Examples of power series 184 10.4. Algebraic operations on power series 188 10.5. Differentiation of power series 193 10.6. The exponential function 195 10.7. * Smooth versus analytic functions 197 Chapter 11. The Riemann Integral 205 11.1. The supremum and infimum of functions 206 11.2. Definition of the integral 208 11.3. The Cauchy criterion for integrability 215 11.4. Continuous and monotonic functions 219 11.5. Linearity, monotonicity, and additivity 222 11.6. Further existence results 230 11.7. * Riemann sums 234 11.8. * The Lebesgue criterion 238 Chapter 12. Properties and Applications of the Integral 241 12.1. The fundamental theorem of calculus 241 12.2. Consequences of the fundamental theorem 246 12.3. Integrals and sequences of functions 251 12.4. Improper Riemann integrals 255 12.5. * Principal value integrals 261 12.6. The integral test for series 265 12.7. Taylor's theorem with integral remainder 268 Chapter 13. Metric, Normed, and Topological Spaces 271 13.1. Metric spaces 271 13.2. Normed spaces 276 vi Contents 13.3. Open and closed sets 279 13.4. Completeness, compactness, and continuity 282 13.5. Topological spaces 287 13.6. * Function spaces 289 13.7. * The Minkowski inequality 293 Bibliography 299 Chapter 1 Sets and Functions We understand a \set" to be any collection M of certain distinct objects of our thought or intuition (called the \elements" of M) into a whole. (Georg Cantor, 1895) In mathematics you don't understand things. You just get used to them. (Attributed to John von Neumann) In this chapter, we define sets, functions, and relations and discuss some of their general properties. This material can be referred back to as needed in the subsequent chapters. 1.1. Sets A set is a collection of objects, called the elements or members of the set. The objects could be anything (planets, squirrels, characters in Shakespeare's plays, or other sets) but for us they will be mathematical objects such as numbers, or sets of numbers. We write x 2 X if x is an element of the set X and x2 = X if x is not an element of X. If the definition of a \set" as a \collection" seems circular, that's because it is. Conceiving of many objects as a single whole is a basic intuition that cannot be analyzed further, and the the notions of \set" and \membership" are primitive ones. These notions can be made mathematically precise by introducing a system of axioms for sets and membership that agrees with our intuition and proving other set-theoretic properties from the axioms. The most commonly used axioms for sets are the ZFC axioms, named somewhat inconsistently after two of their founders (Zermelo and Fraenkel) and one of their axioms (the Axiom of Choice). We won't state these axioms here; instead, we use \naive" set theory, based on the intuitive properties of sets. Nevertheless, all the set-theory arguments we use can be rigorously formalized within the ZFC system. 1 2 1. Sets and Functions Sets are determined entirely by their elements. Thus, the sets X, Y are equal, written X = Y , if x 2 X if and only if x 2 Y: It is convenient to define the empty set, denoted by ?, as the set with no elements. (Since sets are determined by their elements, there is only one set with no elements!) If X 6= ?, meaning that X has at least one element, then we say that X is non- empty. We can define a finite set by listing its elements (between curly brackets). For example, X = f2; 3; 5; 7; 11g is a set with five elements. The order in which the elements are listed or repetitions of the same element are irrelevant. Alternatively, we can define X as the set whose elements are the first five prime numbers. It doesn't matter how we specify the elements of X, only that they are the same. Infinite sets can't be defined by explicitly listing all of their elements. Never- theless, we will adopt a realist (or \platonist") approach towards arbitrary infinite sets and regard them as well-defined totalities. In constructive mathematics and computer science, one may be interested only in sets that can be defined by a rule or algorithm | for example, the set of all prime numbers | rather than by infinitely many arbitrary specifications, and there are some mathematicians who consider infinite sets to be meaningless without some way of constructing them. Similar issues arise with the notion of arbitrary subsets, functions, and relations. 1.1.1. Numbers. The infinite sets we use are derived from the natural and real numbers, about which we have a direct intuitive understanding. Our understanding of the natural numbers 1; 2; 3;::: derives from counting. We denote the set of natural numbers by N = f1; 2; 3;::: g : We define N so that it starts at 1. In set theory and logic, the natural numbers are defined to start at zero, but we denote this set by N0 = f0; 1; 2;::: g. Histori- cally, the number 0 was later addition to the number system, primarily by Indian mathematicians in the 5th century AD. The ancient Greek mathematicians, such as Euclid, defined a number as a multiplicity and didn't consider 1 to be a number either. Our understanding of the real numbers derives from durations of time and lengths in space. We think of the real line, or continuum, as being composed of an (uncountably) infinite number of points, each of which corresponds to a real number, and denote the set of real numbers by R. There are philosophical questions, going back at least to Zeno's paradoxes, about whether the continuum can be represented as a set of points, and a number of mathematicians have disputed this assumption or introduced alternative models of the continuum. There are, however, no known inconsistencies in treating R as a set of points, and since Cantor's work it has been the dominant point of view in mathematics because of its precision, power, and simplicity. 1.1. Sets 3 We denote the set of (positive, negative and zero) integers by Z = f:::; −3; −2; −1; 0; 1; 2; 3;::: g; and the set of rational numbers (ratios of integers) by Q = fp=q : p; q 2 Z and q 6= 0g: The letter \Z" comes from \zahl" (German for \number") and \Q" comes from \quotient." These number systems are discussed further in Chapter 2.
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