Int r oduction to Mathematical Analysis Introduction to Mathematical Analysis MTH 311 MTH 311 Introduction to Mathematical Analysis La errie r e, La errie r e, Nam Beatriz Laerriere Gerardo Laerriere Nguyen Mau Nam © 2015 Portland State University ISBN: 978-1-312-74284-0 This work is licensed under a Creative Commons Attribution-NonCommercial 4.0 International License Published by Portland State University Library Portland, OR 97207-1151 Cover design by Matt Eich Contents 1 TOOLS FOR ANALYSIS ............................................7 1.1 BASIC CONCEPTS OF SET THEORY7 1.2 FUNCTIONS 10 1.3 THE NATURAL NUMBERS AND MATHEMATICAL INDUCTION 13 1.4 ORDERED FIELD AXIOMS 16 1.5 THE COMPLETENESS AXIOM FOR THE REAL NUMBERS 19 1.6 APPLICATIONS OF THE COMPLETENESS AXIOM 22 2 SEQUENCES ................................................... 27 2.1 CONVERGENCE 27 2.2 LIMIT THEOREMS 32 2.3 MONOTONE SEQUENCES 35 2.4 THE BOLZANO-WEIERSTRASS THEOREM 40 2.5 LIMIT SUPERIOR AND LIMIT INFERIOR 43 2.6 OPEN SETS, CLOSED SETS, AND LIMIT POINTS 47 3 LIMITS AND CONTINUITY ........................................ 53 3.1 LIMITS OF FUNCTIONS 53 3.2 LIMIT THEOREMS 56 3.3 CONTINUITY 65 3.4 PROPERTIES OF CONTINUOUS FUNCTIONS 70 3.5 UNIFORM CONTINUITY 75 3.6 LOWER SEMICONTINUITY AND UPPER SEMICONTINUITY 78 4 DIFFERENTIATION ............................................... 85 4.1 DEFINITION AND BASIC PROPERTIES OF THE DERIVATIVE 85 4.2 THE MEAN VALUE THEOREM 90 4.3 SOME APPLICATIONS OF THE MEAN VALUE THEOREM 96 4.4 L’HOSPITAL’S RULE 98 4.5 TAYLOR’S THEOREM 101 4.6 CONVEX FUNCTIONS AND DERIVATIVES 104 4.7 NONDIFFERENTIABLE CONVEX FUNCTIONS AND SUBDIFFERENTIALS 109 5 Solutions and Hints for Selected Exercises ...................... 119 Preface Our goal in this set of lecture notes is to provide students with a strong foundation in mathematical analysis. Such a foundation is crucial for future study of deeper topics of analysis. Students should be familiar with most of the concepts presented here after completing the calculus sequence. However, these concepts will be reinforced through rigorous proofs. The lecture notes contain topics of real analysis usually covered in a 10-week course: the completeness axiom, sequences and convergence, continuity, and differentiation. The lecture notes also contain many well-selected exercises of various levels. Although these topics are written in a more abstract way compared with those available in some textbooks, teachers can choose to simplify them depending on the background of the students. For instance, rather than introducing the topology of the real line to students, related topological concepts can be replaced by more familiar concepts such as open and closed intervals. Some other topics such as lower and upper semicontinuity, differentiation of convex functions, and generalized differentiation of non-differentiable convex functions can be used as optional mathematical projects. In this way, the lecture notes are suitable for teaching students of different backgrounds. Hints and solutions to selected exercises are collected in Chapter5. For each section, there is at least one exercise fully solved. For those exercises, in addition to the solutions, there are explanations about the process itself and examples of more general problems where the same technique may be used. Exercises with solutions are indicated by a I and those with hints are indicated by a B. Finally, to make it easier for students to navigate the text, the electronic version of these notes contains many hyperlinks that students can click on to go to a definition, theorem, example, or exercise at a different place in the notes. These hyperlinks can be easily recognized because the text or number is on a different color and the mouse pointer changes shape when going over them. BASIC CONCEPTS OF SET THEORY FUNCTIONS THE NATURAL NUMBERS AND MATHEMATICAL INDUC- TION ORDERED FIELD AXIOMS THE COMPLETENESS AXIOM FOR THE REAL NUMBERS APPLICATIONS OF THE COMPLETENESS AXIOM 1. TOOLS FOR ANALYSIS This chapter discusses various mathematical concepts and constructions which are central to the study of the many fundamental results in analysis. Generalities are kept to a minimum in order to move quickly to the heart of analysis: the structure of the real number system and the notion of limit. The reader should consult the bibliographical references for more details. 1.1 BASIC CONCEPTS OF SET THEORY Intuitively, a set is a collection of objects with certain properties. The objects in a set are called the elements or members of the set. We usually use uppercase letters to denote sets and lowercase letters to denote elements of sets. If a is an element of a set A, we write a 2 A. If a is not an element of a set A, we write a 62 A. To specify a set, we can list all of its elements, if possible, or we can use a defining rule. For instance, to specify the fact that a set A contains four elements a;b;c;d, we write A = fa;b;c;dg: To describe the set E containing all even integers, we write E = fx : x = 2k for some integer kg: We say that a set A is a subset of a set B if every element of A is also an element of B, and write A ⊂ B: Two sets are equal if they contain the same elements. If A and B are equal, we write A = B. The following result is straightforward and very convenient for proving equality between sets. Theorem 1.1.1 Two sets A and B are equal if and only if A ⊂ B and B ⊂ A. If A ⊂ B and A does not equal B, we say that A is a proper subset of B, and write A ( B: 8 1.1 BASIC CONCEPTS OF SET THEORY The set /0 = fx : x 6= xg is called the empty set. This set clearly has no elements. Using Theorem 1.1.1, it is easy to show that all sets with no elements are equal. Thus, we refer to the empty set. Throughout this book, we will discuss several sets of numbers which should be familiar to the reader: • N = f1;2;3;:::g, the set of natural numbers or positive integers. • Z = f0;1;−1;2;−2;:::g, the set of integers (that is, the natural numbers together with zero and the negative of each natural number). • Q = fm=n : m;n 2 Z;n 6= 0g, the set of rational numbers. • R, the set of real numbers. • Intervals. For a;b 2 R, we have [a;b] = fx 2 R : a ≤ x ≤ bg, (a;b] = fx 2 R : a < x ≤ bg, [a;¥) = fx 2 R : a ≤ xg, (a;¥) = fx 2 R : a < xg, and similar definitions for (a;b), [a;b), (−¥;b], and (−¥;b). We will say more about the symbols ¥ and −¥ in Section 1.5. Since the real numbers are central to the study of analysis, we will discuss them in great detail in Sections 1.4, 1.5, and 1.6. For two sets A and B, the union, intersection, difference, and symmetric difference of A and B are given respectively by A [ B = fx : x 2 A or x 2 Bg; A \ B = fx : x 2 A and x 2 Bg; A n B = fx : x 2 A and x 2= Bg;and ADB = (A n B) [ (B n A): If A \ B = /0,we say that A and B are disjoint. The difference of A and B is also called the complement of B in A. If X is a universal set, that is, a set containing all the objects under consideration, then the complement of A in X is denoted simply by Ac. Theorem 1.1.2 Let A;B, and C be subsets of a universal set X. Then the following hold: (1) A [ Ac = X; (2) A \ Ac = /0; (3) (Ac)c = A; (4) (Distributive law) A \ (B [C) = (A \ B) [ (A \C); (5) (Distributive law) A [ (B \C) = (A [ B) \ (A [C); (6) (DeMorgan’s law) A n (B [C) = (A n B) \ (A nC); (7) (DeMorgan’s law) A n (B \C) = (A n B) [ (A nC); (8) A n B = A \ Bc: Proof: We prove some of the results and leave the rest for the exercises. (1) Clearly, A [ Ac ⊂ X since both A and Ac are subsets of X. Now let x 2 X. Then either x is an element of A or it is not an element of A. In the first case, x 2 A and, so, x 2 A [ Ac. In the second case, x 2 Ac and, so, x 2 A [ Ac. Thus, X ⊂ A [ Ac. 9 (2) No element of x can be simultaneously in A and not in A. Thus, A \ Ac = /0. (4) Let x 2 A \ (B [C). Then x 2 A and x 2 B [C. Therefore, x 2 B or x 2 C. In the first case, since x is also in A we get x 2 A \ B and, hence, x 2 (A \ B) [ (A \C). In the second case, x 2 A \C and, hence, x 2 (A \ B) [ (A \C). Thus, in all cases, x 2 (A \ B) [ (A \C). This shows A \ (B [C) ⊂ (A \ B) [ (A \C). Now we prove the other inclusion. Let x 2 (A \ B) [ (A \C). Then x 2 A \ B or x 2 A \C. In either case, x 2 A. In the first case, x 2 B and, hence, x 2 B [C. It follows in this case that x 2 A\(B[C). In the second case, x 2 C and, hence, x 2 B[C. Again, we conclude x 2 A\(B[C).