Problems and Solutions for Undergraduate Real Analysis

Description

The aim of Problems and Solutions for Undergraduate Real Analysis I, as the name reveals, is to assist undergraduate students or first-year students who study mathematics in learning their first rigorous real analysis course. The wide variety of problems, which are of varying difficulty, include the following topics: Elementary Set Algebra The Real Number System Countable and Uncountable Sets Elementary Topology on Metric Spaces Sequences in Metric Spaces Series of Numbers Limits and Continuity of Functions Differentiation the Riemann-Stieltjes Integral Furthermore, the main features of this book are listed as follows: The book contains 230 problems, which cover the topics mentioned above, with detailed and complete solutions. As a matter of fact, my solutions show every detail, every step and every theorem that I applied. Each chapter starts with a brief and concise note of introducing the notations, terminologies, basic mathematical concepts or important/famous/frequently used theorems (without proofs) relevant to the topic. Three levels of difficulty have been assigned to problems so that you can sharpen your mathematics step-by-step. Different colors are used frequently in order to highlight or explain problems, examples, remarks, main points/formulas involved, or show the steps of manipulation in some complicated proofs. (ebook only) An appendix about mathematical logic is included. It tells students what concepts of logic (e.g. techniques of proofs) are necessary in advanced mathematics. Preface List of Figures List of Tables Contents Chapter 1. Elementary Set Algebra 1.1 Fundamental Concepts 1.2 Sets, Functions and Relations Chapter 2. The Real Number System 2.1 Fundamental Concepts 2.2 Rational and Irrational Numbers 2.3 Absolute Values 2.4 The Completeness Axiom Chapter 3. Countable and Uncountable Sets 3.1 Fundamental Concepts 3.2 Problems on Countable and Uncountable Sets Chapter 4. Elementary Topology on Metric Spaces 4.1 Fundamental Concepts 4.2 Open Sets and Closed Sets 4.3 Compact Sets 4.4 The Heine-Borel Theorem 4.5 Connected Sets Chapter 5. Sequences in Metric Spaces 5.1 Fundamental Concepts 5.2 Convergence of Sequences 5.3 Upper and Lower Limits 5.4 Cauchy Sequences and Complete Metric Spaces 5.5 Recurrence Relations Chapter 6. Series of Numbers 6.1 Fundamental Concepts 6.2 Convergence of Series of Nonnegative Terms 6.3 Alternating Series and Absolute Convergence 6.4 The Series Σ_{n=1}^∞ a_n b_n and Multiplication of Series 6.5 Power Series Chapter 7. Limits and Continuity of Functions 7.1 Fundamental Concepts 7.2 Limits of Functions 7.3 Continuity and Uniform Continuity of Functions 7.4 The Extreme Value Theorem and the Intermediate Value Theorem 7.5 Discontinuity of Functions 7.6 Monotonic Functions Chapter 8. Differentiation 8.1 Fundamental Concepts 8.2 Properties of Derivatives 8.3 The Mean Value Theorem for Derivatives 8.4 L’Hôspital’s Rule 8.5 Higher Order Derivatives and Taylor’s Theorem Chapter 9. The Riemann-Stieltjes Integral 9.1 Fundamental Concepts 9.2 Integrability of Real Functions 9.3 Applications of Integration Theorems 9.4 The Mean Value Theorems for Integrals Chapter 10. Sequences and Series of Functions 10.1 Fundamental Concepts 10.2 Uniform Convergence for Sequences of Functions 10.3 Uniform Convergence for Series of Functions 10.4 Equicontinuous Families of Functions 10.5 Approximation by Polynomials Chapter 11. Improper Integrals 11.1 Fundamental Concepts 11.2 Evaluations of Improper Integrals 11.3 Convergence of Improper Integrals 11.4 Miscellaneous Problems on Improper Integrals Chapter 12. Lebesgue Measure 12.1 Fundamental Concepts 12.2 Lebesgue Outer Measure 12.3 Lebesgue Measurable Sets 12.4 Necessary and Sufficient Conditions for Measurable Sets Chapter 13. Lebesgue Measurable Functions 13.1 Fundamental Concepts 13.2 Lebesgue Measurable Functions 13.3 Applications of Littlewood’s Three Principles Chapter 14. Lebesgue Integration 14.1 Fundamental Concepts 14.2 Properties of Integrable Functions 14.3 Applications of Fatou’s Lemma Chapter 15. Differential Calculus of Functions of Several Variables 15.1 Fundamental Concepts 15.2 Differentiation of Functions of Several Variables 15.3 The Mean Value Theorem for Differentiable Functions 15.4 The Inverse Function Theorem and the Implicit Function Theorem 15.5 Higher Order Derivatives Chapter 16. Integral Calculus of Functions of Several Variables 16.1 Fundamental Concepts 16.2 Jordan Measurable Sets 16.3 Integration on ℝ^n 16.4 Applications of the Mean Value Theorem 16.5 Applications of the Change of Variables Theorem Appendix A. Language of Mathematics A.1 Fundamental Concepts A.2 Statements and Logical Connectives A.3 Quantifiers and their Basic Properties A.4 Necessity and Sufficiency A.5 Techniques of Proofs Index Bibliography [1]-[16] [17]-[34]