Invitation to Real Analysis

Description

This book is an introduction to real analysis for a one-semester course aimed at students who have completed the calculus sequence and preferably one other course, such as linear algebra. It does not assume any specific knowledge and starts with all that is needed from sets, logic, and induction. Then there is a careful introduction to the real numbers with an emphasis on developing proof-writing skills. It continues with a logical development of the notions of sequences, open and closed sets (including compactness and the Cantor set), continuity, differentiation, integration, and series of numbers and functions. A theme in the book is to give more than one proof for interesting facts; this illustrates how different ideas interact and it makes connections among the facts that are being learned. Metric spaces are introduced early in the book, but there are instructions on how to avoid metric spaces for the instructor who wishes to do so. There are questions that check the readers' understanding of the material, with solutions provided at the end. Topics that could be optional or assigned for independent reading include the Cantor function, nowhere differentiable functions, the Gamma function, and the Weierstrass theorem on approximation by continuous functions. Contents Preface Chapter 0. Preliminaries: Sets, Functions, and Induction 0.1. Notation on Sets and Functions 0.2. Basic Logic: Statements and Logical Connectives Exercises: Basic Logic 0.3. Sets Exercises: Sets 0.4. Functions Exercises: Functions 0.5. Mathematical Induction Exercises: Mathematical Induction 0.6. More on Sets: Axioms and Constructions Exercises: More on Sets Chapter 1. The Real Numbers and the Completeness Property 1.1. Field and Order Properties of Exercises: Field and Order Properties of 1.2. Completeness Property of Exercises: Completeness Property of 1.3. Countable and Uncountable Sets Exercises: Countable and Uncountable Sets 1.4. Construction of the Real Numbers Exercises: Construction of the Real Numbers 1.5. The Complex Numbers Exercises: The Complex Numbers Chapter 2. Sequences 2.1. Limits of Sequences Exercises: Limits of Sequences 2.2. Three Consequences of Order Completeness Exercises: Three Consequences of Order Completeness 2.3. The Cauchy Property for Sequences Exercises: The Cauchy Property for Sequences Chapter 3. Topology of the Real Numbers and Metric Spaces 3.1. Metrics Exercises: Metrics 3.2. Open and Closed Sets in Exercises: Open and Closed Sets in 3.3. Open and Closed Sets in Metric Spaces Exercises: Open and Closed Sets in Metric Spaces 3.4. Compactness in Exercises: Compactness in 3.5. The Cantor Set Exercises: The Cantor Set 3.6. Connected Sets in Exercises: Connected Sets in 3.7. Compactness, Connectedness, and Completeness in Metric Spaces Exercises: Compactness and Completeness in Metric Spaces Chapter 4. Continuous Functions 4.1. Continuous Functions on Exercises: Continuous Functions on 4.2. Intermediate Value and Extreme Value Theorems Exercises: Intermediate Value and Extreme Value Theorems 4.3. Limits Exercises: Limits 4.4. Uniform Continuity Exercises: Uniform Continuity 4.5. Continuous Functions on Metric Spaces Exercises: Continuous Functions on Metric Spaces Chapter 5. Differentiable Functions 5.1. Differentiable Functions on Exercises: Differentiable Functions on 5.2. Mean Value Theorem Exercises: Mean Value Theorem 5.3. Taylor’s Theorem Exercises: Taylor’s Theorem Chapter 6. Integration 6.1. The Riemann Integral Exercises: The Riemann Integral 6.2. The Fundamental Theorem of Calculus Exercises: The Fundamental Theorem of Calculus 6.3. Improper Riemann Integrals Exercises: Improper Riemann Integrals Chapter 7. Series 7.1. Series of Real Numbers Exercises: Series of Real Numbers 7.2. Alternating Series and Absolute Convergence Exercises: Alternating Series and Absolute Convergence Chapter 8. Sequences and Series of Functions 8.1. Pointwise Convergence Exercises: Pointwise Convergence 8.2. Uniform Convergence Exercises: Uniform Convergence 8.3. Series of Functions Exercises: Series of Functions 8.4. Power Series Exercises: Power Series 8.5. Taylor Series Exercises: Taylor Series 8.6. Weierstrass Approximation Theorem Exercises: Weierstrass Approximation Theorem 8.7. The Complex Exponential Exercises: The Complex Exponential Solutions to Questions Bibliographical Notes Bibliography Index Página em branco