An Introduction to Analysis

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Completely revised and updated, the second edition of An Introduction to Analysis presents a concise and sharply focused introduction to the basic concepts of analysis-from the development of real numbers through uniform convergences of a sequence of functions-and includes coverage both of the analysis of functions of more than one variable and of differential equations. This student-friendly text maintains a cautious and deliberate pace, and examples and figures are used extensively to assist the reader in understanding the concepts and then applying them. Students will become actively engaged in the learning process with a broad and comprehensive collection of problems found at the end of each section. Key Features Includes an introduction to multivariable calculus with three new sections integrated into the development of the corresponding single variable material. Includes a new final chapter on differential equations A wealth of exercises, varying in style and degree of difficulty, are found in each section, and enable students to test their understanding of key material. Hints are provided for the more difficult problems, and a number of problems are broken down into more manageable parts. An introductory note speaks to instructors who may not have time to cover all material in the text, and includes comments on the interrelationship between topics and how to modify the presentation to fit course needs. Solutions to the text’s exercises are available to qualify instructors. Table of Contents Chapter 1: The Real Numbers Sets Functions Algebraic and order properties The positive integers The least upper bound axiom Chapter 2: Sequences Sequences and limits Limit theorems Monotonic sequences Sequences defined inductively Subsequences Cauchy sequences Infinite limits Chapter 3: Functions and Continuity Limit of a function Limit theorems Other Limits Continuity Intermediate values, extreme values Uniform continuity Functions of two variables Chapter 4: The Derivative Definition of the derivative Rules for differentiation The Mean Value Theorem Inverse functions Differentiability in R2 Chapter 5: The Integral The definition of the integral Properties of the Integral Existence theory The Fundamental Theorem of Calculus Improper integrals Double integrals Chapter 6: Infinite Series Basic Theory Absolute convergence Power series Taylor series Chapter 7: Sequences and Series of Functions Uniform Convergence Consequences of uniform convergence Two examples Chapter 8: Introduction to Differential Equations Elementary first order differential equations Existence and uniqueness Power series solutions Appendix: Solutions and Hints for Selected Problems List of Symbols