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E-book
Author Yau, Donald

Title A First Course in Analysis
Published Singapore : World Scientific, 2012
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World Scientific    View Resource Record  

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Description 1 online resource (206 pages)
Contents Preface; Contents; 1. Sets, Functions, and Real Numbers; 1.1 Sets; 1.1.1 Set Notations; 1.1.2 Subsets; 1.1.3 Operations on Sets; 1.1.4 Exercises; 1.2 Functions; 1.2.1 Functional Notations; 1.2.2 Special Functions; 1.2.3 Inverse Functions; 1.2.4 Composition; 1.2.5 Exercises; 1.3 Real Numbers; 1.3.1 Ordered Field; 1.3.2 Absolute Value; 1.3.3 Upper and Lower Bounds; 1.3.4 The Completeness Axiom; 1.3.5 Lots of Rationals and Irrationals; 1.3.6 Intervals; 1.3.7 Exercises; 1.4 Mathematical Induction; 1.4.1 Induction; 1.4.2 Strong Induction; 1.4.3 Exercises; 1.5 Countability
1.5.1 Finite and infinite sets1.5.2 Countable and Uncountable Sets; 1.5.3 Exercises; 1.6 Additional Exercises; 2. Sequences; 2.1 Sequences of Real Numbers; 2.1.1 Definition of a Sequence; 2.1.2 Convergent Sequences; 2.1.3 Divergent Sequences; 2.1.4 Uniqueness of Limits; 2.1.5 Exercises; 2.2 Properties of Limits; 2.2.1 Bounded Sequences; 2.2.2 Monotone Sequences; 2.2.3 Arithmetics of Sequences; 2.2.4 Excercises; 2.3 The Bolzano-Weierstrass Theorem; 2.3.1 Subsequences; 2.3.2 Monotone Subsequences; 2.3.3 Cauchy Sequences; 2.3.4 Exercises; 2.4 Limit Superior and Limit Inferior
2.4.1 Subsequential Limits2.4.2 Size of Subsequential Limits; 2.4.3 Limit Superior and Limit Inferior are Subsequential Limits; 2.4.4 Convergence in Terms of Limit Superior and Limit Inferior; 2.4.5 Exercises; 2.5 Additional Exercises; 3. Series; 3.1 Convergence of Series; 3.1.1 Definition of a Series; 3.1.2 Cauchy Convergence Criterion for Series; 3.1.3 Harmonic and Geometric Series; 3.1.4 Cauchy Condensation Test; 3.1.5 Exercises; 3.2 Comparison Tests; 3.2.1 Comparison Test; 3.2.2 Limit Comparison Test; 3.2.3 Exercises; 3.3 Alternating Series Test; 3.3.1 Exercises; 3.4 Absolute Convergence
3.4.1 Absolute and Conditional Convergence3.4.2 Root Test; 3.4.3 Ratio Test; 3.4.4 Exercises; 3.5 Rearrangement of Series; 3.5.1 Exercises; 3.6 Additional Exercises; 4. Continuous Functions; 4.1 Limit Points; 4.1.1 Exercises; 4.2 Limits of Functions; 4.2.1 Limits; 4.2.2 Uniqueness of Limits; 4.2.3 -- Characterization; 4.2.4 One-Sided Limits; 4.2.5 Exercises; 4.3 Continuity; 4.3.1 Sequential Definition of Continuity; 4.3.2 -- Characterization of Continuity; 4.3.3 Exercises; 4.4 Extreme and Intermediate Value Theorems; 4.4.1 Exercises; 4.5 Uniform Continuity; 4.5.1 Exercises
4.6 Monotone and Inverse Functions4.6.1 Monotone Functions; 4.6.2 Continuity of Inverse Functions; 4.6.3 Points of Discontinuity; 4.6.4 Exercises; 4.7 Functions of Bounded Variation; 4.7.1 Variation; 4.7.2 Variations on Different Intervals; 4.7.3 Characterization; 4.7.4 Exercises; 4.8 Additional Exercises; 5. Differentiation; 5.1 The Derivative; 5.1.1 Definition of Derivative; 5.1.2 Differentiability and Continuity; 5.1.3 Arithmetics of Derivatives; 5.1.4 Chain Rule; 5.1.5 Derivatives of Inverse Functions; 5.1.6 Exercises; 5.2 Mean Value Theorem; 5.2.1 Two Preliminary Theorems
Summary This book is an introductory text on real analysis for undergraduate students. The prerequisite for this book is a solid background in freshman calculus in one variable. The intended audience of this book includes undergraduate mathematics majors and students from other disciplines who use real analysis. Since this book is aimed at students who do not have much prior experience with proofs, the pace is slower in earlier chapters than in later chapters. There are hundreds of exercises, and hints for some of them are included
Bibliography Includes bibliographical references (page 191) and index
Notes Print version record
Subject Mathematical analysis.
Form Electronic book
ISBN 9789814417860
9814417866