First Course In Real Analysis, A

Preface 0 Preliminaries 0.1 Sets 0.2 The Set of Natural Numbers 0.3 The Real Line 0.4 Mathematical Proofs 0.5 Absolute Values 0.6 The Completeness Axiom 0.7 Cardinality 0.8 Problems 1 Sequences of Real Numbers 1.1 Convergence 1.2 Algebra of Limits 1.3 Monotone Sequences 1.4 Nested Sequence of Intervals 1.5 Subsequences 1.6 Cauchy Sequences 1.7 Contractive Sequences 1.8 Limsup and Liminf 1.9 Problems 2 Functions 2.1 Topology of R 2.2 Limit of a Function 2.3 Monotone Functions 2.4 Continuous Functions 2.5 Uniform Continuity 2.6 The Cantor Set 2.7 Problems 3 Differentiation 3.1 The Derivative 3.2 The Algebra of Derivatives 3.3 Mean Value Theorems 3.4 L'Hopital's Rule 3.5 Problems 4 Integration 4.1 The Riemann Integral 4.2 Riemann Sums 4.3 Properties of Integrals 4.4 Mean Value Theorems 4.5 The Fundamental Theorem of Calculus 4.6 Improper Riemann Integrals 4.7 Problems 5 Series of Real Numbers 5.1 Introduction 5.2 Series with Nonnegative Terms 5.3 Absolute Convergence 5.4 Problems 6 Sequences and Series of Functions 6.1 Pointwise Convergence 6.2 Uniform Convergence 6.3 Uniform Convergence of Series 6.4 Power Series 6.5 Problems Exercises Chapter 1 Chapter 2 Chapter 3 Chapter 4 Chapter 5 Chapter 6 Index

In the early calculus course, the chief emphasis is on developing computational techniques and applications. In real analysis, the emphasis is primarily on understanding the concepts and developing the ability to prove results. A student should not only learn manipulative skills but also argumentative skills. It is assumed that the reader has completed courses in elementary calculus covering one- two- and three-dimensional calculus where he/she is exposed to only the mechanics of calculus without much theory. A course based on this text is suitable not only for mathematics majors, but also for science and engineering majors who need a careful introduction to the concepts and methods of analytical proofs.