Introduction to Analysis

1 Sets, Functions, and Proofs 1.1 Logic, and an Introduction to Proof 1.2 Sets and their Operations 1.3 Mathematical Induction 1.4 Functions 1.5 Cardinality 1.6 Troubleshooting Guide for Sets, Functions, and Proofs 2 The Real Numbers 2.1 The Real and Extended Real Number System 2.2 The Supremum and In mum 2.3 The Completeness Axiom 2.4 Troubleshooting Guide for The Real Numbers 3 Sequences and their Limits 3.1 Sequences 3.2 Limits and Convergence 3.3 Limit Theorems 3.4 Subsequences 3.5 Monotone Sequences 3.6 The Bolzano-Weierstrass Theorems 3.7 Cauchy Sequences 3.8 In nite Limits 3.9 The lim sup and lim inf 3.10 Troubleshooting guide for Sequences and Their Limits 4 Series of Real Numbers 4.1 In nite Series 4.2 Convergence Tests for Series with Non-negative Terms 4.3 Other Convergence Tests for Series 4.4 Rearrangements 4.5 Troubleshooting Guide for Series of Real Numbers 5 Limits and Continuity 5.1 Limits of Functions 5.2 Properties of Limits 5.3 In nite Limits 5.4 Continuity and Discontinuity 5.5 Consequences of Continuity 5.6 Uniform Continuity 5.7 Troubleshooting Guide for Limits and Continuity 6 Differentiation 6.1 The Derivative 6.2 The Mean Value Theorem 6.3 Taylor Polynomials 6.4 L'H^opital's Rule 6.5 Troubleshooting Guide for Di erentiation 7 Sequences and Series of Functions 7.1 Sequences of Functions 7.2 Consequences of Uniform Convergence 7.3 In nite Series of Functions 7.4 Power Series 7.5 Troubleshooting Guide for Series of Real Numbers A List of Commonly Used Symbols Bibliography Index Contents