Real Analysis Preface 1 Introduction 1.1 What is Analysis? 2 Notions from Set Theory 2.1 Introduction to Sets 2.2 Algebra of Sets 2.3 Ordering & Bounding 3 The Real Numbers 3.1 Why do we Need the Real Numbers? 3.2 Construction of the Reals 3.3 Subsets of the Real Line 3.4 Absolute Value 3.5 Inequalities 4 Sequences 4.1 Sequences & Convergence 4.2 Properties of Limits 4.3 Monotone Sequences 4.4 Subsequences 4.5 Cauchy Sequences 5 Series 5.1 Series 5.2 The Comparison Test 5.3 Series of Positive & Negative Terms 5.4 Power Series 6 Continuous Functions 6.1 The Basic Definition of a Function 6.2 Operations on Functions 6.3 Classes of Functions 6.4 Limits 6.5 Continuity 7 Differentiation 7.1 The Derivative 7.2 Important Theorems about Derivatives 7.3 Higher Derivatives