Contents 1 TOOLS FOR ANALYSIS. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1 Basic Concepts of Set Theory 7 1.2 Functions 11 1.3 The Natural Numbers and Mathematical Induction 15 1.4 Ordered Field Axioms 19 1.5 The Completeness Axiom for the Real Numbers 23 1.6 Applications of the Completeness Axiom 27 2 SEQUENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 2.1 Convergence 33 2.2 Limit Theorems 41 2.3 Monotone Sequences 45 2.4 The Bolzano-Weierstrass Theorem 51 2.5 Limit Superior and Limit Inferior 54 3 LIMITS AND CONTINUITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 3.1 Limits of Functions 61 3.2 Limit Theorems 67 3.3 Continuity 74 3.4 Properties of Continuous Functions 78
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