﻿ Introduction to Mathematical Analysis I - Second Edition

# Introduction to Mathematical Analysis I - Second Edition

 TOOLS FOR ANALYSIS 8 Basic Concepts of Set Theory 8 Functions 12 The Natural Numbers And Mathematical Induction 16 Ordered Field Axioms 19 The Completeness Axiom for the Real numbers 23 Applications of the Completeness Axiom 27 SEQUENCES 32 Convergence 32 Limit Theorems 38 Monotone Sequences 42 The Bolzano-Weierstrass Theorem 48 Limit Superior and Limit Inferior 51 Open Sets, Closed Sets, Compact Sets, and Limit Points 56 LIMITS AND CONTINUITY 64 Limits of Functions 64 Limit Theorems 68 Continuity 75 Properties of Continuous Functions 78 Uniform Continuity 84 Limit Superior and Limit Inferior of Functions 90 Lower Semicontinuity and Upper Semicontinuity 95 DIFFERENTIATION 102 Definition and Basic Properties of The Derivative 102 The Mean Value Theorem 108 Some Applications of The Mean Value Theorem 115 L'Hospital's Rule 118 Taylor's Theorem 124 Convex Functions and Derivatives 127 Nondifferentiable Convex Functions and Subdifferentials 133 Solutions and Hints for Selected Exercises 144
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