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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