Introduction to Mathematical Analysis I 3rd Edition

TOOLS FOR ANALYSIS	8
Basic Concepts of Set Theory	8
Functions	12
The Natural Numbers and Mathematical Induction	16
Ordered Field Axioms	20
The Completeness Axiom for the Real Numbers	24
Applications of the Completeness Axiom	28
SEQUENCES	34
Convergence	34
Limit Theorems	42
Monotone Sequences	46
The Bolzano-Weierstrass Theorem	52
Limit Superior and Limit Inferior	55
LIMITS AND CONTINUITY	62
Limits of Functions	62
Limit Theorems	68
Continuity	75
Properties of Continuous Functions	79
Uniform Continuity	84
DIFFERENTIATION	90
Definition and Basic Properties of the Derivative	90
The Mean Value Theorem	96
Some Applications of the Mean Value Theorem	103
L'Hôpital's Rule	105
Taylor's Theorem	112
ADDITIONAL TOPICS	116
Topology of the Real Line	116
Continuity and Compactness	123
Limit Superior and Limit Inferior of Functions	126
Lower Semicontinuity and Upper Semicontinuity	132
Convex Functions and Derivatives	137
Nondifferentiable Convex Functions and Subdifferentials	142
References	152
Solutions and Hints for Selected Exercises	154

RkJQdWJsaXNoZXIy NTc4NTAz