Introduction to Mathematical Analysis I - Second Edition

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 18 1.5 THE COMPLETENESS AXIOM FOR THE REAL NUMBERS 22 1.6 APPLICATIONS OF THE COMPLETENESS AXIOM 26 2 SEQUENCES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 2.1 CONVERGENCE 31 2.2 LIMIT THEOREMS 37 2.3 MONOTONE SEQUENCES 41 2.4 THE BOLZANO-WEIERSTRASS THEOREM 47 2.5 LIMIT SUPERIOR AND LIMIT INFERIOR 50 2.6 OPEN SETS, CLOSED SETS, COMPACT SETS, AND LIMIT POINTS 55 3 LIMITS AND CONTINUITY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.1 LIMITS OF FUNCTIONS 63 3.2 LIMIT THEOREMS 67 3.3 CONTINUITY 74