COMPANION NOTES: A WORKING EXCURSION TO ACCOMPANY BABY RUDIN

By O. BEDAN

CONTENTS 1 THE FIELD OF REALS AND BEYOND 1 1 1 FIELDS 1 1 2 ORDERED FIELDS 11 1 2 1 SPECIAL SUBSETS OF AN ORDERED FIELD 15 1 2 2 BOUNDING PROPERTIES 17 1 3 THE REAL FIELD 25 1 3 1 DENSITY PROPERTIES OF THE REALS 27 1 3 2 EXISTENCE OF NTH ROOTS 29 1 3 3 THE EXTENDED REAL NUMBER SYSTEM 33 1 4 THE COMPLEX FIELD 34 1 4 1 THINKING COMPLEX 39 1 5 PROBLEM SET A 43 2 FROM FINITE TO UNCOUNTABLE SETS 49 2 1 SOME REVIEW OF FUNCTIONS 49 2 2 A REVIEW OF CARDINAL EQUIVALENCE 54 2 2 1 DENUMERABLE SETS AND SEQUENCES 61 2 3 REVIEW OF INDEXED FAMILIES OF SETS 62 2 4 CARDINALITY OF UNIONS OVER FAMILIES 65 2 5 THE UNCOUNTABLE REALS 69 2 6 PROBLEM SET B 70 3 METRIC SPACES AND SOME BASIC TOPOLOGY 73 3 1 EUCLIDEAN N SPACE 73 III IV CONTENTS 3 2 METRIC SPACES 77 3 3 POINT SET TOPOLOGY ON METRIC SPACES 82 3 3 1 COMPLEMENTS AND FAMILIES OF SUBSETS OF METRIC SPACES 87 3 3 2 OPEN RELATIVE TO SUBSETS OF METRIC SPACES 96 3 3 3 COMPACT SETS 98 3 3 4 COMPACTNESS IN EUCLIDEAN N SPACE 105 3 3 5 CONNECTED SETS 111 3 3 6 PERFECT SETS 114 3 4 PROBLEM SET C 118 4 SEQUENCES AND SERIES FIRST VIEW 123 4 1 SEQUENCES AND SUBSEQUENCES IN METRIC SPACES 124 4 2 CAUCHY SEQUENCES IN METRIC SPACES 135 4 3 SEQUENCES IN EUCLIDEAN K SPACE 139 4 3 1 UPPER AND LOWER BOUNDS 145 4 4 SOME SPECIAL SEQUENCES 149 4 5 SERIES OF COMPLEX NUMBERS 152 4 5 1 SOME ABSOLUTE CONVERGENCE TESTS 156 4 5 2 ABSOLUTE CONVERGENCE AND CAUCHY PRODUCTS 169 4 5 3 HADAMARD PRODUCTS AND SERIES WITH POSITIVE AND NEGATIVE TERMS 173 4 5 4 DISCUSSING CONVERGENCE 176 4 5 5 REARRANGEMENTS OF SERIES 177 4 6 PROBLEM SET D 178 5 FUNCTIONS ON METRIC SPACES AND CONTINUITY 183 5 1 LIMITS OF FUNCTIONS 183 5 2 CONTINUOUS FUNCTIONS ON METRIC SPACES 197 5 2 1 A CHARACTERIZATION OF CONTINUITY 200 5 2 2 CONTINUITY AND COMPACTNESS 203 5 2 3 CONTINUITY AND CONNECTEDNESS 206 5 3 UNIFORM CONTINUITY 208 5 4 DISCONTINUITIES AND MONOTONIC FUNCTIONS 211 5 4 1 LIMITS OF FUNCTIONS IN THE EXTENDED REAL NUMBER SYSTEM 221 5 5 PROBLEM SET E 225