Contents
Preface....................................................... V
I - Sets and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§ 1. Set Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1 - Membership, equality, empty set . . . . . . . . . . . . . . . . . . . . . . 7
2 - The set defined by a relation. Intersections and unions . .. 10
3 - Whole numbers. Infinite sets. . . . . . . . . . . . . . . . . . . . . . . . .. 13
4 - Ordered pairs, Cartesian products, sets of subsets . . . . . .. 17
5 - Functions, maps, correspondences . . . . . . . . . . . . . . . . . . . .. 19
6 - Injections, surjections, bijections . . . . . . . . . . . . . . . . . . . . .. 23
7 - Equipotent sets. Countable sets. . . . . . . . . . . . . . . . . . . . . .. 25
8 - The different types of infinity. . . . . . . . . . . . . . . . . . . . . . . .. 28
9 - Ordinals and cardinals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
§2. The logic of logicians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39
II - Convergence: Discrete variables . . . . . . . . . . . . . . . . . . . . . . . . .. 45
§1. Convergent sequences and series. . . . . . . . . . . . . . . . . . . . . . . . . .. 45
o - Introduction: what is a real number? .................. 45
1 - Algebraic operations and the order relation: axioms of ~. 53
2 - Inequalities and intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56
3 - Local or asymptotic properties. . . . . . . . . . . . . . . . . . . . . . .. 59
4 - The concept of limit. Continuity and differentiability .... 63
5 - Convergent sequences: definition and examples. . . . . . . . .. 67
6 - The language of series ............................ . .. 76
7 - The marvels of the harmonic series. . . . . . . . . . . . . . . . . . .. 81
8 - Algebraic operations on limits .. . . . . . . . . . . . . . . . . . . . . .. 95
§2. Absolutely convergent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98
9 - Increasing sequences. Upper bound of a set of real numbers 98
10 - The function logx. Roots of a positive number ......... 103
11 - What is an integral? ................................ 110
12 - Series with positive terms ........................... 114
13 - Alternating series .................................. 119
14 - Classical absolutely convergent series ................. 123
15 - Unconditional convergence: general case ............... 127
Preface....................................................... V
I - Sets and Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
§ 1. Set Theory. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1 - Membership, equality, empty set . . . . . . . . . . . . . . . . . . . . . . 7
2 - The set defined by a relation. Intersections and unions . .. 10
3 - Whole numbers. Infinite sets. . . . . . . . . . . . . . . . . . . . . . . . .. 13
4 - Ordered pairs, Cartesian products, sets of subsets . . . . . .. 17
5 - Functions, maps, correspondences . . . . . . . . . . . . . . . . . . . .. 19
6 - Injections, surjections, bijections . . . . . . . . . . . . . . . . . . . . .. 23
7 - Equipotent sets. Countable sets. . . . . . . . . . . . . . . . . . . . . .. 25
8 - The different types of infinity. . . . . . . . . . . . . . . . . . . . . . . .. 28
9 - Ordinals and cardinals. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 31
§2. The logic of logicians. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 39
II - Convergence: Discrete variables . . . . . . . . . . . . . . . . . . . . . . . . .. 45
§1. Convergent sequences and series. . . . . . . . . . . . . . . . . . . . . . . . . .. 45
o - Introduction: what is a real number? .................. 45
1 - Algebraic operations and the order relation: axioms of ~. 53
2 - Inequalities and intervals. . . . . . . . . . . . . . . . . . . . . . . . . . . .. 56
3 - Local or asymptotic properties. . . . . . . . . . . . . . . . . . . . . . .. 59
4 - The concept of limit. Continuity and differentiability .... 63
5 - Convergent sequences: definition and examples. . . . . . . . .. 67
6 - The language of series ............................ . .. 76
7 - The marvels of the harmonic series. . . . . . . . . . . . . . . . . . .. 81
8 - Algebraic operations on limits .. . . . . . . . . . . . . . . . . . . . . .. 95
§2. Absolutely convergent series. . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 98
9 - Increasing sequences. Upper bound of a set of real numbers 98
10 - The function logx. Roots of a positive number ......... 103
11 - What is an integral? ................................ 110
12 - Series with positive terms ........................... 114
13 - Alternating series .................................. 119
14 - Classical absolutely convergent series ................. 123
15 - Unconditional convergence: general case ............... 127