Table of Contents of Volume I
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
0 – Introduction: what is a real number? . . . . . . . . . . . . . . . . . . 45
1 – Algebraic operations and the order relation: axioms of R . 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 log x. 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
315
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
0 – Introduction: what is a real number? . . . . . . . . . . . . . . . . . . 45
1 – Algebraic operations and the order relation: axioms of R . 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 log x. 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
315