56
II - Convergence: Discrete variables
(IV) Let E be a nonempty set of real numbers. Suppose that there exist numbers M E IR such that x ::; M for all x E E. Then the set of these
numbers possesses a least element.
This least of all the majorants or upper bounds M of E is called the least
upper bound of E.
If for example E is the set of truncated decimal expansions for 7r, this
number will be 7r itself. The necessity of axiom (IV) will become clear in nO 9
if it is not yet so at this stage.
If one defines the real numbers by means of cuts as explained at the end
of nO 0, then axiom (IV) becomes a theorem, like the others, even almost
trivial. For let E be a set of real numbers, positive (for simplicity), and
bounded above. By definition, every x E E is a subset of Ql+ satisfying the
conditions (a), (b) and (c) of nO 0, the relation x ::; y being, by definition,
equivalent to the inclusion relation xC y. The least upper bound of E is then
the union u in Ql+ of all the sets x E E. It is indeed obvious that the set u
satisfies the conditions (a), (b), (c) stated at the end of nO 0, that u ~ x (i.e.
u:J x) for all x E E, and that y ~ u (Le. y :J u) for all majorants y of E, i.e.
for every cut such that y :J x for all x E Ej this is even almost the definition
of a union of sets: the smallest set which contains them all.
2 - Inequalities and intervals
The handling of inequalities is absolutely fundamental in analysis, since they
govern the approximation calculations in constant use. We shall not prove
them in detail: everyone knows them, and sceptics, if any there are, may
find their proofs in the first volume of Dieudonne's Treatise on Analysis,
(Academic Press, 1960) for example.
To avoid confusions with the anglo-saxon textbooks, as one calls them
in the Quai d'Orsay, let us be very careful to make clear that for us the
relation x ~ 0 means that x is positive (in the wide sense), while the relation
x > 0 means that x is strictly positive. The anglophones say non negative
and positive, the Germans say positiv for x > 0 and negativ for x < 0,
the number 0 being, for them, neither the one nor the other. Despite the
hymns to "the French exception" , the French will probably end up imitating
the Americans, since roughly 60% of global mathematical production is in
English, and America provides about 40% of the total. But as a contemporary
sociologist has remarked, one cannot change society by decree.
The first essential point is the triangle inequality
(2.1)
Ix + yl ::; Ixl + IYI,
valid for all complex x and y, and obvious geometrically (one can also prove
it ... ). One can generalise it to
(2.2)
II - Convergence: Discrete variables
(IV) Let E be a nonempty set of real numbers. Suppose that there exist numbers M E IR such that x ::; M for all x E E. Then the set of these
numbers possesses a least element.
This least of all the majorants or upper bounds M of E is called the least
upper bound of E.
If for example E is the set of truncated decimal expansions for 7r, this
number will be 7r itself. The necessity of axiom (IV) will become clear in nO 9
if it is not yet so at this stage.
If one defines the real numbers by means of cuts as explained at the end
of nO 0, then axiom (IV) becomes a theorem, like the others, even almost
trivial. For let E be a set of real numbers, positive (for simplicity), and
bounded above. By definition, every x E E is a subset of Ql+ satisfying the
conditions (a), (b) and (c) of nO 0, the relation x ::; y being, by definition,
equivalent to the inclusion relation xC y. The least upper bound of E is then
the union u in Ql+ of all the sets x E E. It is indeed obvious that the set u
satisfies the conditions (a), (b), (c) stated at the end of nO 0, that u ~ x (i.e.
u:J x) for all x E E, and that y ~ u (Le. y :J u) for all majorants y of E, i.e.
for every cut such that y :J x for all x E Ej this is even almost the definition
of a union of sets: the smallest set which contains them all.
2 - Inequalities and intervals
The handling of inequalities is absolutely fundamental in analysis, since they
govern the approximation calculations in constant use. We shall not prove
them in detail: everyone knows them, and sceptics, if any there are, may
find their proofs in the first volume of Dieudonne's Treatise on Analysis,
(Academic Press, 1960) for example.
To avoid confusions with the anglo-saxon textbooks, as one calls them
in the Quai d'Orsay, let us be very careful to make clear that for us the
relation x ~ 0 means that x is positive (in the wide sense), while the relation
x > 0 means that x is strictly positive. The anglophones say non negative
and positive, the Germans say positiv for x > 0 and negativ for x < 0,
the number 0 being, for them, neither the one nor the other. Despite the
hymns to "the French exception" , the French will probably end up imitating
the Americans, since roughly 60% of global mathematical production is in
English, and America provides about 40% of the total. But as a contemporary
sociologist has remarked, one cannot change society by decree.
The first essential point is the triangle inequality
(2.1)
Ix + yl ::; Ixl + IYI,
valid for all complex x and y, and obvious geometrically (one can also prove
it ... ). One can generalise it to
(2.2)