§1. Convergent sequences and series
55
The binomial formula can be written in the form
(1.4)
(x + y)n In! = L xPyq Ip!q!
p+q=n
where the L is taken over all the ordered pairs of integers p, q E N such that
p + q = n. If one writes generally
(1.5)
(divided powers), one then has
(1.6)
(x + y)[n1 = L x[ply[ql.
In this form, the relation extends to a sum of any number of terms; for
example,
(1.7)
(x + y + z + u)[nl =
p+q+r+s=n
where, here again, the L means that one must give the letters p, q, r, s all
positive integer values such that p + q + r + s = n and then calculate the
sum of all the corresponding expressions x[Ply[qlz[r1u[sl. In a relation such as
(6), the letter n denotes a fully determined integer, while the letters p, ... , s
denote bound variables, or phantoms, whose only function is to serve as a
logical link between the symbol L and the monomial x[Ply[qlz[r1u[sl. One can
of course denote them by letters other than p, ... , s so long as one does not
use the letter n, which has a totally different sense; an expression such as
n
is meaningless.
A second group of formulae involves the order relation x ::; y in lR or Q:
(11.1) the relations x ::; y and y ::; z imply x ::; z;
(11.2) the relation {(x ::; y) & (y::; x)} is equivalent to x = y;
(11.3) for all x and y, one has x ::; y or y ::; x;
(11.4) the relation x ::; y implies x + z ::; y + z for all z;
(11.5) the relations 0 ::; x and 0 ::; y imply 0 ::; xy.
Next comes Archimedes' axiom:
(III) given x, y > 0 there exists an n E N such that y < nx.
As we said above, one could, in the preceding, replace lR by the set Q of
rational numbers. The fourth axiom, not true for Q, characterises the real
numbers; in this form or in equivalent forms, it is indispensable to proving
that there is something nontrivial to analysis. One can state it in several
equivalent ways, for example:
55
The binomial formula can be written in the form
(1.4)
(x + y)n In! = L xPyq Ip!q!
p+q=n
where the L is taken over all the ordered pairs of integers p, q E N such that
p + q = n. If one writes generally
(1.5)
(divided powers), one then has
(1.6)
(x + y)[n1 = L x[ply[ql.
In this form, the relation extends to a sum of any number of terms; for
example,
(1.7)
(x + y + z + u)[nl =
p+q+r+s=n
where, here again, the L means that one must give the letters p, q, r, s all
positive integer values such that p + q + r + s = n and then calculate the
sum of all the corresponding expressions x[Ply[qlz[r1u[sl. In a relation such as
(6), the letter n denotes a fully determined integer, while the letters p, ... , s
denote bound variables, or phantoms, whose only function is to serve as a
logical link between the symbol L and the monomial x[Ply[qlz[r1u[sl. One can
of course denote them by letters other than p, ... , s so long as one does not
use the letter n, which has a totally different sense; an expression such as
n
is meaningless.
A second group of formulae involves the order relation x ::; y in lR or Q:
(11.1) the relations x ::; y and y ::; z imply x ::; z;
(11.2) the relation {(x ::; y) & (y::; x)} is equivalent to x = y;
(11.3) for all x and y, one has x ::; y or y ::; x;
(11.4) the relation x ::; y implies x + z ::; y + z for all z;
(11.5) the relations 0 ::; x and 0 ::; y imply 0 ::; xy.
Next comes Archimedes' axiom:
(III) given x, y > 0 there exists an n E N such that y < nx.
As we said above, one could, in the preceding, replace lR by the set Q of
rational numbers. The fourth axiom, not true for Q, characterises the real
numbers; in this form or in equivalent forms, it is indispensable to proving
that there is something nontrivial to analysis. One can state it in several
equivalent ways, for example: