§1. Convergent sequences and series
49
(a) it is bounded above, i.e. there exist numbers in Q greater than all the
xEX,
(b) the relations x E X and 0 ::; y < x imply y E X,
(c) for every nonzero element x of X there exists ayE X such that x < y.
Intuitively, and leaving aside the set X = {OJ - which satisfies (a), (b) and
(c) -, a cut is the set of all the positive rational numbers which are strictly
less than a given (possibly rational) positive real number, in other words, the
set of all its rational positive approximations from below, to an unspecified
precision: the number 10- 123 is a rational approximation from below to the
number 7r as much as 3.14159 is.
The correct definition of the real numbers to which we have alluded then
consists of saying that a positive real number is purely and simply a cut;
in other words, that there is no difference in nature between a real number
and the set of all the rational numbers which are strictly less than it. This
definition suggests no mystical, metaphysical or physical interpretation of the
number 7r for example, but it allows one to argue rationally about all the real
numbers, even the irrational ones. For a start, one can very simply define the
two fundamental algebraic operations and the inequality relation from this:
(1) the sum X + Y of two cuts is the set of z = x+y with x E X and y E Y;
(2) the product XY of two cuts is the set of z = xy with x EX, Y E Y;
(3) the inequality X::; Y means simply that X c Y.
A posteriori explanations: (1) and (2) mean that if a and b are two real
positive numbers, then every rational positive number < a + b (resp. ab)
is the sum (resp. the product) of two rational positive numbers < a and
< b respectively. (3) means that if a and b are two real numbers, the weak
inequality a ::; b is equivalent to the fact that every rational x < a is also < b
(strict inequalities).
Naturally one has to verify that these definitions lead to sets satisfying
(a), (b) and (c), then that the "obvious" expected properties of addition,
of multiplication and of inequalities (see the following nO) are satisfied. One
must also show that every rational positive number x can be considered as
a real number: to do this one associates to it the set C(x) of y E Q+ such
that y < x if x > 0, or else the cut {OJ if x = o. All this requires only most
elementary reasoning about the rational numbers, patience, and sometimes a
little ingenuity. This done, one can forget this quite abstract construction for
defining the real numbers and, as everyone has always done, restrict oneself to
reasoning from the fundamental properties that we will state in nO 1. The only
interest in this construction is to prove the existence of a mathematical object
rational numbers from the irrational in the construction of R. The concept of a
cut, in Dedekind, has a slightly different meaning: for him, it is a partition of Q
into two nonempty sets X and Y such that x < y for all x E X and all y E Y;
there is then a unique real number "between" X and Y. Exercise: deduce from
(b) that x < y if x E X and y ¢ X, y > o.
49
(a) it is bounded above, i.e. there exist numbers in Q greater than all the
xEX,
(b) the relations x E X and 0 ::; y < x imply y E X,
(c) for every nonzero element x of X there exists ayE X such that x < y.
Intuitively, and leaving aside the set X = {OJ - which satisfies (a), (b) and
(c) -, a cut is the set of all the positive rational numbers which are strictly
less than a given (possibly rational) positive real number, in other words, the
set of all its rational positive approximations from below, to an unspecified
precision: the number 10- 123 is a rational approximation from below to the
number 7r as much as 3.14159 is.
The correct definition of the real numbers to which we have alluded then
consists of saying that a positive real number is purely and simply a cut;
in other words, that there is no difference in nature between a real number
and the set of all the rational numbers which are strictly less than it. This
definition suggests no mystical, metaphysical or physical interpretation of the
number 7r for example, but it allows one to argue rationally about all the real
numbers, even the irrational ones. For a start, one can very simply define the
two fundamental algebraic operations and the inequality relation from this:
(1) the sum X + Y of two cuts is the set of z = x+y with x E X and y E Y;
(2) the product XY of two cuts is the set of z = xy with x EX, Y E Y;
(3) the inequality X::; Y means simply that X c Y.
A posteriori explanations: (1) and (2) mean that if a and b are two real
positive numbers, then every rational positive number < a + b (resp. ab)
is the sum (resp. the product) of two rational positive numbers < a and
< b respectively. (3) means that if a and b are two real numbers, the weak
inequality a ::; b is equivalent to the fact that every rational x < a is also < b
(strict inequalities).
Naturally one has to verify that these definitions lead to sets satisfying
(a), (b) and (c), then that the "obvious" expected properties of addition,
of multiplication and of inequalities (see the following nO) are satisfied. One
must also show that every rational positive number x can be considered as
a real number: to do this one associates to it the set C(x) of y E Q+ such
that y < x if x > 0, or else the cut {OJ if x = o. All this requires only most
elementary reasoning about the rational numbers, patience, and sometimes a
little ingenuity. This done, one can forget this quite abstract construction for
defining the real numbers and, as everyone has always done, restrict oneself to
reasoning from the fundamental properties that we will state in nO 1. The only
interest in this construction is to prove the existence of a mathematical object
rational numbers from the irrational in the construction of R. The concept of a
cut, in Dedekind, has a slightly different meaning: for him, it is a partition of Q
into two nonempty sets X and Y such that x < y for all x E X and all y E Y;
there is then a unique real number "between" X and Y. Exercise: deduce from
(b) that x < y if x E X and y ¢ X, y > o.