102
II - Convergence: Discrete variables
Since Un :::; M for all n, the sequence Un tends to a limit u, by virtue of
Theorem 2. Let us show that U satisfies the conditions (SUP 1) and (SUP 2')
which characterise the least upper bound of E.
For all x E E we have x :::; Un + I/IO n for any n, since the right hand
side majorises E. On the other hand, it converges to u. The remarks at the
beginning of this nO then show that x :::; u, whence (SUP 1).
Since Un does not majorise E, for each n there is an Xn E E such that
Un < Xn· One has also Xn :::; Un +I/IO n since the right hand side majorises E.
In consequence, lim Xn = lim Un = u, whence (SUP 2'), and this completes
the proof.
We have quite deliberately used decimal numbers in the preceding proof,
to construct successive terminating decimal expansions uo, UI, U2, .•. for u.
To be precise, if one writes the default decimal expansion of each x E E in
the form
x = XO.XIX2 ...
with an integer part Xo and decimals XI,X2, ..• between 0 and 9, then one
obtains Uo, UI, etc. by the following procedure: Uo is the maximum value
taken by Xo when x varies in E; UI has integer part Uo and its first decimal
(the following are zeros) is the maximum value of Xl when x runs through
the set Eo of x E E such that Xo = uo; the decimal expansion of U2 starts
like that of UI, but has one more decimal, namely the maximum value of X2
when x runs through the set EI CEo c E of x E E such that XO,XI = UI,
and so on. In other words, one considers the x E E whose integer part is a
maximum, then, among them, those whose first decimal is a maximum, then,
among these, those whose second decimal is a maximum, etc. On pursuing
this construction indefinitely we find the successive decimals of the least upper
bound of E which we seek.
This construction allows one to prove the theorem "without knowing anything" subject to accepting that any nonterminating expansion decimal corresponds to a real number; but this comes back to accepting either axiom
(IV) of n° 1, or Theorem 2 which, as we have seen, is equivalent to it. More
ingenious arguments will never let you escape this. On the contrary, it is axiom (IV) which justifies the decimal representation of the real numbers.
The concept of least upper bound also applies to the case of a family (Ui),
i E I, of real numbers - in other words, up to notation, of a map of the set I
into JR.; the least upper bound of the set E of Ui (x E E <:==> there exists an
i such that x = Ui) is denoted by
SUPUi
iEI
or simply SUP(Ui) if there is no fear of ambiguity. This is the least number
which majorises all the Ui or again, among the numbers which majorise it,
II - Convergence: Discrete variables
Since Un :::; M for all n, the sequence Un tends to a limit u, by virtue of
Theorem 2. Let us show that U satisfies the conditions (SUP 1) and (SUP 2')
which characterise the least upper bound of E.
For all x E E we have x :::; Un + I/IO n for any n, since the right hand
side majorises E. On the other hand, it converges to u. The remarks at the
beginning of this nO then show that x :::; u, whence (SUP 1).
Since Un does not majorise E, for each n there is an Xn E E such that
Un < Xn· One has also Xn :::; Un +I/IO n since the right hand side majorises E.
In consequence, lim Xn = lim Un = u, whence (SUP 2'), and this completes
the proof.
We have quite deliberately used decimal numbers in the preceding proof,
to construct successive terminating decimal expansions uo, UI, U2, .•. for u.
To be precise, if one writes the default decimal expansion of each x E E in
the form
x = XO.XIX2 ...
with an integer part Xo and decimals XI,X2, ..• between 0 and 9, then one
obtains Uo, UI, etc. by the following procedure: Uo is the maximum value
taken by Xo when x varies in E; UI has integer part Uo and its first decimal
(the following are zeros) is the maximum value of Xl when x runs through
the set Eo of x E E such that Xo = uo; the decimal expansion of U2 starts
like that of UI, but has one more decimal, namely the maximum value of X2
when x runs through the set EI CEo c E of x E E such that XO,XI = UI,
and so on. In other words, one considers the x E E whose integer part is a
maximum, then, among them, those whose first decimal is a maximum, then,
among these, those whose second decimal is a maximum, etc. On pursuing
this construction indefinitely we find the successive decimals of the least upper
bound of E which we seek.
This construction allows one to prove the theorem "without knowing anything" subject to accepting that any nonterminating expansion decimal corresponds to a real number; but this comes back to accepting either axiom
(IV) of n° 1, or Theorem 2 which, as we have seen, is equivalent to it. More
ingenious arguments will never let you escape this. On the contrary, it is axiom (IV) which justifies the decimal representation of the real numbers.
The concept of least upper bound also applies to the case of a family (Ui),
i E I, of real numbers - in other words, up to notation, of a map of the set I
into JR.; the least upper bound of the set E of Ui (x E E <:==> there exists an
i such that x = Ui) is denoted by
SUPUi
iEI
or simply SUP(Ui) if there is no fear of ambiguity. This is the least number
which majorises all the Ui or again, among the numbers which majorise it,