§2. Absolutely convergent series
101
upper bound is established. The problem is solved by axiom (IV) of nO 1
which the majority of authors call Bolzano's "Theorem" because he was the
first, it seems, to formulate it more or less clearly, in 1817. He did not prove
it, no doubt because it seemed too obvious to him, and, in reality, because no
one of his time yet had a sufficiently clear idea of the concept of real number
to be able to provide a correct proof. Hardly a surprising situation: if you
want to prove such a "theorem" you have to rely on other previous results;
now the axioms (I), (II) and (III) of nO 1 clearly have not the least chance
of sufficing, since, if such were the case, they would prove that Bolzano's
Theorem is valid in Q; the invention of real numbers would then be totally
superfluous. If one really wants to make the existence of least upper bounds a
theorem, one needs to have either an axiom valid in JR but not in Q, as do those
who prefer the "nested intervals" axiom (for us, a theorem of Chap. III), or a
rigorous construction of JR, for example that provided by Dedekind sections
of which we have spoken in the introduction and at the end of n° 1.
It may all the same be interesting to show that Bolzano's Theorem, as it
is called, which trivially implies Theorem 2, is, conversely, a consequence of
it, so that Theorem 2 might have also have been taken as axiom (IV).
Bolzano's "Theorem" (1817). Every nonempty bounded-above subset E
of JR possesses one and only one least upper bound.
Uniqueness is clear: one cannot see how a set of numbers, the majorants
of E, which possesses a least element could possess two different ones, for
each has to be smaller than the other. Paul Klee once depicted two naked
bureaucrats bowing to each other with their backs at the horizontal, each
thinking the other of a higher rank than his own. This has not prevented some
authors, including the present one formerly, from providing a textbook proof
of the uniqueness of the least upper bound. Now let us prove its existence.
Let M be a majorant of E. There are integers n E Z which majorise E, for
example those larger than M. There are also integers which do not majorise
E since E is nonempty; they are all < M, so one can consider the largest of
them, say uo. It does not majorise E, but Uo + 1 majorises E, for otherwise
Uo would not be the largest possible.
Among the numbers Uo + n/lO, where n is an integer 2: 0, let Ul be the
largest of those which do not majorise E; one has n ~ 9 for this number since
Uo + 10/10 majorises E. So Uo ~ Ul. Ul does not majorise E, but Ul + 1/10
does.
Similarly, let U2 be the largest of the numbers of the form Ul + n/l00
which do not majorise E. One has n ~ 9 since Ul + 10/100 majorises E,
Ul ~ U2, U2 does not majorise E, but U2 + 1/100 does.
On repeating this construction indefinitely, one obtains an increasing sequence of numbers Un possessing the following properties:
(i) Un does not majorise E, (ii) Un + 1/lO n majorises E.
101
upper bound is established. The problem is solved by axiom (IV) of nO 1
which the majority of authors call Bolzano's "Theorem" because he was the
first, it seems, to formulate it more or less clearly, in 1817. He did not prove
it, no doubt because it seemed too obvious to him, and, in reality, because no
one of his time yet had a sufficiently clear idea of the concept of real number
to be able to provide a correct proof. Hardly a surprising situation: if you
want to prove such a "theorem" you have to rely on other previous results;
now the axioms (I), (II) and (III) of nO 1 clearly have not the least chance
of sufficing, since, if such were the case, they would prove that Bolzano's
Theorem is valid in Q; the invention of real numbers would then be totally
superfluous. If one really wants to make the existence of least upper bounds a
theorem, one needs to have either an axiom valid in JR but not in Q, as do those
who prefer the "nested intervals" axiom (for us, a theorem of Chap. III), or a
rigorous construction of JR, for example that provided by Dedekind sections
of which we have spoken in the introduction and at the end of n° 1.
It may all the same be interesting to show that Bolzano's Theorem, as it
is called, which trivially implies Theorem 2, is, conversely, a consequence of
it, so that Theorem 2 might have also have been taken as axiom (IV).
Bolzano's "Theorem" (1817). Every nonempty bounded-above subset E
of JR possesses one and only one least upper bound.
Uniqueness is clear: one cannot see how a set of numbers, the majorants
of E, which possesses a least element could possess two different ones, for
each has to be smaller than the other. Paul Klee once depicted two naked
bureaucrats bowing to each other with their backs at the horizontal, each
thinking the other of a higher rank than his own. This has not prevented some
authors, including the present one formerly, from providing a textbook proof
of the uniqueness of the least upper bound. Now let us prove its existence.
Let M be a majorant of E. There are integers n E Z which majorise E, for
example those larger than M. There are also integers which do not majorise
E since E is nonempty; they are all < M, so one can consider the largest of
them, say uo. It does not majorise E, but Uo + 1 majorises E, for otherwise
Uo would not be the largest possible.
Among the numbers Uo + n/lO, where n is an integer 2: 0, let Ul be the
largest of those which do not majorise E; one has n ~ 9 for this number since
Uo + 10/10 majorises E. So Uo ~ Ul. Ul does not majorise E, but Ul + 1/10
does.
Similarly, let U2 be the largest of the numbers of the form Ul + n/l00
which do not majorise E. One has n ~ 9 since Ul + 10/100 majorises E,
Ul ~ U2, U2 does not majorise E, but U2 + 1/100 does.
On repeating this construction indefinitely, one obtains an increasing sequence of numbers Un possessing the following properties:
(i) Un does not majorise E, (ii) Un + 1/lO n majorises E.