§2. Absolutely convergent series
99
(9.6) If an increasing sequence (un) converges, then the set of its
majomnts possesses a least element, namely the limit of the given
sequence.
Conversely:
(9.7) Let (un) be an increasing sequence, and bounded. Suppose that
the set of its majomnts possesses a least element u. Then Un converges to u.
Take a number r > O. Since u is the least number which majorises the sequence, the number u - r does not majorise it. There is therefore an index p
such that u-r < up. Since the sequence is increasing, one again has u-r < Un
for all n 2: p. But since u majorises the sequence, one has finally
(9.8)
u - r < Un ::::; u for all n 2: p,
which establishes (7).
We have no alternative. When a "magnitude", as one used to call it,
increases constantly but not indefinitely, i.e. remains below a certain finite
value, then common sense - the most widely spread out thing in the world
according to a philosopher and mathematician who believed only what he
could prove or verify himself -, common sense, then, indicates that this magnitude must necessarily accumulate towards a limit. In mathematical terms:
every increasing bounded-above sequence converges.
For example, common sense indicates that the sequence
3
3.1
3.14
3.141
3.1415
3.14159
must converge to something. Alas, this something is not rational, so common
sense will not help us at all if we know only about Q. This will not wipe out
any of the banalities which we have already established in this chapter, since
they rely only on axioms (I), (II) and (III) common to Q and to JR, including
(6) and (7); but to go further one clearly needs an axiom specific to JR so
as not to have false theorems such as: "every bounded increasing sequence
of rational numbers converges to a rational limit" or, what would hardly be
better, under penalty of being unable to attribute a limit to almost all the
convergent sequences that one meets in analysis.
It is clearly the axiom (IV) of nO 1 that we lack. This affirms that if a
nonempty set E c JR, for example the set of numbers of the form Un in what
precedes, is bounded above, then the set of numbers which majorise it, its
majorants, possesses a least element, its least upper bound. This makes the
following theorem obvious, by (7):
Theorem 2. For an increasing sequence of real numbers to converge it is
necessary and sufficient that it be bounded above. Its limit is then the least
number which majorises it, i.e. the least upper bound of the set of its terms.
99
(9.6) If an increasing sequence (un) converges, then the set of its
majomnts possesses a least element, namely the limit of the given
sequence.
Conversely:
(9.7) Let (un) be an increasing sequence, and bounded. Suppose that
the set of its majomnts possesses a least element u. Then Un converges to u.
Take a number r > O. Since u is the least number which majorises the sequence, the number u - r does not majorise it. There is therefore an index p
such that u-r < up. Since the sequence is increasing, one again has u-r < Un
for all n 2: p. But since u majorises the sequence, one has finally
(9.8)
u - r < Un ::::; u for all n 2: p,
which establishes (7).
We have no alternative. When a "magnitude", as one used to call it,
increases constantly but not indefinitely, i.e. remains below a certain finite
value, then common sense - the most widely spread out thing in the world
according to a philosopher and mathematician who believed only what he
could prove or verify himself -, common sense, then, indicates that this magnitude must necessarily accumulate towards a limit. In mathematical terms:
every increasing bounded-above sequence converges.
For example, common sense indicates that the sequence
3
3.1
3.14
3.141
3.1415
3.14159
must converge to something. Alas, this something is not rational, so common
sense will not help us at all if we know only about Q. This will not wipe out
any of the banalities which we have already established in this chapter, since
they rely only on axioms (I), (II) and (III) common to Q and to JR, including
(6) and (7); but to go further one clearly needs an axiom specific to JR so
as not to have false theorems such as: "every bounded increasing sequence
of rational numbers converges to a rational limit" or, what would hardly be
better, under penalty of being unable to attribute a limit to almost all the
convergent sequences that one meets in analysis.
It is clearly the axiom (IV) of nO 1 that we lack. This affirms that if a
nonempty set E c JR, for example the set of numbers of the form Un in what
precedes, is bounded above, then the set of numbers which majorise it, its
majorants, possesses a least element, its least upper bound. This makes the
following theorem obvious, by (7):
Theorem 2. For an increasing sequence of real numbers to converge it is
necessary and sufficient that it be bounded above. Its limit is then the least
number which majorises it, i.e. the least upper bound of the set of its terms.