§ 1. Direct construction
329
and, on the other hand, that one can choose u and v so that the differences
I-(s) - aU and a V - I+(s) are arbitrarily small. So it all reduces to proving
the following lemma, which will also, a little later, assure us of the continuity
of the map s f---* as on IR (see also Chap. III, nO 11, (IV bis)):
Lemma. Let a and s be two real numbers; assume that a > 1. Then, lor any
number r > 0, there exist rational numbers u and v such that
(2.8)
u < s < v,
For any n E N we can indeed find a number Un in Q such that
Un < s < Un + lin = vn • Then
(2.9)
a Vn - a Un = a Un (a 1 / n - 1) ::; I-(s). (a 1 / n -1)
•.. by (7). Now lima 1 / n = 1 as we saw in Chap. II, nO 5, example 7, where we
inoffensively anticipated the existence of nth roots. The left hand side of (9)
. is thus ::; r for n large, qed.
We may now define the number
(2.10)
unambiguously, and it remains to establish that it satisfies all the conditions
we require.
First, I(s) = as for SEQ. We see this 3 from the lemma above, valid for
SEQ as for s E 1R: in Q, we already know that
and, since 1- (s) and 1+ (s) also lie between aU and a v, the lemma shows that
as = f-(s) = 1+(8).
On the other hand f(8) < f(t) for all real s and t such that s < t. To
see this, consider rational numbers u and v such that s ::; u < v ::; tj we
have i+(s) ::; aU and a V ::; f-(t) by definition of these expressions, whence
f(s) = 1+(s) ::; aU < a V ::; f-(t) = f(t), hence the result. (The reader will be
interested scrupulously to distinguish the strict from the weak inequalities in
these proofs ... )
The uniqueness of the monotone function f was shown above.
Since we now know that f(u) = aU for u E Q, we can replace aU and
a V by f(u) and f(v) in the lemma abovej on the interval [u,vJ, i.e. on a
neighbourhood of s, the function f, being monotone, lies between aU and aVo
It is therefore equal to f(8) to within r on a neighbourhood of s, whence the
continuity of f.
3 This is not obvious: the definition of 1(8) for real 8 involves only rational exponents different from 8, even il8 E Q.
329
and, on the other hand, that one can choose u and v so that the differences
I-(s) - aU and a V - I+(s) are arbitrarily small. So it all reduces to proving
the following lemma, which will also, a little later, assure us of the continuity
of the map s f---* as on IR (see also Chap. III, nO 11, (IV bis)):
Lemma. Let a and s be two real numbers; assume that a > 1. Then, lor any
number r > 0, there exist rational numbers u and v such that
(2.8)
u < s < v,
For any n E N we can indeed find a number Un in Q such that
Un < s < Un + lin = vn • Then
(2.9)
a Vn - a Un = a Un (a 1 / n - 1) ::; I-(s). (a 1 / n -1)
•.. by (7). Now lima 1 / n = 1 as we saw in Chap. II, nO 5, example 7, where we
inoffensively anticipated the existence of nth roots. The left hand side of (9)
. is thus ::; r for n large, qed.
We may now define the number
(2.10)
unambiguously, and it remains to establish that it satisfies all the conditions
we require.
First, I(s) = as for SEQ. We see this 3 from the lemma above, valid for
SEQ as for s E 1R: in Q, we already know that
and, since 1- (s) and 1+ (s) also lie between aU and a v, the lemma shows that
as = f-(s) = 1+(8).
On the other hand f(8) < f(t) for all real s and t such that s < t. To
see this, consider rational numbers u and v such that s ::; u < v ::; tj we
have i+(s) ::; aU and a V ::; f-(t) by definition of these expressions, whence
f(s) = 1+(s) ::; aU < a V ::; f-(t) = f(t), hence the result. (The reader will be
interested scrupulously to distinguish the strict from the weak inequalities in
these proofs ... )
The uniqueness of the monotone function f was shown above.
Since we now know that f(u) = aU for u E Q, we can replace aU and
a V by f(u) and f(v) in the lemma abovej on the interval [u,vJ, i.e. on a
neighbourhood of s, the function f, being monotone, lies between aU and aVo
It is therefore equal to f(8) to within r on a neighbourhood of s, whence the
continuity of f.
3 This is not obvious: the definition of 1(8) for real 8 involves only rational exponents different from 8, even il8 E Q.