340
IV - Powers, Exponentials, Logarithms, Trigonometric FUnctions
Let us start by making h tend to 0 through positive values. We shall see
that the ratio (1) is an increasing7 function of h for h > O. Since it is bounded
below by 0 (we know that a h > 1 for a > 1 and h> 0), the existence of the
limit as h decreases to 0 will be assured by the general theorems (Chap. III,
nO 3) on monotone functions.
We now need to show that
(7.2)
at! - 1
a V - 1
O - - < - -
-
u
-
v '
in other words, that, for u and v real,
(7.3)
0< u ~ v implies vat! - ua v ~ v - u.
It is enough to do this for u, v rational. Approximating u and v by sequences
(un) and (vn) of rational numbers, we have indeed
by the continuity of the functions in question; if the relation (3) can be
established for Un and vn , it then holds in the limit for u and v.
Assuming u and v rational, put u = pin, v = qln with p, q, n integers,
n > 0, and p < q since u < v. On dividing by n the relation (2) becomes
aP/ n - 1
a q / n - 1
- - - < - - -
p
q
Putting a 1 / n = b > 1, this can again be written
(7.4)
bP-l
b q - l
- - < - - .
P -
q
This has reduced (2) to the case where the exponents u and v in the formula
(2) are integers.
Now put b = 1 +e with e > o. The algebraic binomial formula immediately
yields the relation
bP - 1 = e + (p _ l)e[2] + (p _ l)(p _ 2)e[3] + ... + (p -1) ... 1 elP]
p
and a similar relation for q. If p < q, it is clear that the coefficient of
e[k] = d< I k! in the formula relative to p is smaller than the corresponding coefficient in the formula relative to q. What is more, the latter contains
more terms than the first. Since all these terms are positive, the relation (4) is
proved, so also (2). [A similar argument to that in Chap. II, nO 10, example 2.]
7 For h of the form lin, this is the result which allowed us to define the function
log in Chap. II, nO 10, Theorem 3.
IV - Powers, Exponentials, Logarithms, Trigonometric FUnctions
Let us start by making h tend to 0 through positive values. We shall see
that the ratio (1) is an increasing7 function of h for h > O. Since it is bounded
below by 0 (we know that a h > 1 for a > 1 and h> 0), the existence of the
limit as h decreases to 0 will be assured by the general theorems (Chap. III,
nO 3) on monotone functions.
We now need to show that
(7.2)
at! - 1
a V - 1
O
-
u
-
v '
in other words, that, for u and v real,
(7.3)
0< u ~ v implies vat! - ua v ~ v - u.
It is enough to do this for u, v rational. Approximating u and v by sequences
(un) and (vn) of rational numbers, we have indeed
by the continuity of the functions in question; if the relation (3) can be
established for Un and vn , it then holds in the limit for u and v.
Assuming u and v rational, put u = pin, v = qln with p, q, n integers,
n > 0, and p < q since u < v. On dividing by n the relation (2) becomes
aP/ n - 1
a q / n - 1
- - - < - - -
p
q
Putting a 1 / n = b > 1, this can again be written
(7.4)
bP-l
b q - l
- - < - - .
P -
q
This has reduced (2) to the case where the exponents u and v in the formula
(2) are integers.
Now put b = 1 +e with e > o. The algebraic binomial formula immediately
yields the relation
bP - 1 = e + (p _ l)e[2] + (p _ l)(p _ 2)e[3] + ... + (p -1) ... 1 elP]
p
and a similar relation for q. If p < q, it is clear that the coefficient of
e[k] = d< I k! in the formula relative to p is smaller than the corresponding coefficient in the formula relative to q. What is more, the latter contains
more terms than the first. Since all these terms are positive, the relation (4) is
proved, so also (2). [A similar argument to that in Chap. II, nO 10, example 2.]
7 For h of the form lin, this is the result which allowed us to define the function
log in Chap. II, nO 10, Theorem 3.