414 Chapter 7 • The Riemann Integral
Then Vx E [a, b], mg(x) :::; f(x)g(x) :::; Mg(x), so by Theorems 7.5.2 (d) and
7.5.l (a),
(29)If l: g = 0, inequality (1) shows that l: f g = 0, in which case any c E (a, b)
will do. So suppose l: g =f. 0. Then l: g > 0, so inequality (29) becomes
m < l: Jg< M.
- l: g -
Thus, by the intermediate value theorem (5.3.9), :3 c E (a, b), such that
from which it follows that l: f g = f(c) l: g.
Case 2 (g :::; 0 throughout [a, b ]): Assume this hypothesis and apply Case
1 to -g. •*IRRATIONALITY OF ere AND 7rWe can use the Fundamental Theorem of Calculus to prove that ex is irra-
tional for all nonzero rational numbers x , and that n is irrational. It may come
as a surprise that derivatives and integrals are involved in proving these facts.
Since n can be defined and understood without reference to any mathematics
beyond elementary algebra and geometry, one would not expect that proving
the irrationality of 7r will require the concepts and techniques of real analysis.
Before we can proceed with the proofs, we need to develop a few properties
of the following functions.
*Definition 7.6.19 In the remainder of this section, Vn EN, define
'l/Jn(x) = xn(l - x)n
n!
*Theorem 7.6.20 For all n EN, the function 'l/Jn(x) has the following prop-
erties:
l 2n
(a) 'l/Jn(x) =I 2:: CkXk for some integers Ck·
n. k=n