The Riemann Integral

10 1. The Riemann Integral
Proof. If P = {I 1 ,I 2 ,...,I n } is a partition of [a,b], then
n∑
[ ]
U (f;P)−L(f;P) = supf −inff
·|I k |
I k
I k
=
≤ C
k=1
n∑
k=1
osc
I k
f ·|I k |
n∑
k=1
osc
I k
g ·|I k |
≤ C[U(g;P)−L(g;P)].
Thus, f satisfies the Cauchy criterion in Theorem 1.14 if g does, which proves that
f is integrable if g is integrable.
□
We can also give a sequential characterization of integrability.
Theorem 1.17. A bounded function f : [a,b] → R is Riemann integrable if and
only if there is a sequence (P n ) of partitions such that
In that case,
∫ b
a
lim [U(f;P n)−L(f;P n )] = 0.
n→∞
f = lim
n→∞ U(f;P n) = lim
n→∞ L(f;P n).
Proof. First, suppose that the condition holds. Then, given ǫ > 0, there is an
n ∈ N such that U(f;P n ) − L(f;P n ) < ǫ, so Theorem 1.14 implies that f is
integrable and U(f) = L(f).
Furthermore, since U(f) ≤ U(f;P n ) and L(f;P n ) ≤ L(f), we have
0 ≤ U(f;P n )−U(f) = U(f;P n )−L(f) ≤ U(f;P n )−L(f;P n ).
Since the limit of the right-hand side is zero, the ‘squeeze’ theorem implies that
It also follows that
lim U(f;P n) = U(f) =
n→∞
lim L(f;P n) = lim U(f;P n)− lim [U(f;P n)−L(f;P n )] =
n→∞ n→∞ n→∞
Conversely, if f is integrable then, by Theorem 1.14, for every n ∈ N there
exists a partition P n such that
∫ b
0 ≤ U(f;P n )−L(f;P n ) < 1 n ,
a
f
∫ b
a
f.
and U(f;P n )−L(f;P n ) → 0 as n → ∞.
□
Note that if the limits of U(f;P n ) and L(f;P n ) both exist and are equal, then
lim [U(f;P n)−L(f;P n )] = lim U(f;P n)− lim L(f;P n),
n→∞ n→∞ n→∞
so the conditions of the theorem are satisfied. Conversely, the proof of the theorem
shows that if the limit of U(f;P n ) −L(f;P n ) is zero, then the limits of U(f;P n )