382 Chapter 7 • The Riemann Integral
Proof. (a) By definition, l: f =sup mu, P) : Pis a partition of [a, b]}.
Let c: > 0. By the c:-criterion for supremum (Theorem 1.6.6) 3 partition P
of [a, b] such that
Q_(f, P) > l:f -~-
By Theorem 7.3.11, 3 regular partition Qm of [a, b] such that
c:
Q_(f, Qm) > Q_(f, P) - 2> l:f -c:.
Therefore, sup {Q_(f, Qm) : Qm is a regular partition of [a, b]} :::0: l: f. (9)
On the other hand,sup {Q_(f, Qm) : Qm is a regular partition of [a, b]}
:::; sup {Q_(f, P) : P is a partition of [a, b]} = l: f. (10)Putting (9) and (10) together, we have the desired result.(b) Exercise 16. •With the help of these results, we can prove modified forms of Riemann's
condition in terms of regular partitions. In what follows, the notation Qn will
represent the regular n-partition of [a, b].
Theorem 7.3.13 (Regular Partition Riemann's Criterion for Integra-
bility) A bounded function f :[a, b] --->JR is integrable on [a, b] if and only if
I \ic: > 0, 3n EN 3 S(f, Qn) - Q_(f, Qn) < c:.1
Proof. Exercise 17. •Theorem 7.3.14 (Regular Partition Equivalent Form of Riemann's
Criterion) A bounded function f :[a, b] ---> JR is integrable on [a, b] {::} there is
one and only one number I such that \i regular partitions Q of [a, b], Q_(f, Q) :::;
I:::; S(f, Q). (1n this case, I= l: J.)
Proof. Exercise 18. •Similarly, we can restate the Riemann/Darboux criterion for integrability
in terms of regular partitions. When we do so it becomes clear that, when ex-
pressed in terms of regular partitions, the attention is focused on the sequences
{Q_(f, Qn)} and {S(f, Qn) }.