7.3 The Integral as a Limit of Riemann Sums 375Lemma 7.3.4 For any partition P of [a, b], and any selection of tags xi, x;_, · · · , x~
in their respective subintervals [xi-l, xi], we have
S.(f, P) :S R(f, P*) :S S(f, P).
That is, for a given partition P of [a, b], all Riemann sums for f over P
fall between the upper and lower Darboux sums for f over P.
Proof. Exercise 2. •Theorem 7.3.5 (Limit Criterion for Integrability) Given any
f : [a, b] => JR, f is integrable over [a, b] and l: f = I {::::::?
Ve> 0, 3 8 > 0 3 V tagged partitions P* of [a , b],
llP*ll < 8 => IR(f, P*) - II < E:.That is, for all tagged partitions of sufficiently small mesh, the Riemann
sum is within E: of I.
[Equivalently, 3 k > 0 3 Ve > 0, 3 8 > 0 3 V tagged partitions P of [a, b],
llPll < 8 => IR(f, P*) - II< kc .]
Proof. Part 1 ( => ): Suppose f is integrable over [a, b] and l: f =I.
Let E: > 0. By the Riemann/Darboux criterion (Theorem 7.3.2) 3 8 > 0 3llPll < 8 => S(f, P) - S.(f, P) < c.
Let P* be a tagged partition of [a, b] 3 llP* II < 8. ThenS.(f, P) :S R(f, P*) :S S(f, P) by Lemma 7.3.4, so
-S(f, P) :S -R(f, P*) :S -S.(f, P). Also,
S.(f, P) ::=:;I::=:; S(f, P) by Theorem 7.2.15.Adding the last two inequalities, we have- (S(f, P) - S.(f, P)) :::; I - R(f, P*) :S S(f, P) - S.(f, P),
so IR(f, P*) - II :S S(f, P) - S.(f, P)
< E: since llPll < 8.
Hence, 38 > 0 3 llPll < 8 => IR(f, P) - II< E:.