1549901369-Elements_of_Real_Analysis__Denlinger_

380 Chapter 7 • The Riemann Integral

The next theorem and its consequences provide conclusive evidence that

regular partitions are sufficient for Riemann integrability.

Theorem 7.3.11 Suppose f is defined and bounded on [a, b], where a < b.
Then, for every partition P of [a, b] and ' 0, :J regular m-partition Qm of
[a, b] such that

S.(f, Qm) > S.(f, P) - € and S(f, Qm) < S(f, P) + €.

Proof. Suppose f is defined and bounded on [a, b], where a < b, and
P = {xo,x1,x2,· · · ,xn} is any partition of [a,b]. Then :JB > 0 3 'ix E [a,b],
lf(x)I < B. Let 60 = min{61, 62, · · · , l::::.n}·

{

b-a n(n+l)B(b-a)}

Let c > O. Choose any natural number m > max

60

,

2

c ·

Let Qm = {yo, Y1, y2, · · · , Ym} be the regular m-partition of [a, b]. The condi-
b - a b-a
tion m > --;:::--assures us that llQmll = --< 60 , and thus each subinterval
uo m
[xi-1, Xi] created by P contains at least one point of Q. To see the relationship
between S.(f, P) and S.(f, Qm) we introduce some notation:
n
S.(f, P) = L mi!::::.i, where mi= inf f ([xi-1, xi]) and 6i =Xi - Xi-1,
i=l

and for each i = 1 , 2, · · · , n, let ki =min{ k : Yk ;::: xi}·

As suggested by Figure 7. 6,

m161:::; m16 + m26 + · · · + m1o 1 - 16 +Bf::::.

= (m1 + m2 + · · · + m1o 1 )6 + (B - m1o 1 )6

:::; (m1 + m2 + · · · + m1o 1 )6 + (B - (-B))!::::.

:::; (m1 + m2 + · · · + m1o,)6 + 2B6.

Continuing, with the help of Figure 7.6, we see that

m262:::; Bf::::.+ m1o 1 +i6 + m1o 1 +26 + · · · + m1or16 +Bf::::..

= (m1o 1 +1 + m1o 1 +2 + · · · + m1o 2 )6 + (B + B - m1oJ6

:::; (m1o 1 +1 + m1o 1 +2 + · · · + m1o 2 )6 + (B + B - (-B))!::::.

:::; (m1o 1 +i + mk,+2 + · · · + mk 2 )6 + 3B6.

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