1549901369-Elements_of_Real_Analysis__Denlinger_

376 Chapter 7 11 The Riemann Integral

Part 2 ( {:::: ): Suppose that Ve > 0, 3 8 > 0 3 V tagged partitions P* of [a, b],

llP II < 8 =? IR(f, P ) - II < E:

i.e., I -c < R(f, P*) < I+ c. (5)

Let E: > 0. Choose 8 as guaranteed by our hypothesis.

Let P be any partition of [a, b] 3 llPll < 8. By the E:-criterion for infimum

(Theorem 1.6.7), for each i = 1,2, · · · ,n, we can select tags x: E [xi-i,xi] 3

1 (x:) < mi + -b E:.

-a

For this choice of tags Xi, the Riemann sum satisfies

n

R(f, P *) = L 1 (x:)6i

i=n l

< L (mi + b ~ a) 6i

i=l n n

= ~ mi6i + _E:_ ~ 6i

L b-aL

i=l i=l

= S_(f, P) + b:a (b -a)

= S_(f, P) + c.

Thus, S_(f, P) > R(f, P*) -c

S_(f, P) > (I - c) - E: by (5) above.

S_(f, P) > I - 2c. Therefore,

l:l > I - 2c.

Thus, Ve > 0, I < l: 1+2c. Therefore, by the forcing principle,

By a similar argument (Exercise 3), we can show that

l:l ~I.

Thus we have,

from which we conclude that 1 is integrable over [a, b], and l: 1 = I. •