1549901369-Elements_of_Real_Analysis__Denlinger_

7.3 The Integral as a Limit of Riemann Sums 373

Proof. Suppose f is defined and bounded on [a, b]. Then 3M > 0 3 \:/x E

[a, b], lf(x)I :::; M.

Part 1 (~):Suppose that Ve> 0, 36 > 0 3 \:/partitions P of [a,b],

llPll < o::::? S(f, P) -S_(f, P) < c. Let c > 0. For the o > 0 guaranteed by our

b-a

hypothesis, choose any n E N 3 --< o, and let P be the partition of [a, b]

n

into n subintervals of equal l ength. Then llPll = b - a < o, so by hypothesis,

n

S(f, P) -S_(f, P) < c. Hence, by Riemann's criterion for integrability (7.2.14),

f is integrable over [a, b].

Part 2 (::::?): Suppose f is integrable over [a,b]. Let c > 0. By Riemann's

criterion (Theorem 7.2.14), 3 partition P = {xo,x 1 ,x2, · · · ,xn} of [a,b] 3

• c
  S(f, P) -S_(f, P) < "2.

Let this partition remain fixed throughout the remainder of the proof.

c

Leto= 16nM·

Now, consider any partition Q of [a, b] such that II Qll < o. Let R=Q UP.
If R has one more point than Q, then (see Exercise 1)

(1)

S_(f, R) -S_(f, Q):::; 2MllQll - (-2M)llQll = 4MllQll (2)

Since R has at most n points not in Q, we can use mathematical induction
to show that

S_(f, R) - S_(f, Q) :::; 4nMll Qll

< 4nMo

c

=4nM·--

c 16nM

• 4

Now, R 2 Q and R 2 P. Thus, by Theorem 7.2.4,

c

S_(f, P) - S_(f, Q) :::; S_(f, R) - S_(f, Q) <

4

, so

c

-S_(f, Q) < -S_(f, P) + 4·

Using a similar argument with upper sums, we can show that

• 
  
  
  
  • c
    S(f, Q) < S(f, P) +
    4
  
  
  
  

.

(3)

(4)