10 1. The Regulated and Riemann Integrals

Therefore, whenever p, q ≥ M,

|zp − zq | =

b

a

fp(x) − fq(x) dx

≤

b

a

|fp(x) − fq (x)| dx

≤

b

a

ε

b − a

dx = ε,

where the first inequality comes from the absolute value property

of Proposition 1.2.1 and the second follows from the monotonicity

property and equation (1.5.1). This shows that the sequence {zm} is

Cauchy and hence converges.

Now suppose that {gm} is another sequence of step functions

which also converges uniformly to f. Then for any ε 0 there is an

M such that for all x,

|f(x) − fm(x)| ε and |f(x) − gm(x)| ε

whenever m ≥ M. It follows that

|fm(x) − gm(x)| ≤ |fm(x) − f(x)| + |f(x) − gm(x)| 2ε.

Hence, using the absolute value and monotonicity properties, we see

b

a

fm(x) − gm(x) dx ≤

b

a

|fm(x) − gm(x)| dx

≤

b

a

2ε dx = 2ε(b − a),

for all m ≥ M. Since ε is arbitrarily small we may conclude that

lim

m→∞

b

a

fm(x) dx −

b

a

gm(x) dx

= lim

m→∞

b

a

fm(x) − gm(x) dx = 0.

This implies

lim

m→∞

b

a

fm(x) = lim

m→∞

b

a

gm(x) dx.

This result enables us to define the regulated integral.