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
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.
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