22 1. The Regulated and Riemann Integrals
Since
un(x) vn(x) = gn(x) +
1/2n
(gn(x)
1/2n)
=
1/2n−1,
we have
b
a
un(x) dx
b
a
vn(x) dx =
b
a
un(x) vn(x) dx
=
b
a
1
2n−1
dx
=
b a
2n−1
.
Hence, we may apply Theorem 1.7.3 to conclude that f is Riemann
integrable.
Also,
lim
n→∞
b
a
gn(x) dx = lim
n→∞
b
a
vn(x) +
1
2n
dx = lim
n→∞
b
a
vn(x) dx,
and
lim
n→∞
b
a
gn(x) dx = lim
n→∞
b
a
un(x)
1
2n
dx = lim
n→∞
b
a
un(x) dx.
Since for all n,
b
a
vn(x) dx
b
a
f(x) dx
b
a
un(x) dx,
we conclude that
lim
n→∞
b
a
gn(x) dx =
b
a
f(x) dx.
That is, the regulated integral equals the Riemann integral.
Theorem 1.7.6. The set R of bounded Riemann integrable functions
on an interval I = [a, b] is a vector space containing the vector space
of regulated functions.
Proof. We have already shown that every regulated function is Rie-
mann integrable. Hence, we need only show that whenever f, g R
and r R we also have (f + g) R and rf R. We will do only the
sum and leave the product as an exercise.
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