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.