1.7. The Riemann Integral 19
a step function v less than f such that the difference of the integrals
of u and v is less than ε.
Theorem 1.7.3. A bounded function f : [a, b] R is Riemann
integrable if and only if, for every ε 0 there are step functions v0
and u0 such that v0(x) f(x) u0(x) for all x [a, b] and
b
a
u0(x) dx
b
a
v0(x) dx ε.
Proof. Suppose the functions v0 L(f) and u0 U(f) have integrals
within ε of each other. Then
b
a
v0(x) dx sup
v∈L(f )
b
a
v(x) dx
inf
u∈U(f )
b
a
u(x) dx

b
a
u0(x) dx,
where the second inequality follows from Proposition 1.7.1.
This implies
inf
u∈U(f )
b
a
u(x) dx sup
v∈L(f )
b
a
v(x) dx ε.
Since this is true for all ε 0, we conclude that f is Riemann inte-
grable.
Conversely, if f is Riemann integrable, then from the properties
of the infimum there exists a step function u0 U(f) such that
b
a
u0(x) dx inf
u∈U(f )
b
a
u(x) dx +
ε
2
=
b
a
f(x) dx +
ε
2
.
Thus,
b
a
u0(x) dx
b
a
f(x) dx
ε
2
.
Similarly, there exists a step function v0 L(f) such that
b
a
f(x) dx
b
a
v0(x) dx
ε
2
.
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