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

.