Prove that
where is a nonnegative, increasing function defined for all
.
Use this to deduce the inequalities
by taking .
Proof. Since is increasing we know
(where
denotes the greatest integer less than or equal to
) for all
. Define step functions
and
by
Then and
are constant on the open subintervals of the partition
So,
Since is integrable we must have
for every pair of step functions . Hence,
Next, if we take (which is nonnegative and increasing on
) we have
From this we then get two inequalities
Therefore,