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
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