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,