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
Consider a bounded, monotonic, real-valued function on the interval . The define sequences
- Prove that
- Prove that the two sequences and converge to .
- State and prove a generalization of the above to interval .
- Proof.First, we define two step functions,
where denotes the greatest integer less than or equal to . Then we define a partition of ,
For any we have
So, and are constant on the open subintervals of the partition .
Since is monotonically increasing and for all (by the definition of and ) we have
- Proof. From part (a) we have
since . Since does not depend on we have
- Claim: If is a real-valued function that is monotonic increasing and bounded on the interval , then
for and defined as follows:
be a partition of the interval . Then, define step functions and with and for . By these definitions we have for all (since is monotonic increasing). Since is bounded and monotonic increasing it is integrable, and
And, since , and we have