Consider a bounded, monotonic, real-valued function on the interval . The define sequences
- Prove that
and 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
Therefore,
- Claim: If is a real-valued function that is monotonic increasing and bounded on the interval , then
for and defined as follows:
Proof. Let
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
You can prove (c) much more easily by defining a new function g(x)=f((b-a)x+a) and using parts (a) and (b) on g