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# Prove some statements about integrals of bounded monotonic increasing functions

Consider a bounded, monotonic, real-valued function on the interval . The define sequences

1. Prove that

and that

2. Prove that the two sequences and converge to .
3. State and prove a generalization of the above to interval .

1. 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

2. Proof. From part (a) we have

since . Since does not depend on we have

Therefore,

3. 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