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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 ### One comment

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