Basically is the fact that and are not Riemann sums they key idea here?
Lemma. Let , and let be a function. Let be a partition of , and let be a real number. Suppose that whenever and are Riemann sums of on , then . Then these statements follow:
1. is bounded on .
3. If is any refinement of , and and are Riemann sums, then
Proof. (1) Suppose is unbounded. Then there is a sequence in that converges to such that for every , . Then the set of Riemann sums of corresponding to is an unbounded set of real numbers. (2) and don't have to be Riemann sums. And since we are taking a "maximum" difference, we can get to . (3) I think that . And so the inequaliy follows.
Is this correct?