Are there any direct ways of proving that if is real-valued and Riemann integrable on , then is bounded?

Because the usual proof would be by contraposition:

1. There exists a sequence in that converges to , such that for every , .Outline of "Usual Proof"

2. Let be a partition of . Then the set of Riemann sums of corresponding to is an unbounded set of real numbers.

3. Thus cannot be Riemann integrable on .