Prove the following: Let be a generalized interval, and let be a function which is uniformly continuous on . Then is Riemann integrable.

We know that is bounded. We need to show that (e.g upper and lower integrals are equal). Let . Then there exists a such that whenever . Skipping some steps, we have for all where are the subintervals of with length . Then . And so or .

From here, how would I conclude that ?