The product of two Riemann integrable functions is integrable

This is not a problem I would assign as homework (at least, not without substantial guidance). Rather, it is one of the fundamental results of the subject -- the subject being advanced calculus / elementary real analysis -- and as such I would expect any instructor / textbook to supply a proof. For instance, Rudin's Principles of Mathematical Analysis covers this. Or see for instance the chapter on integration here.

As Robin says, the result also follows from Lebesgue's criterion of Riemann integrability: now that's something -- I mean the deduction from Lebesgue's Criterion, not the proof of Lebesgue's Criterion! -- I would leave as an exercise, since finding this short argument on one's own helps to drive home the power of the Lebesgue criterion.

This follows from Lebesgue's characterization of Riemann integrable functions as bounded functions continuous outside a set of Lebesgue measure zero. This characterization is usually the swiftest way of deciding on the Riemannn integrability of a function.