The goal is to show that the product of two Riemann integrable functions is integrable.
First step is to use the identity $f\cdot g = \frac{1}{4} \left[(f+g)^2 – (f-g)^2\right]$ so that we only need to consider squares of functions.
The second step is to reduce to positive valued functions because $f(x)^2=\left|f(x)\right|^2$.
The third step is to use that if $0 \leq f(x) \leq M$ on $\left[a,b\right]$, $$f^2(x) – f^2(y) \leq 2M \left(\,f(x)-f(y)\right)$$
How should I go about implementing the above steps?
Best Answer
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.