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?