Product of increasing functions is integrable in two dimensions

Hint: Given the cancellations that you have already performed, along with $f(x)\leq f(1)$ and $g(y)\leq g(1)$ for all $x,y\in[0,1]$, the triangle inequality implies $$ \left\lvert\sum_R v(R)(M_R(f)-m_R(f))\right\rvert\leq\frac{4nf(1)g(1)}{n^2}. $$