This is from page 90 of Munkres. We say $f: [0,1] \to \mathbb{R}$ is increasing if $f(x_1) \geq f(x_2)$ whenever $x_1 \geq x_2$. If $f,g : [0,1] \to \mathbb{R}$ are increasing and non-negative, show that the function $h(x,y) = f(x)g(y)$ is integrable over $[0,1]$.
My attempt: Consider that if either $f$ or $g$ are constantly zero, then the function is constant and integrable. Therefore, we assume that both are zero only at countably many or finitely many points but certainly not all. The function $h(x,y)$ is also increasing.
Take an equal partition $P$ where each cube has area $\dfrac{1}{N^2}$ was an idea but I'm not sure