Approximating the Riemann integral for a certain product

Question

Suppose that we have two continuous maps $f,g\colon I\to\mathbb{R}$ on the interval $I:=[0,1]$. Now suppose that $0=x_{0}<x_{2}<\ldots<x_{n}=1$ is some partition of $I$ and that for each $i$ we have a number $y_{i}=y_{i}(x_{i-1},x_{i})\in[x_{i-1},x_{i}]$. I am trying to prove that $$\sup\sum_{i=1}^{n}|f(x_{i-1})g(y_{i})|(x_{i}-x_{i-1})\leq\int_{0}^{1}|f(x)g(x)| \ dx,$$ where the supremum ranges over all such partitions $x_{1},x_{2},\ldots,x_{n}$ of $I$. I see that the left hand side looks a lot like an approximation for the Riemann integral. However, since the inputs of $f$ and $g$ differ, I don't see how to make this precise. Any suggestions are greatly appreciated!