Exercise 11, § 1

The hypotheses being the same as in Exercise 10, now prove that, for every $\varepsilon > 0$, there exists a number $\rho > 0$ such that for every subdivision of $[a, b]$ into intervals $[x_i, x_{i+1}]$ of length $\leq \rho$ ($0 \leq i \leq n - 1$), one has

whatever the points $u_i, v_i$ in $[x_i, x_{i+1}]$ may be, for each index $i$ (reduce to the case where $f$ and $g$ are step functions).