= Solution
It suffices by part a to treat a nondecreasing right-continuous integrator, whose increments define a finite Lebesgue–Stieltjes measure $\mu_f$ on $[0,1]$. Let $\alpha_n$ be the left-endpoint step approximation on the dyadic intervals. The displayed sum is exactly
$$
\int_{(0,1]}\alpha_n\,d\mu_f.
$$
The <continuous function> $\alpha$ is uniformly continuous on the compact interval, so $\|\alpha_n-\alpha\|_\infty\to0$. Therefore
$$
\left|\int(\alpha_n-\alpha)\,df\right|
\leq\|\alpha_n-\alpha\|_\infty V(1)\longrightarrow0.
$$
Apply this separately to $g$ and $h$ to obtain the asserted <Lebesgue-Stieltjes integral> limit.
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