Convergence of sequence of Riemann-Stieltjes integrals to Riemann-Stieltjes integral In connection with my post Convergence to Riemann-Stieltjes integral of sequence of Riemann-Stieltjes-like sums with changing integrand and integrator, an alternative approach to my main objective would be considering the convergence of the sequence of Riemann-Stieltjes (RS) integrals $\int_0^1 \, f_N(x) \, \mathrm{d}F_N(x)$ to the RS integral $\int_0^1 \, f(x) \, \mathrm{d}F(x)$. The properties of the functions involved are as defined in the aforementioned post. I repeat them in the following paragraph for convenience, though. 
The functions are all real of a single real variable. Those in the sequence $(f_N)_N$ are continuous and bounded in $[0,1]$, and the sequence converges to a continuous function $f$ bounded in the same interval. In turn, those in $(F_N)_N$ are monotonically increasing step functions bounded in $[0,1]$ and the sequence is uniformly convergent to a function $F$ that is a cumulative distribution function. 
Any hints on how to prove the above statement of convergence will be welcome.   
 A: Hints:
Note that
$$\left|\int_0^1f_N \, dF_N - \int_0^1 f \, dF  \right| \leqslant \left|\int_0^1f_N \, dF_N - \int_0^1 f \, dF_N  \right|+ \left|\int_0^1f \, dF_N - \int_0^1 f \, dF  \right|$$
(1) We can estimate the first term on the RHS and prove convergence to $0$, if $F_N$ has bounded variation $V_0^1(F_N)$, using
$$\left|\int_0^1f_N \, dF_N - \int_0^1 f \, dF_N  \right| \leqslant \int_0^1|f_N - f|\, dV_0^x(F_N), $$
although with the simplification that $F_N$ is montonically increasing we have
$$\left|\int_0^1f_N \, dF_N - \int_0^1 f \, dF_N  \right| \leqslant \int_0^1|f_N - f|\, dF_N $$
With uniform convergence of $f_n \to f$ and uniform boundedness of $F_N$ it is easy to progress. Given that $|F_N(x)| \leqslant M$ uniformly in $N$ and $x$ -- which is true if $F_N$ converges uniformly to a bounded function $F$ --  then for all sufficiently large $N$ we have $|f_N(x) - f(x)| < \epsilon/(2M)$ and
$$\left|\int_0^1f_N \, dF_N - \int_0^1 f \, dF_N  \right| < \frac{\epsilon}{2M}[F_N(1) - F_N(0)] < \frac{\epsilon}{2M}2M = \epsilon  $$
(2) Estimate the second term on the RHS most easily using Rieman sums.
