Show that if $f:[a, b]\subset \mathbb{R} \longrightarrow \mathbb{R}$ is a Lipschitzian function, then the Riemann-Stieltjes integral is independent of the choice of $\xi_i $, for all i = 1,2, ..., n.
I started like this, since $f$ is a Lipschitzian function, then
$$|f(x)-f(y)|\leq c|x-y|, \forall \:x,y \in [a,b]$$
for some $c>0$. Thus, $$R_{\Gamma}(\Gamma,f,\phi)=\sum_{i=1}^{m} f(\xi_i)(\phi_i - \phi_{i-1})$$ where $\Gamma=\{a=x_0,x_1,...,x_m\}$ is a partition of $[a,b]$ and $\xi_i$ is such that $\xi_i \in [x_{i-1},x_i]$.
But now I do not know how to use Lipschtiz's condition.