An inequality for the Riemann-Stieltjes integral

Let $g$ and $h$ be real valued and continuous on the interval $[a,b]$, and suppose that the variation, $V[h]$, of $h$ on $[a,b]$ is finite. By completely elementary methods, it is shown that $V[h] \cdot {\sup _{_{a \leqq \alpha < \beta \leqq b}}}(g(\beta ) - g(\alpha ))$ is an upper bound for $\int _a^b {(h - \inf h)dg}$.