I'm having trouble with this problem
Let $X\in \mathcal{C}^{\alpha}[0,T]$ and $Y\in\mathcal{C}^{\beta}[0,T]$ with $\alpha+\beta>1$. (Here $\mathcal{C}^{\gamma}[0,T]$ is the space of Hölder continuous functions of order $\gamma$ in $[0,T]$). Let $\int_{s}^{t}Y(r)dX(r)$ the integral in the Riemann-Stieltjes sense. Prove that exists a constant $C>0$ such that $$\left|\int_{s}^{t}Y(r)dX(r)-Y(s)(X(t)-X(s))\right|\leq C|t-s|^{\alpha+\beta}.$$
I tried to estimate the difference $\sum_{[u,v]\in\Pi}Y(u)(X(v)-X(u))-Y(s)(X(t)-X(s))=\sum_{[u,v]\in\Pi}(Y(u)-Y(s))(X(v)-X(u))$, where $\Pi$ is a partition of $[s,t]$, but I cannot bound it.