Apply mean value theorem to get $\min\{x_n,y_n\}\leq \xi_n\leq \max\{x_n,y_n\}$ such that $f(x_n)-f(y_n)=f'(\xi_n)(x_n-y_n)$, and the squeeze theorem gives that $\xi_n\to x_0$. Now you can conclude.