Given a Feller semigroup $(T_t)$ on locally compact, separable metric space $S$, I wish to show that, for $\lambda > 0$, the resolvent $R_{\lambda}f(x) := \int_0^{\infty}e^{-\lambda t}T_tf(x)dt$ is continuous + vanishes at infinity, whenever $f$ has these properties.
What I have so far is that to show $|R_{\lambda}f(x_n) - R_{\lambda}f(x)| = |\int_0^{\infty}e^{-\lambda t}[T_tf(x_n)-T_tf(x)]dt| \rightarrow 0$, for $(x_n) \rightarrow x$ in $S$. I want to use now the fact that $T_tf(x)$ is continuous and vanishes at infinity, to possibly show that $\int_0^{\infty}|T_tf(x_n) - T_tf(x)|dt \rightarrow 0$ (as this would imply the result), but I am not sure how to do this, because (i) I am not sure how to bound the integrand at its upper tail end, and (ii) I am not sure how I could use DCT here possibly. Any ideas?