Show that for all $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ and all $x \in \mathbf{R}$, $$\lim_{t \rightarrow +\infty} P_{t}(f)(x) = \int_{-\infty}^{+\infty} f(y) \varphi(y) \mathrm{d}y$$
Show that for all $f \in C^{0}(\mathbf{R}) \cap CL(\mathbf{R})$ and all $x \in \mathbf{R}$,
$$\lim_{t \rightarrow +\infty} P_{t}(f)(x) = \int_{-\infty}^{+\infty} f(y) \varphi(y) \mathrm{d}y$$