Show that $C^{0}(\mathbf{R}) \cap CL(\mathbf{R}) \subset L^{1}(\varphi)$, where $\varphi(x) = \frac{1}{\sqrt{2\pi}}\mathrm{e}^{-x^2/2}$ and $L^1(\varphi) = \{f \in C^0(\mathbf{R}),\, f\varphi \text{ integrable on } \mathbf{R}\}$.
Show that $C^{0}(\mathbf{R}) \cap CL(\mathbf{R}) \subset L^{1}(\varphi)$, where $\varphi(x) = \frac{1}{\sqrt{2\pi}}\mathrm{e}^{-x^2/2}$ and $L^1(\varphi) = \{f \in C^0(\mathbf{R}),\, f\varphi \text{ integrable on } \mathbf{R}\}$.