Let $\Omega \subset \mathbb{R}^{d}$ be open. Let $U_1$ and $U_2$ be open subsets of $\Omega$ such that $U_2 \subset \bar{U_2} \subset U_1$. Then there is a $\phi: \Omega \to \mathbb{R}$ such that $\phi$ is smooth, $\phi \equiv 1$ on $U_2$, and $\phi \equiv 0$ on $\Omega \backslash U_1$.
Correct me if I'm wrong: We can use the Urysohn lemma to find a continuous $\phi$ with the above conditions, but that doesn't give us that $\phi$ is smooth. What tools can I use to find such a smooth function $\phi$?