Let $A_1,A_2\subseteq \mathbb{R}^d$ two domains such that $A_1\subset \subset A_2$. Why exists a function $f\in C_c^{\infty}(A_2)$ such that $f_{|A_1}$ is constant 1?
My idea is to define $f=\rho \star \chi_{\Omega}$, where $\chi_{\Omega}$ is a characteristic function of a suitable set, maybe $A_1\subset \subset \Omega \subset \subset A_2$, and $\rho\in C_c^{\infty}(A_2)$ is a convolution kernel. Because then follows: $f\in C_c^{\infty}(A_2)$. But I don't know how to find $\Omega$ (maybe Tietze/Urysohn's lemma?) and how to do it in detail.. How to do it? I appreciate your help.