The setting is the following:
Let $\Omega \subset \mathbb{R}^n$ be open. For each $m \in \mathbb{N}$, define the sets $$\Omega_m =\{ x \in \Omega : |x| < m \text{ and } dist(x, \partial\Omega) > \frac{1}{m}\} $$ and set $\Omega_0 = \emptyset$ and define $U_m := \Omega_{m+1} \setminus \overline{\Omega_{m-1}}$. We then have that $\Omega = \cup_{m \in \mathbb{N}} U_m$ is an open covering of $\Omega$.
Now how does one choose (or construct) a $C^\infty$ locally finite partition of unity $(\phi_m)_{m \in \mathbb{N}} \subset C_c^\infty(\Omega)$ such that $\text{supp}(\phi_m) \subset U_m$, $0 \leq \psi_m \leq 1$ and $\sum_{m \in \mathbb{N}}\phi_m = 1$?
Thanks a lot in advance