A symbol $p \in S^m(\Omega)$ where $\Omega \subset \mathbb{R^n}$, or its corresponding operator $p(x,D) \in \Psi^m(\Omega)$ is said to be elliptic of order m if for every compact $A\subset \Omega$ there are positive constants $c\_A,C\_A$ such that  

$|p(x,\xi)|\ge c\_A|\xi|^m$ for $x \in A$ and $|\xi|\ge C\_A$. 


Using this definition, I need to prove the following lemma:  

If $p \in S^m(\Omega)$ is elliptic, there exists $\zeta \in C^{\infty}(\Omega \times \mathbb{R^n})$ with the following property: For any compact $A \subset \Omega$ there are positive constants $c,C$ such that for $x \in A$ we have  

a. $\zeta(x,\xi)=1$ when $|\xi|\ge C$  

b.$|p(x,\xi)| \ge c|\xi|^m$ when $\zeta(x,\xi) \not = 0$ 


My try: By definition, zeroes of $p(x,\xi)$ for $x$ in any compact set $A$ are contained within the ball $|\xi|\le C\_A$ with respect to $\xi$ and we have $\frac{1}{|p(x,\xi)|} \le \frac{1}{c\_AC\_A^m}$ for $x \in A$ and $|\xi|\ge C\_A$. I tried to use paracompactness of $\Omega$ to bound the zeroes of $p(x,\xi)$ within some ball centred at $0$ w.r.t $\xi$ (but could not attain it) so that $\zeta(x,\xi)$ can be defined to be $1$ outside such a ball and $0$ in some neighbourhood of zeroes of $\zeta(x,\xi)$.  

Any help is deeply acknowledged.



Definition of Elliptic pseudodifferential operators
In the first place, you can write
$
\mathbf 1\_\Omega=\sum\_{k\ge 1}\chi\_k, \ \chi\_k\in C^\infty\_c(\Omega).
$
Let $\nu\in C^\infty(\mathbb R^n)$, vanishing on $B(0,1/2)$, equal to 1 on $B(0,1)^c$: let us define
$$
\zeta(x,\xi)=\sum\_{k\ge 1}\chi\_k(x) \nu(\xi/R\_k),
$$
where the positive $R\_k$ is chosen so that $p(x,\xi)\ge c\_k\vert\xi\vert^m$ for $\vert \xi\vert\ge R\_k$ for a positive $c\_k$.


