This is Exercise 3.4) in Peter Hintz's Introduction to microlocal analysis
Which I am using for exam preparation.
Let $\Omega \subset \mathbb{R}^n$ open.
Consider for $m \in \mathbb{Z}$ the space $S^m(\Omega \times \mathbb{R}^n)$ of symbols of Hörmander-type (1,0). Let $S^{m}_{cl}$ be the space of classical symbols of order $m$.
The claim is that $S^{-\infty}$, which is the intersection of all $S^m$ is closed in $S^{m}_{cl}$ for any $m$.
My ideas:
We start with a sequence $(a_j)_{j \in \mathbb{N}} \subset S^{-\infty}$ that has a limit in $a \in S^{m}$. This means
$$
|\partial^{\alpha}_{x} \partial^{\beta}_{\xi}(a - a_j)|\cdot (1+|\xi|)^N) \to 0 \text{ as } j \to \infty
$$
for all multiindices $\alpha, \beta$ and $N$ arbitrary. We want to show that $a \in S^{-\infty}$. We further know that, since $a$ is a classical symbol, we have an asymptotic expansion of $a$:
$$
a \sim \sum_{j = 0}^{\infty} \tilde{a}_{m-j}.
$$
But I don't know how to continue from here. Any help would be appreciated!