Show that any polynomial $P(x) \in E(\Omega) = C^{\infty}(\Omega)$ but $P(x)\notin S(\Omega)$, where $S(\Omega)$ is the Schwartz space.
I think I have a rough idea on how to right the proof, I want to start the proof by showing that $E(\Omega)$ is complete and from there since every Cauchy has a limit in in $E(\Omega)$, then perhaps I could extend it to the to show that for every compact $K \subset \Omega$ I can find $\partial_{x}^{\alpha} \varphi_{n} \rightrightarrows \partial_{x}^{\alpha} \varphi$.
Since we define the space of all smooth functions equipped with topology as
$\varphi_{n} \rightarrow \varphi \quad$ in $\quad \mathscr{E}(\Omega) \quad$ as $\quad n \rightarrow \infty$
if for every compact $K \subset \Omega$ and multi-index $\alpha$
$\partial_{x}^{\alpha} \varphi_{n} \rightrightarrows \partial_{x}^{\alpha} \varphi \quad$ uniformly on $\quad K \quad$ as $\quad n \rightarrow \infty$
which we can equally define using seminorms, such that for every compact $K$ and number $l$ the semi-norm
$p_{K, \ell}(\varphi) :=\sum_{|\alpha| \leq \ell} \sup _{K}\left|D^{\alpha} \varphi\right|$
thus we have semi-norms
$\varphi \rightarrow \sum_{|\alpha| \leq k} \sup _{k}\left|D^{\alpha} \varphi\right|, \qquad \varphi \in \mathscr{E}(\Omega)$
where K ranges over all compact subsets of $\Omega$ and $l$ over all integers $\geq 0$
Any help or hints would be greatly appreciated.