Let $\Omega \subset \mathbb{R}^{n}$ be some open set. Let $f\_{n},f\in C^{\infty}(\Omega)$. My question is: What does the following phrase mean? $f\_{n}$ converges to $f$ in $C^{\infty}\_{loc}(\Omega)$. What is the exact definition of such a convergence. 


Does it mean the following? For each compact $K \subset \Omega$ and each integer $m \in \mathbb{N}\_{0}$ there exists a subsequence of the sequence $(f\_{n})$ which converges to $f$ in $C^{m}(K)$.
If no, what is the right definition?


Ben



$C^{\infty}_{loc}$-convergence - right definition
I would read it as: 



> 
> For every compact set $K \subset \Omega$, every $m \ge 0$, and every indices $i\_1, i\_2, \dots, i\_m$, we have $\partial\_{i\_1} \partial\_{i\_2} \dots \partial\_{i\_m} f\_n \to \partial\_{i\_1} \partial\_{i\_2} \dots \partial\_{i\_m} f$ uniformly on $K$.
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