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