I'm trying to prove some properties of Sobolev spaces and it would help me a lot if the following statement was true.
Let $\space \Omega \subset {\mathbb{R}}^n$ be open, bounded, non-empty set and let $f \in C^k(\overline{\Omega})$, i.e. $f \in C^k({\Omega})$ and all the partial derivatives up to order $\space k \in \mathbb{N}$ admit continuous extensions to $\overline{\Omega}$.
Then there is a sequence of functions $\left ( f_n \right )_{n=1}^{\infty}$ in $\space C^{\infty}(\overline{\Omega})$ converging uniformly to $f$ on $\overline{\Omega}$ such that also all the partial derivatives of $f_n$ converge uniformly to the correspondent partial derivative of $f \space$.
Is it true? I struggle finding a proof.
Thank you.