I wonder if there is a result that tells us in what conditions a sequence $\phi_n:\Omega\to\mathbb{R}$ of functions in $C^k(\Omega)$, $\Omega$ being a compact subset of $\mathbb{R}^n$, contains a subsequence that converges to $\phi\in C^k(\Omega)$? ($k\in\mathbb{N}^*$) We consider the convergence in the norm:
$$||\phi||_{C^k}=\max_{|\alpha|\leq k} \sup_{x\in \Omega}\left |D^{\alpha} \phi (x)\right |$$
It is sufficient for the sequence to be bounded in the $C^k$ norm and for the $k$th derivatives of the $\phi_n$ to be equicontinuous. Indeed, by Arzela-Ascoli, this immediately implies there is a subsequence on which the $k$th derivatives converge uniformly. Then observe that the $k$th derivatives being uniformly bounded implies the $(k-1)$st derivatives are equicontinuous, and so we can further pass to a subsequence where the $(k-1)$st derivatives converge uniformly. Iterating this process we obtain a subsequence where the $j$th derivatives converge uniformly for all $j\leq k$ and thus the subsequence converges in the $C^k$ norm.