Could anyone please help me understand the compact embedding of $C^{k,2+\alpha}$ space?
I am reading a paper about second order elliptic pde on $C^{k,2+\alpha}$ space. Here is the link of the paper https://arxiv.org/pdf/1210.6727.pdf. (page 55)
When it comes to the proof of the uniqueness of the solution it says:
By applying the Arzela-Ascoli Theorem, we can extract a subsequence, which we continue to denote by $\{ u_n\}_{n∈N}$, which converges in $C^{k,2+β} (\bar{\rm S})$, for all $β < α$, to a limit function $u\in C^{k,2+α} (\bar{\rm S})$ as $n\rightarrow\infty$.
Could anyone please give a proof of this?
Thank you so much. Appreciate your time and effort of reading my post.