Consider $\Omega$ a bounded open set in $R^n$ with $ \partial \Omega$ smooth and $u_n \in C^{1,\alpha}(\Omega)$ a bounded sequence in $C^{1,\alpha}(\Omega)$.
Suppose that each $u_n$ is p - harmonic (in the weak sense) . the proof of the theorem that i am studying says : "Then we can easily show that on every compact of subset of $\Omega $ , $u_n$ converges in $C^{1,\alpha} $ norm to a p- harmonic function (in the weak sense) $u$". (*)
I believe that is a subsequence and not the sequence.
My ideas : the inclusion $C^{1,\alpha} \subset C^{0}$ is compact, then exists a funtion $u$ in $C^{0}$ where $u_n \rightarrow u$ in $C^{0}$ . I believe that this the that i am searching...
Someone can help me to prove the affirmation in $(*)$ ?. Thanks in advance .