Solution
= Solution
Fix $B_r\Subset B_R\Subset B$. Local uniform convergence bounds $|u_k|_{0;B_R}$ uniformly. Interior <Schauder estimate>[Schauder estimates] therefore bound $u_k$ uniformly in $C^{2,\alpha}(B_r)$. The <Arzela-Ascoli theorem> gives a subsequence converging in $C^2(B_r)$. Its limit must be the locally uniform limit $v$, and passing to the limit in $Lu_k=0$ gives $Lv=0$. A diagonal argument over $r\uparrow1$ proves
$$
\boxed{v\in C^2(B),\qquad Lv=0.}
$$