 Solution

SetPart (b.iii), applied on each compactly contained ball, gives uniform $C^{2,\alpha }$ bounds for $v_k$. A diagonal use of the Arzela-Ascoli theorem therefore gives a subsequence converging in $C^2(K)$ on every compact $K⋐B_1$ to a $C^2$ function $w$.

The same estimate applied to $u_k$ gives $D{u}_k\to 0$ locally uniformly because $\Vert u_k{\Vert }_2\to 0$. Dividing the minimal-surface equation by $\Vert u_k{\Vert }_2$ shows thatThe coefficients converge locally uniformly to ${\delta }_{ij}$. Passing to the $C^2$ limit yieldsThus $w$ is harmonic and the required normalized subsequence converges to it locally in $C^2$.