= Solution
Set
$$
v_k=\frac{u_k}{\|u_k\|_{L^2(B_1)}}.
$$
Part (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 $K\Subset B_1$ to a $C^2$ function $w$.
The same estimate applied to $u_k$ gives $Du_k\to0$ locally uniformly because $\|u_k\|_2\to0$. Dividing the minimal-surface equation by $\|u_k\|_2$ shows that
$$
\left(\delta_{ij}-\frac{D_iu_kD_ju_k}{1+|Du_k|^2}\right)D_{ij}v_k=0.
$$
The coefficients converge locally uniformly to $\delta_{ij}$. Passing to the $C^2$ limit yields
$$
\boxed{\Delta w=0.}
$$
Thus $w$ is harmonic and the required normalized subsequence converges to it locally in $C^2$.
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