I'm reading Schoen and Yau's 1979 paper on the Positive Mass theorem. I'm having trouble understanding the proof of how they extracted a minimal surface as the limit of solutions to the Plateau problem.
So far (at the end of page 7 in the PDF) , for a fixed end $N_k$ (identified with $\mathbb{R}^3$ minus an open ball) of the Riemannian 3-manifold $(N,g)$, they found a sequence of embedded smooth surfaces $S_{\sigma}$, which are g-area minimizers spanning $C_{\sigma}$: The euclidean circle of radius $\sigma$ lying on the x-y plane. Defining the "cylinder" $$A_q = \{x \in N_k : (x^1)^2 + (x^2)^2 \leq q^2\})$$ and the "slab" $$E_h = \{x \in \mathbb{R}^3 : |x^3| \leq h \}$$ They were able to show that, for any $\sigma > q$: $$S_{\sigma} \cap A_{q}\subseteq E_h \cap A_{q} $$
They next quote the following interior regularity estimate, where $U_{r}(x)$ is the geodesic ball of radius $r$:
There exists a number $r_{0}>0$ so that for any $x_0 \in S_{\sigma}$ with $U_{r_0}(x_0)\cap C_{\sigma} =\phi $, $S_{\sigma} \cap U_{r_0}(x_0)$ can be expressed as the graph of a $C^3$ function whose derivatives up to order three are uniformly bounded for any $\sigma$.
Using this, they claim that we can choose a sequence $S_{\sigma}^{q}\cap A_{q}$ converging in the $C^2$ topology, as $\sigma \to \infty$. I'm stuck at this point. I think the Ascoli-Arzela theorem has something to do with this, but can't seem to figure out the details. Any help is much appreciated.