I'm studying by myself Mean Curvature Flow and I'm reading the paper "Interior estimates for hypersurfaces moving by mean curvature" by Klaus Ecker and Gerhard Huisken and I'm stuck in the following theorem:
$\textbf{Theorem 5.1}$: $M_0 \equiv F_0(\mathbb{R}^n)$ be a locally Lipschitz continuous entire graph over $\mathbb{R}^n$, then the initial value problem $(1)$ has a smooth solution $M_t = F_t(\mathbb{R}^n)$ for all $t > 0$. Moreover, each $M_t$ is an entire graph over $\mathbb{R}^n$.
I didn't understand what approximation argument exactly was used in the the last paragraph of the proof:
From Corollary $3.5 (ii)$ we then conclude for any integer $m \geq 0$
$$\sup_{B_{R_0}(0) \times [0,T]} \left| D^m w_R \right| \leq c_m,$$
where $c_m = c_m(m,n,R_0,c_0, c_1)$.
We can therefore select a sequence of solutions $(w_{R_k})$ for $R_k \rightarrow \infty$ ($R_k > R_1$ for any $k \geq 2$) s.t. $w_{R_k} \rightarrow w$ in $\mathcal{C}^{\infty}$ uniformly on $\Omega \times [0,T]$. Since $\Omega$ and $T > 0$ were arbitrary this establishes the existence of a family of entire graphs $M_t = \text{graph} \ w(\cdot, t)$ solving (1) where $w \in \mathcal{C}^{\infty}(\mathbb{R}^n \times (0,\infty))$. As the second and higher order derivative estimates for $w$ on each compact subset of $\mathbb{R}^n$ depend only on the initial height and gradient on a slightly larger subset, an approximation argument yields a smooth solution of (1) also for locally Lipschitz initial data.
This last paragraph leads me to think that I need some corollary of Arzela-Ascoli theorem and proceed as the section "4.4 - Curvature explodes" of this thesis, which teach how construct a global solution. This thesis construct a global solution for the curve shortening flow and the author of thesis states that uniform limit of the subsequence there it's a smooth map and it's clear for me, because this result:
$\textbf{Theorem A.4.2}$ Let $\Omega \subset \mathbb{R}^s$ be a closed bounded set. There is a function $u \in \mathcal{C}^{\infty}(\Omega)$ and a sequence $m_j \rightarrow \infty$ with
$$\max_{x,t} \left| \frac{\partial^{p+q}}{\partial x^p \partial t^q} u_{m_j}(x,t) - \frac{\partial^{p+q}}{\partial x^p \partial t^q} u(x,t) \right| \rightarrow 0 \ \text{as} \ m_j \rightarrow \infty.$$
This theorem can be found on Appendix $4$ of the book "Initial-Boundary Value Problems and the Navier-Stokes Equations" by Heinz-Otto Kreiss and Jens Lorenz, but the problem is that I don't have uniform bounds for $\frac{\partial^{p+q}}{\partial x^p \partial t^q} w_{R_k}$, so I don't know how to proceed here. Is the approximation argument commented by authors really an argument based on Arzela-Ascoli theorem? Anyone can give some details about this argument?
Thanks in advance!
