Proposition $3.4$ of Mean curvature flow with surgeries of two–convex hypersurfaces

Question

I'm studying by myself "Mean curvature flow with surgeries of two–convex hypersurfaces" and I'm trying understand the last three lines of the proof of the proposition $3.4$ on page $149$:

Then it follows easily that $M$ is a graph over this sphere with the desired properties since the extrinsic curvature and its derivatives up to order $k$ control the function $u$ and its derivatives up to order $k + 2$.

My doubts are

1. Why $M$ is a graph over the sphere constructed in the proof?

2. How control $u$ based on the condition of the $(\varepsilon,k)$-parallelism? I can see how to do this control for its derivatives using the mean value theorem which can be read on page $71$ of "Some nonlinear problems in Riemannian geometry" by Thierry Aubin, but I can't use this theorem to control $u$.

Thanks in advance!