I'm reading a paper by Wan-Xiong Shi "Deforming the metric on complete Riemannian manifolds". And there is a statement without proof. It can be summarized as follows:
Let $B(x_{0},\gamma)$ be a geodesic ball of radius $\gamma$ and centered at $x_{0}$. Since $\overline{B(x_{0},\gamma)}$ is compact, there exists a constant $c(g_{ij})>0$ depending on the metric $g$, such that $$\left | \nabla Rm \right |\leqslant c(g_{ij})$$
The manifold itself is assumed non-compact but complete.
I actually have two questions regarding this:
Why closure of geodesic ball is compact? Can the ball itself be non-compact then?;
Why $\left | \nabla Rm \right |$ is bounded? Is it because continuous functions on compact domains are bounded and thus $\left | \nabla Rm \right |$ is bounded as well? Or there is another reason?
Completeness of g implies compactness of closed geodesic balls.
Yes, since metrics are usually assumed to be infinitely differentiable. Your explanation of boundedness is correct.