Why does $f$ is $C^2(\mathbb{R^n})$ bound below and $x$ in compact set imply Lipschitz Continuos Hessian

Question

I read in this paper that is about some numerical optimization method, that if $f$ is $C^2(\mathbb{R^n})$ and the iterates $x_k$ are in a compact set, then $\nabla^2 f$ satisfies a Lipschitz condition. The objective $f$ is furthermore assumed to be bound below in this set but no convexity assumption or similar is made.

Does anyone know, how this property comes about?

Thanks a lot!