I want to show the following. If $f$ is $C^2(X,\mathbb{R})$, where $X\subseteq \mathbb{R}^k$ is compact then;
\begin{equation} \lim_{n \rightarrow \infty}\frac{|\nabla f(x_n)-\nabla f(x)|}{|x_n-x|} \end{equation} converges for all sequences $\{x_n\}$ tending to $x$.
I think I must use a Taylor expansion, but I am a bit confused in the higher dimensions on how this works. I know that at some point I must use the fact that $X$ is compact and the derivatives are continuous to say that all derivatives are bounded but I'm confused about how to write this formally. Also, is there a simple expression for what the above converges to?