Let $f(x):\Re^n\rightarrow \Re$ be a proper and closed convex function. Its Moreau-Yosida regularization is defined as
$F(x)=\min_y\Big(f(y)+\frac{1}{2}\|y-x\|_2^2\Big)$
$\operatorname{Prox}_f(x)=\arg\min_y \Big(f(y)+\frac{1}{2}\|y-x\|_2^2\Big)$
Lots of literature say $F(x)$ is Lipschtiz continuous and give explicitly the expression of $\nabla F(x)$ involving $\operatorname{Prox}_f(x)$. But I have no idea how to calculate $\nabla F(x)$. Can anyone provide a straightforward method? I know Rockafellar's book gives a proof. But it assumes too much prior knowledge. I am wondering if there is a more elementary method to prove the Lipschtiz continuity and calculate its gradient.