[P] Lipschitz continuity and convex functions

I have this:

Sigma: R->R is a non-decreasing L-Lipschitz function, W € R^kxd, and b €R^k there exists a convex L||W||_2^2 smooth function F_(w,b) such that
Nabla_x F_(w b)(x) =W^T sigma(Wx +b)

and we have a convex potential layer
z = x - (2*W^T sigma(Wx + b) )/ ||W||_2^2


Now, can anyone help me with the rigorous proof of how F_(w,b) is L||W||_2^2 smooth? And, is z differentiable?

If not the solution, then can someone else suggest some reading material …

continuity function functions layer machinelearning