Distance to a set and normal cone

We consider a nonempty set $S\subset\mathbb{R}^n$ and $x\notin S$. We define the set

$$P_{S}(x) =\{z\in S : \lVert z - x\rVert =d(x,S)\}$$

I would like to prove that for all $z\in P_{S}(x)$ we have $x-z\in N_{S}(z)$ where

$$N_{S}(z) = \{y\in\mathbb{R}^n : \langle w,y\rangle\leq 0, \forall w\in T_{S}(z)\}$$

and where $T_{S}(z)$ is composed of elements $w\in\mathbb{R}^n$ such that there exist sequences $(x_n)_n\subset S$ and $(\varepsilon_n)_n$ a sequence of positive real numbers such that $x_n$ converges to $z$ and $\varepsilon_n$ converges to $0$ and

$$\lim_{n\to\infty}\frac{x_n - z}{\varepsilon_n} = w$$

Clearly if $P_{S}(x)$ is empty it is vacuously true. However, if $P_{S}(x)\neq\emptyset$ I don't know how to proceed. If the set $S$ is convex it is just a consequence of the fact that $\langle x-z,y-z \rangle\leq 0$ for all $y\in S$ if $\lVert z - x\rVert =d(x,S)$.

Do you have some ideas on how to tackle the proof please ?

Thank you a lot !