Let $s: \mathbb{R} \rightarrow [a,b]$ be the saturation function, i.e., $s(x) = a$ if $x \leq a$, $s(x)=x$ if $a < x < b$, $s(x) = b$ if $x \geq b$.
Consider the vector saturation function $S: \mathbb{R}^n \rightarrow [a,b]^n$ defined as $$ S(x) = \left[ \begin{matrix} s(x_1) \\ \vdots \\ s(x_n) \end{matrix} \right].$$
$S$ is Lipschitz continuous with Lipschitz constant $1$.
Prove or disprove that $S$ is Lipschitz continuous with Lipschitz constant $1$ in any weighted Euclidean space, i.e., for any $A$ positive definite, $$ \left\| S(x) - S(y)\right\|_A \leq \left\| x-y\right\|_A$$ for all $x,y \in \mathbb{R}^n$, where the norm $\left\| z \right\|_A$ is defined as $\sqrt{z^\top A z}$.