Question: Assume you have a metric space $(E,d)$ and $A\subset E$ that is nonempty. Define $d_A:x \in E \rightarrow d(x,A)$. Show that $d_A$ is lipschitz and compute its Lipschitz seminorm.
My Question: My question is whether or not it is true that if you take $x,y,z \in E$ then $d_A(d(x,y),d(z,y)) \leq d(x,z)$.