(5) (10 %) If $E$ is a nonempty subset of a metric space $X$, define the distance from $x \in X$ to $E$ by $\rhoE(x) = \inf{z \in E} d(x, z)$. Prove that $\rho_E$ is a uniformly continuous function on $X$, by showing that

$$|\rhoE(x) - \rhoE(y)| \leq d(x, y)$$

for all $x, y \in X$.