A Equivalence of Uniform Continuity Using Distance (Rudin's exercise 4.20) 
If $E$ is a nonempty subset of a metric space $X$, define distance from $x\in X$ to $E$ by $$\rho_E(x) := \inf_{z\in E}d(x,z)$$
  Prove that $\rho_E$ is uniformly continuous function on $X$ by showing that $$|\rho_E(x) - \rho_E(y)| \le d(x,y)$$ for all $x,y\in X$.

I know how to do it, I just can't prove how the last inequality implies uniform convergence, because uniform convergence says that given $\varepsilon > 0$ there is a $\delta>0$ such that $d(x,y) \Rightarrow |f(x) - f(y)| < \varepsilon$, but I can't choose that $\delta$ freely in order to have $|f(x) - f(y)|$ as small as I want.

The argument proves that $\rho_E$ is Lipschitz which implies in particular that it is uniformly continuous.
