How can I show that $f$ is uniformly continuous in $X$?

Question

Let $(X,\rho)$ a metric space, and $x_0 \in X$ an arbitrary point. It defines $f:X\rightarrow \mathbb{R}$ through: $$f(x)=\rho(x,x_0).$$ prove that $f$ is uniformly continuous in $X$

Answer 1

Hint : Prove that if $\rho$ is a metric, then $|\rho(x,z) - \rho(y,z)| \leq \rho(x,y)$.

Answer 2

Hint: It's actually Lipschitz continuous.