The Distance to a Set is Nonexpansive

Let $(X,d)$ be a metric space and let $A\subseteq X$ be nonempty, and write $\operatorname{dist}_d(z,A)$ for the distance from a point $z\in X$ to $A$ in $(X,d)$. Let $\mathbb{R}$ denote the real numbers, with the order, addition and additive inverses of their ordered field structure; write $s-t$ for $s+(-t)$, write $s<t$ to mean that $s\le t$ and $s\ne t$, let $|s|$ be the absolute value of $s$, and let $d_{\mathbb{R}}$ be given by $d_{\mathbb{R}}(s,t)=|s-t|$, which is a metric on $\mathbb{R}$ by The Absolute Value Metric on the Real Line.

Then the following hold for all $x,y\in X$ and every $a\in A$.

1. $0\le\operatorname{dist}_d(x,A)$.

2. $\operatorname{dist}_d(x,A)\le d(x,a)$.

3. $\operatorname{dist}_d(x,A)\le d(x,y)+\operatorname{dist}_d(y,A)$.

4. $|\operatorname{dist}_d(x,A)-\operatorname{dist}_d(y,A)|\le d(x,y)$.

5. The map $f:X\to\mathbb{R}$ defined by $f(z)=\operatorname{dist}_d(z,A)$ is uniformly continuous on $X$ as a map from $(X,d)$ to $(\mathbb{R},d_{\mathbb{R}})$, and is continuous on $X$.