$\mathrm{\mid }d(x,A)-d(y,A)\mathrm{\mid }\le d(x,y)$, so the distance to a fixed nonempty set is $1$-Lipschitz

Let $(X,d)$ be a metric space (Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric), let $A\subseteq X$ be nonempty and let $x,y\in X$. Then

$\mathrm{\mid }d(x,A)-d(y,A)\mathrm{\mid }\le d(x,y),$

with $d(\cdot ,A)$ the distance to a nonempty set (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space). Thus the real-valued function $u\mapsto d(u,A)$ changes by at most $d(u,v)$ between $u$ and $v$: it is $1$-Lipschitz.

Metric space: $d(x,y)=0$ iff $x=y$, symmetry, and the triangle inequality; pseudometric and ultrametric

Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space

• Where the nonemptiness is used. Twice, and both times essentially: it is what makes $d(x,A)$ and $d(y,A)$ exist at all (Bounded subset, diameter, distance from a point to a set, and distance between two sets in a metric space), and it is what makes $E(x)$ and $E(y)$ nonempty so that "greatest lower bound" has content. For $A=\mathrm{\varnothing }$ the statement has no meaning in this library, since $\inf \mathrm{\varnothing }$ is undefined.
• The point case is The reverse triangle inequality $\mathrm{\mid }d(x,z)-d(y,z)\mathrm{\mid }\le d(x,y)$ in any metric space: taking $A=\{z\}$ gives $E(u)=\{d(u,z)\}$, whose infimum is $d(u,z)$, and the conclusion becomes $\mathrm{\mid }d(x,z)-d(y,z)\mathrm{\mid }\le d(x,y)$.
• The constant $1$ is best possible in general: on $\mathbb{R}$ with $A=\{0\}$ the function is $u\mapsto \mathrm{\mid }u\mathrm{\mid }$, and $\mid \mathrm{\mid }x\mathrm{\mid }-\mathrm{\mid }y\mathrm{\mid }\mid =\mathrm{\mid }x-y\mathrm{\mid }$ whenever $x$ and $y$ have the same sign.