Proposition 7.2.2. Let $(X, d)$ be a pseudometric space, $A \subset X$, and
\[d_{A}: X \to [0, \infty) \quad x \mapsto d(x, A)\]
then $d_{A} \in UC(X; [0, \infty))$, where for any $x, y \in X$, $|d_{A}(x) - d_{A}(y)| \le d(x, y)$.
Proof. Let $x, y \in X$, then
\[d(y, A) = \inf_{a \in A}d(y, a) \le d(x, y) + \inf_{a \in A}d(x, a) = d(x, y) + d(x, A)\]
As the argument is symmetric, $|d_{A}(x) - d_{A}(y)| \le d(x, y)$.$\square$