Exercise 5, § 2

Let $X$ be a metric space, $d$ its metric. For each pair $(x, y)$ of points of $X$, let $d_0(x, y)$ denote the greatest lower bound of the real numbers $\alpha > 0$ such that $x$ and $y$ can be joined by a $V_\alpha$-chain (Chapter II, § 4, no. 4). Show that $d_0$ is a pseudometric on $X$, and that the metric space associated with the uniform space defined by the pseudometric $d_0$ on $X$ (no. 1) is an ultrametric space (Exercise 4).