Lemma 5.3.10. Let $X$ be a set and $d: X \times X \to [0, \infty)$ be a pseudometric, then the pseudometric
\[\td d: X \times X \to [0, \infty) \quad (x, y) \mapsto d(x, y) \wedge 1\]
is equivalent to $d$.
Proof. For any $r \in (0, 1]$, $E(d, r) = E(\td d, r)$. Since sets of the above form generate the uniformity induced by $d$ and $\td d$, their induced uniformities coincide.$\square$