Corollary 3.2.8.5. Let $S$ be a nonempty linearly ordered set. Then the nerve $\operatorname{N}_{\bullet }(S)$ is weakly contractible.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. By virtue of Corollary 3.2.8.4, we may assume without loss of generality that $S$ is finite. In this case, there is an isomorphism $S \simeq [n]$ for some integer $n \geq 0$, so that $\operatorname{N}_{\bullet }(S)$ is isomorphic to the standard simplex $\Delta ^ n$. $\square$