Proposition 4.7.1.27. Let $(T, \leq )$ be a linearly ordered set. There exists a well-ordered subset $S \subseteq T$ for which the inclusion map $S \hookrightarrow T$ is cofinal.
$\Newextarrow{\xhookrightarrow}{10,10}{0x21AA}$
Proof. Let $\{ S_ q \} _{q \in Q}$ be the collection of all well-ordered subsets of $T$. We regard $Q$ as a partially ordered set, where $q \leq q'$ if the set $S_{q}$ is an initial segment of $S_{q'}$. This partial ordering satisfies the hypotheses of Zorn's lemma, and therefore contains a maximal element $S_{\mathrm{max}}$. To complete the proof, it will suffice to show that the inclusion $S_{ \mathrm{max}} \hookrightarrow T$ is cofinal. Assume otherwise: then there exists an element $t \in T$ satisfying $s < t$ for each $s \in S_{\mathrm{max}}$. Then $S_{\mathrm{max}}$ is an initial segment of the well-ordered subset $S_{\mathrm{max}} \cup \{ t \} \subseteq T$, contradicting the maximality of $S_{\mathrm{max}}$. $\square$