relationship between stongly inductive poset and zorn's lemma

Question

In appendix A of Prof. Richard Elman's lecture notes link, at first, he gives a proof in Proposition A.3 for an equivalence statement of zorn's lemma, i. e. every nonempty strongly inductive poset has a maximal element . However, I find the proof there has something wrong. Set $\mathfrak{C}$ here contains all chains of $S$. I think there must be something uncorrect in Line 4 starting from 'Let $C \in \mathfrak{C}$'. But I don't know how to fix the problem here. Hope that someone can help.

Answer 1

Nothing is incorrect. The collection of chains in a partial order is itself a strongly inductive poset, when ordered by $\subseteq$.

By the assumption, this strongly inductive poset has a maximal element, which is, by definition, a maximal chain in the original poset $S$. Since every chain has an upper bound in $S$, so must this maximal chain have an upper bound. But because it is maximal, the upper bound must be an element of the chain and a maximal element of $S$.