Exercise 6, § 2

Let E be an ordered set and let $f$ be a mapping of E into E such that $f(x) \geqslant x$ for all $x \in E$.

(a) Let $\mathfrak{G}$ be the set of subsets M of E with the following properties : (1) the relation $x \in M$ implies $f(x) \in M$; (2) if a non-empty subset of M has a least upper bound in E, then this least upper bound belongs to M. For each $a \in E$, show that the intersection $C_a$ of the sets of $\mathfrak{G}$ which contain $a$ also belongs to $\mathfrak{G}$; that $C_a$ is well-ordered; and that if $C_a$ has an upper bound $b$ in E, then $b \in C_a$ and $f(b) = b$. $C_a$ is said to be the chain of $a$ (with respect to the function $f$). (Consider the set $\mathfrak{M}$ whose elements are the empty set and the subsets X of E which contain $a$ and have a least upper bound $m$ in E such that either $m \notin X$ or $f(m) > m$, any apply Lemma 3 of no. 3 to the set $\mathfrak{M}$.)

(b) Deduce from (a) that if E is inductive, then there exists $b \in E$ such that $f(b) = b$.