Exercise 4, § 2

An ordered set F is said to be partially well-ordered if every totally ordered subset of F is well-ordered. Show that in every ordered set E there exists a partially well-ordered subset which is cofinal in E. (Consider the set $\mathfrak{F}$ of partially well-ordered subsets of E, and the order relation "$X \subset Y$ and no element of $Y - X$ is bounded above by any element of X" between elements X and Y of $\mathfrak{F}$. Show that $\mathfrak{F}$ is inductive with respect to this order relation.)