Exercise 2, § 2

Let E be an ordered set and let $\mathfrak{B}$ be the set of subsets of E which are well-ordered by the induced ordering. Show that the relation "X is a segment of Y" on $\mathfrak{B}$ is an order relation between X and Y and that $\mathfrak{B}$ is inductive with respect to this order relation. Deduce that there exist well-ordered subsets of E which have no strict upper bound in E.