Note that we cannot directly apply Exercise 2, as there is a condition in (a)(i) of that exercise that requires $h$ to map $J$ onto $E$ or a section of $E$ . And this assumption is not given for $k$ . In fact, this is exactly what we want from an order preserving map from $J$ into $E$ , so, we construct another order preserving function $h$ satisfying this condition, as suggested in the hint.

First, we need to formally show that $h$ is well-defined. This is pretty much straightforward using Exercise 1. For any $S_{\alpha}\subset J$ and $f:S_{\alpha}\rightarrow E$ define $\rho(f)=smallest[E-f(S_{\alpha})]$ if $f(S_{\alpha})\neq E$ , or $\rho(f)=e_{0}$ otherwise. Then $\rho$ is well-defined, and it uniquely defines $h$ satisfying the recursive formula.

Second, we use the existence of the order-preserving function $k$ to show that a) $h$ must be bounded above by $k$ , b) no section is mapped by $h$ onto $E$ , and c) using Exercise 2, $h$ is an order-preserving function that maps $J$ onto $E$ or its section.

a) Let $B$ be the subset of $\alpha\in J$ such that the inequality $h(\alpha)\le k(\alpha)$ holds. If $S_{\alpha}\subset B$ then for every $\beta\in S_{\alpha}$ , $k(\alpha)>k(\beta)\ge h(\beta)$ , so $k(\alpha)$ is an element that is greater than all elements in $h(S_{\alpha})$ , but $h(\alpha)$ is the smallest element that is greater than all elements in $h(S_{\alpha})$ provided there is at least one such element, and there is one, namely, $k(\alpha)$ , hence, $h(\alpha)\le k(\alpha)$ . By transfinite induction, $B=J$ .

b) If for some $\alpha\in J$ , $h(S_{\alpha})=E$ , then there is some $\beta<\alpha$ such that $k(\alpha)=h(\beta)\le k(\beta)$ . Contradiction. Therefore, $h(S_{\alpha})\neq E$ for all $\alpha\in J$ .

c) Now, since $h(S_{\alpha})\neq E$ for all $\alpha\in J$ , $h$ is defined by the expression 2(a)(ii), and by Exercise 2(a), $h$ is an order preserving function mapping $J$ onto $E$ or its section. Since $h$ is order preserving, it is injective, so $h$ is an order preserving bijection from $J$ onto $E$ or its section.