Exercise 19, § 2

Let $\alpha$ and $\beta$ be two ordinals, and let E, F be two well-ordered sets such that $\operatorname{Ord}(E)=\alpha$ and $\operatorname{Ord}(F)=\beta$. In the set $E^F$ of mappings of F into E, consider the subset G of mappings $g$ such that $g(y)$ is equal to the least element of E for all but a finite number of elements $y\in F$. If $F^*$ is the ordered set obtained by endowing F with the opposite order, show that G is a connected component with respect to the relation “$x$ and $y$ are comparable” (Chapter II, § 6, Exercise 10) in the product $E^{F^*}$ endowed with the ordering defined in Exercise 12, and show that G is well-ordered. Furthermore, prove that $\operatorname{Ord}(G)=\alpha^\beta$ (use the uniqueness property of Exercise 17 (b)).

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