Let $J$ and $E$ be well-ordered sets; let $h: J\rightarrow E$. Show the following two statements are equivalent.
- $h$ is order preserving and its image is $E$ or a section of $E$
- $h(\alpha)= smallest(E-h(S_{\alpha}))$ for all $\alpha$
[Hint: Show that each of these conditions implies that $h(S_\alpha$) is a section of $E$; conclude that it must be a section by $h(\alpha)$]
To clarify notation, $S_\alpha$ means the section by $\alpha$.
I got as far as proving the hint suggestion. How do I proceed from there?
Given that you already proved the hint, here is how to proceed.
The first point implying the second is already answered here.
For the other direction, let $e < h(\alpha)$ for $\alpha \in J$. Then $e \in S_{h(\alpha)} = h(S_\alpha)$, so $e$ is in the image of $h$. Hence the image of $h$ must be a section (or $E$ itself).