Munkres Topology: Section 11; Problem 3

Question

Here's the problem:

I've managed to show that $$h(\alpha) \leq k(\alpha),$$ but I can't figure out how it follows that $$h(S_\alpha) \neq E.$$

Answer 1

For all $\beta \in S_\alpha$, $h(\beta) \leq k(\beta) < k(\alpha)$ since $k$ is order-preserving, so $k(\alpha) \in E- h(S_\alpha)$.