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I need help with the following two questions. Let $\alpha$ be a cut. We define $$\alpha^{-1}=\{p \in \mathbb{Q}:\frac{1}{p} \in \alpha^c \text{ and } \frac{1}{p} \text{ is not the least element of } \alpha^c\} \cup \{p \in \mathbb{Q}: p \leq 0\}$$ Also $\alpha$ is called a rational cut if there is $r \in \mathbb{Q}$ such that $\alpha=\{p \in \mathbb{Q}:p.

Question 1: Show that $\alpha^{-1}$ is also a cut.

Question 2: Prove that if $\alpha$ is a rational cut then $\alpha^{-1}$ is also so.

(I don't know if I tag this question correctly.)

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