# Multiplication inverse for dedekind cut

Let $\alpha \in P_R$ be a cut. Since there exists a cut that is not $\{q\in Q\mid q<r\}=r^*$ for every $r\in Q$, $\alpha$ doesn't need to be of the form $r^*$.

Let $$\gamma= 0^* \cup \{0\} \cup \{q\in P_Q\mid\text{ there exists }r\in P_Q\text{ such that }r>q\text{ and }1/r \notin \alpha\}\;.$$

I have proved that $\alpha \gamma$ is a subset of $1^*$. I dont't know how to prove $1^*$ is a subset of $\alpha \gamma$. Help

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