Product of Rational Cuts is Rational Cut

Theorem
Let $p \in\ Q$ and $q \in \Q$ be rational numbers.

Let $p^*$ and $q^*$ denote the rational cuts associated with $p$ and $q$.

Then:
 * $p^* q^* = \paren {p q}^*$

Thus the operation of multiplication on the set of rational cuts is closed.

Proof
From Product of Cuts is Cut, $p^* q^*$ is a cut.