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.

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Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.28$. Theorem: $\text {(b)}$