Ordering of Rational Cuts preserves Ordering of Associated Rational Numbers

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:

$p^* < q^* \iff p < q$

where $p^* < q^*$ denotes the strict ordering on cuts defined as:

$\beta < \gamma \iff \exists p \in \Q: p \in \beta, p \notin \gamma$

Proof

Let $p < q$.

Then $p \notin p^*$ but $q \in q^*$.

Thus $p^* < q^*$ by definition of the strict ordering on cuts .

Let $p^* < q^*$.

Then:

$\exists r \in \Q: r \notin p^*, r \in q^*$

Hence:

$p \le r < q$

and so:

$p < q$

$\blacksquare$

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: $\text {(c)}$