Condition for Rational Cut to be Less than Given Cut

Theorem
Let $\alpha$ be a cut.

Let $p^*$ be the rational cut associated with a rational number $p$.

Then:
 * $p \in \alpha$


 * $p^* < \alpha$
 * $p^* < \alpha$

where $<$ denotes the strict ordering on cuts.

Proof
Let $p$ be a rational number such that $p \in \alpha$.

Then by definition of rational cut:
 * $p \notin p^*$

Thus:
 * $p \in \alpha \implies p^* < \alpha$

Let $p^* < \alpha$.

Then there exists a rational number $q$ such that $q \in \alpha$ and $q \notin p$.

Thus $q \ge p$.

But as $q \in \alpha$ it follows that $p \in \alpha$.