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:

$p \in \alpha$

if and only if:

$p^* < \alpha$

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:

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

$\Box$

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 $p \in \alpha$.

$\blacksquare$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.30$. Theorem