Absolute Value of Cut is Greater Than or Equal To Zero Cut

Definition

Let $\alpha$ be a cut.

Let $\size \alpha$ denote the absolute value of $\alpha$.

Then:

$\size \alpha \ge 0^*$

where:

$0^*$ denotes the rational cut associated with the (rational) number $0$

$\ge$ denotes the ordering on cuts.

Proof

Let $\alpha \ge 0^*$.

Then by definition $\size \alpha = \alpha \ge 0^*$.

Let $\alpha < 0^*$.

Then:

$\exists \beta: \beta + \alpha = 0^*$

Thus:

$\alpha = -\beta$

and it follows that $\beta > 0^*$.

The result follows.

$\blacksquare$

Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.24$. Definition