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