Absolute Value of Cut is Zero iff Cut is Zero

Definition

Let $\alpha$ be a cut.

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

Then:

$\size \alpha = 0^*$ if and only if $\alpha = 0^*$

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

Proof

Let $\alpha 0^*$.

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

Let $\alpha \ne 0^*$.

Then either:

$\alpha > 0^*$ in which case $\size \alpha = \alpha > 0^*$

or:

$\alpha < 0^*$ in which case $\size \alpha = -\alpha > 0^*$

In either case $\size \alpha \ne 0^*$.

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