Ordering on Cuts is Compatible with Addition of Cuts/Corollary
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Theorem
Let $\alpha$ and $\gamma$ be cuts.
Let the operation of $\alpha + \gamma$ be the sum of $\alpha$ and $\gamma$.
Let $0^*$ denote the rational cut associated with the (rational) number $0$.
If:
- $\alpha > 0^*$ and $\gamma > 0^*$
then:
- $\alpha + \gamma > 0^*$
where $>$ denotes the strict ordering on cuts.
Proof
From Ordering on Cuts is Compatible with Addition of Cuts
- $0^* + 0^* < 0^* + \alpha$
- $\alpha + 0^* < \alpha + \gamma$
The result follows from Ordering on Cuts is Transitive.
$\blacksquare$
Sources
- 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.18$. Theorem