Ordering on Cuts is Compatible with Addition of Cuts/Corollary

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