Multiplication of Positive Cuts preserves Ordering

Theorem

Let $0^*$ denote the rational cut associated with the (rational) number $0$.

Let $\alpha$, $\beta$ and $\gamma$ be cuts such that:

$0^* < \alpha < \beta$

$0^* < \gamma$

where $<$ denotes the strict ordering on cuts.

Then

$\alpha \gamma < \beta \gamma$

where $\alpha \gamma$ denotes the product of $\alpha$ and $\gamma$.

Proof

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Sources

• 1964: Walter Rudin: Principles of Mathematical Analysis (2nd ed.) ... (previous) ... (next): Chapter $1$: The Real and Complex Number Systems: Dedekind Cuts: $1.26$. Theorem $\text {(g)}$