Real Zero is Less than Real One

Theorem

The real number $0$ is less than the real number $1$:

$0 < 1$

$\ds 1 \times 1$

$>$

$\ds 0$

$\ds \leadsto \ \ $

$\ds 1$

$>$

$\ds 0$

Real Number Axiom $\R \text M3$: Identity Element for Multiplication

$\ds \leadsto \ \ $

$\ds 0$

$<$

$\ds 1$

Definition of Dual Ordering

$\blacksquare$

2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $2 \ \text{(g)}$