Real Zero is Less than Real One
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Theorem
The real number $0$ is less than the real number $1$:
- $0 < 1$
Proof
| \(\ds 1 \times 1\) | \(>\) | \(\ds 0\) | Square of Non-Zero Real Number is Strictly Positive | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 1\) | \(>\) | \(\ds 0\) | Real Number Axiom $\R \text M3$: Identity Element for Multiplication | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0\) | \(<\) | \(\ds 1\) | Definition of Dual Ordering |
$\blacksquare$
Sources
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $2 \ \text{(g)}$