Minus One is Less than Zero
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Theorem
- $-1 < 0$
Proof
| \(\ds 0\) | \(<\) | \(\ds 1\) | Real Zero is Less than Real One | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds -0\) | \(>\) | \(\ds -1\) | Order of Real Numbers is Dual of Order of their Negatives | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds 0\) | \(>\) | \(\ds -1\) | Negative of Real Zero equals Zero | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds -1\) | \(<\) | \(\ds 0\) | 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 $2 \ \text{(g)}$