Minus One is Less than Zero

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 $2 \ \text{(g)}$
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