Sum of Strictly Positive Real Numbers is Strictly Positive
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Theorem
- $x, y \in \R_{>0} \implies x + y \in \R_{>0}$
Proof
| \(\ds x\) | \(>\) | \(\ds 0\) | ||||||||||||
| \(\, \ds \land \, \) | \(\ds y\) | \(>\) | \(\ds 0\) | Real Number Ordering is Compatible with Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds x + y\) | \(>\) | \(\ds 0 + 0\) | Real Number Inequalities can be Added | ||||||||||
| \(\ds \) | \(=\) | \(\ds 0\) | Real Number Axiom $\R \text A3$: Identity for Addition |
$\blacksquare$
Sources
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $2 \ \text{(b)}$