Real Number Inequalities can be Added
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Theorem
Let $a, b, c, d \in \R$ such that $a > b$ and $c > d$.
Then:
- $a + c > b + d$
Proof 1
| \(\ds a\) | \(>\) | \(\ds b\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a + c\) | \(>\) | \(\ds b + c\) | Real Number Ordering is Compatible with Addition |
| \(\ds c\) | \(>\) | \(\ds d\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b + c\) | \(>\) | \(\ds b + d\) | Real Number Ordering is Compatible with Addition |
Finally:
| \(\ds a + c\) | \(>\) | \(\ds b + c\) | ||||||||||||
| \(\ds b + c\) | \(>\) | \(\ds b + d\) | ||||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a + c\) | \(>\) | \(\ds b + d\) | Trichotomy Law for Real Numbers |
$\blacksquare$
Proof 2
| \(\ds a\) | \(>\) | \(\ds b\) | by hypothesis | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a + c\) | \(>\) | \(\ds b + c\) | Real Number Ordering is Compatible with Addition | ||||||||||
| \(\ds c\) | \(>\) | \(\ds d\) | by hypothesis | |||||||||||
| \(\ds \leadsto \ \ \) | \(\ds b + c\) | \(>\) | \(\ds b + d\) | Real Number Ordering is Compatible with Addition | ||||||||||
| \(\ds \leadsto \ \ \) | \(\ds a + c\) | \(>\) | \(\ds b + d\) | Real Number Ordering is Transitive |
$\blacksquare$
Sources
- 1990: Edwin E. Moise: Elementary Geometry from an Advanced Standpoint (3rd ed.): Chapter $1$ / Section $1.4$: "The Ordering of the Real Numbers."
- 2000: James R. Munkres: Topology (2nd ed.) ... (previous) ... (next): $1$: Set Theory and Logic: $\S 4$: The Integers and the Real Numbers: Exercise $2 \ \text{(a)}$