Real Number Inequalities can be Added/Proof 2
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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
\(\ds a\) | \(>\) | \(\ds b\) | 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 |
$\blacksquare$