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

\(\displaystyle a\) \(>\) \(\displaystyle b\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle a + c\) \(>\) \(\displaystyle b + c\) Real Number Ordering is Compatible with Addition


\(\displaystyle c\) \(>\) \(\displaystyle d\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle b + c\) \(>\) \(\displaystyle b + d\) Real Number Ordering is Compatible with Addition


Finally:

\(\displaystyle a + c\) \(>\) \(\displaystyle b + c\)
\(\displaystyle b + c\) \(>\) \(\displaystyle b + d\)
\(\displaystyle \leadsto \ \ \) \(\displaystyle a + c\) \(>\) \(\displaystyle b + d\) Trichotomy Law for Real Numbers

$\blacksquare$


Sources