# Real Number Inequalities can be Added

## 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

$\blacksquare$