Sample "Smallest Interval" Proof




We will prove for intervals $I$ and $J$, that $(-I)J=-(IJ)$ (page 21 problem 16(e)).

We first note that $-I$ is the smallest interval containing all $-x$ where $x$ lies in $I$. Also, $UJ$ is the smallest interval containing all $uy$ where $u\in U$ and $y\in J$. Replacing $U$ by $-I$ now tells us that:

$(-I)J$ is the smallest interval containing all $(-x)y$ for $x\in I$ and $y\in J$.

Similarly, $-(IJ)$ is the smallest interval containing all $-w$ where $w\in IJ$. Thus:

$-\left( IJ\right) $ is the smallest interval containing all $-\left( xy\right) $ where $x\in I$ and $y\in J$.

But now we know that for rational numbers $x$ and $y$, $(-x)y=-(xy)$. Thus, $(-I)J$ and $-(IJ)$ are both smallest intervals with the same property (i.e. containing the same set of numbers). Thus, by Proposition 1.1.16, they are equal.

This is a model for how to use the smallest interval property to prove, pretty easily, that certain intervals are equal.