Trichotomy Law for Real Numbers

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Theorem

The real numbers obey the Trichotomy Law.

That is, $\forall a, b \in \R$, exactly one of the following holds:

\((1)\)   $:$   $a$ is greater than $b$:    \(\displaystyle a > b \)             
\((2)\)   $:$   $a$ is equal to $b$:    \(\displaystyle a = b \)             
\((3)\)   $:$   $a$ is less than $b$:    \(\displaystyle a < b \)             


Proof 1

This follows directly Real Numbers form Totally Ordered Field.

$\blacksquare$


Proof 2

$\le$ is a total ordering on $\R$.

The trichotomy follows directly from Trichotomy Law.

$\blacksquare$


Sources