Trichotomy Law for Real Numbers/Proof 2
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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 $b$: | \(\ds a > b \) | ||||||
\((2)\) | $:$ | $a$ is equal to $b$: | \(\ds a = b \) | ||||||
\((3)\) | $:$ | $a$ is less than $b$: | \(\ds a < b \) |
Proof
$\le$ is a total ordering on $\R$.
The trichotomy follows directly from Trichotomy Law.
$\blacksquare$