Trichotomy Law for Real Numbers

Theorem

The real numbers obey the Trichotomy Law. That is, $\forall a, b \in \R$, exactly one of the following holds:

$(1): \quad a > b$ ($a$ is greater than $b$)

$(2): \quad a = b$ ($a$ is equal to $b$)

$(3): \quad a < b$ ($a$ is less than $b$).

Note that $a > b \iff b < a$.

We also use the following notation:

$(a): \quad a \le b \iff a < b \lor a = b$ ($a$ is less than or equal to $b$)

$(b): \quad a \ge b \iff a > b \lor a = b$ ($a$ is greater than or equal to $b$).

The following also holds:

$\forall a, b, c \in \R: a < b \land b < c \implies a < c$

Proof

This follows directly from the fact that the real numbers form a totally ordered field.

$\blacksquare$

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 1.4$