Commensurability is Transitive

Theorem
Let $a$, $b$, $c$ be three real numbers.

If $a$ and $b$ is commensurable and so is $b$ and $c$, then $a$ and $c$ is commensurable.

Proof
From the definition of commensurablility:
 * $\displaystyle \frac a b, \frac b c \in \Q$

where $\Q$ denotes the set of all rational numbers.

From Rational Multiplication is Closed:
 * $\displaystyle \frac a b \times \frac b c \in \Q$

Cancelling $b$, we have:
 * $\displaystyle\frac a c \in \Q$

Hence the result.