Set of Liouville Numbers is Uncountable

Theorem
The set of Liouville numbers is uncountable.

Proof
By Corollary to Liouville's Constant is Transcendental, all numbers of the form:

where
 * $a_1, a_2, a_3, \ldots \in \set {1, 2, \ldots, 9}$

are Liouville numbers.

Therefore each sequence in $\set {1, 2, \ldots, 9}$ defines a unique Liouville number.

By Set of Infinite Sequences is Uncountable, there are uncountable sequences in $\set {1, 2, \ldots, 9}$.

As the set of Liouville numbers has an uncountable subset, it is also uncountable by Sufficient Conditions for Uncountability.