Set of Liouville Numbers is Uncountable
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Theorem
The set of Liouville numbers is uncountable.
Proof
By Corollary to Liouville's Constant is Transcendental, all numbers of the form:
\(\ds \sum_{n \mathop \ge 1} \frac {a_n} {10^{n!} }\) | \(=\) | \(\ds \frac {a_1} {10^1} + \frac {a_2} {10^2} + \frac {a_3} {10^6} + \frac {a_4} {10^{24} } + \cdots\) |
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.
$\blacksquare$