Set of Liouville Numbers is Uncountable
Jump to navigation
Jump to search
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.
$\blacksquare$