Abstract: 
Liouville numbers are a class of transcendental numbers that can be approximated
particularly well with rational numbers. Historically, they were the first
numbers whose transcendence was proven.
In this entry, we define the concept of Liouville numbers as well as the
standard construction to obtain Liouville numbers (including Liouville's
constant) and we prove their most important properties: irrationality and
transcendence.
The proof is very elementary and requires only standard arithmetic, the Mean
Value Theorem for polynomials, and the boundedness of polynomials on compact
intervals.

BibTeX: 
@article{Liouville_NumbersAFP,
author = {Manuel Eberl},
title = {Liouville numbers},
journal = {Archive of Formal Proofs},
month = dec,
year = 2015,
note = {\url{https://isaafp.org/entries/Liouville_Numbers.html},
Formal proof development},
ISSN = {2150914x},
}
