Liouville Numbers are Irrational

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Theorem

Liouville numbers are irrational.


Proof

Let $x$ be a Liouville number.

Aiming for a contradiction, suppose $x$ were rational, that is:

$x = \dfrac a b$

with $a, b \in \Z$ and $b > 0$.

By definition of a Liouville number, for all $n \in \N$, there exist $p,q \in \Z$ (which may depend on $n$) with $q > 1$ such that:

$0 < \left \lvert x - \dfrac p q \right \rvert < \dfrac 1 {q^n}$

Let $n$ be a positive integer such that $2^{n - 1} > b$.

Let $p$ and $q$ be any integers with $q > 1$.

We have:

$\left \lvert x - \dfrac p q \right \rvert = \dfrac {\left \lvert a q - b p \right \rvert} {b q}$

If $\left \lvert a q - b p \right \rvert = 0$, this would violate the first inequality.

If $\left \lvert a q - b p \right \rvert \ne 0$, then:

\(\displaystyle \left \lvert x - \frac p q \right \rvert\) \(\ge\) \(\displaystyle \frac 1 {b q}\) as $\left \lvert a q - b p \right \rvert$ is a positive integer
\(\displaystyle \) \(>\) \(\displaystyle \frac 1 {2^{n - 1} q}\) by our choice of $n$
\(\displaystyle \) \(\ge\) \(\displaystyle \frac 1 {q^n}\) as $q > 1$ by definition

which would violate the second inequality.

Therefore, in any case, if $n$ is sufficiently large, there cannot exist integers $p$ and $q$ with $q > 1$ satisfying the two inequalities.

This is a contradiction.

Thus $x$ is irrational.

$\blacksquare$