Liouville's Constant is Transcendental

Theorem
Liouville's constant:

is transcendental.

Proof
Let $q = 10^{n!}$ and write:


 * $\displaystyle L = \frac p q + \sum_{k \mathop = n + 1}^\infty \frac 1 {10^{k!} }$

for some suitable $p \in \Z$.

Then:

Thus, by definition, $L$ is a Liouville number.

Therefore, by Liouville's Theorem, $L$ is transcendental.