Liouville's Constant is Transcendental/Corollary

Corollary to Liouville's Constant is Transcendental
All real numbers of the form:

where
 * $a_1, a_2, a_3, \ldots \in \set {1, 2, \ldots, 9}$

are transcendental.

Proof
Let $n \in \N$.

For $n = 1$, let $p = a_1$ and $q = 10$. Then:
 * $\size {L - \dfrac p q} = \displaystyle \sum_{k \mathop \ge 2} \dfrac {a_k} {10^{k!} } < \dfrac 1 {10} = \dfrac 1 q$

For $n > 1$, let $q = 10^{n!}$ and write:


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

for some suitable $p \in \Z$.

Then:

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

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