Liouville's theorem and examples

Theorem. If $\alpha$ is algebraic of degree $d\ge1$, then there exists $c>0$ such that for all $p,q\in\mathbb{Z}$ with $q>0$:
$$|\alpha-p/q|\ge\frac{c}{q^d}.$$

Construction. $\alpha=\sum_{n\ge1}10^{-n!}$ can be approximated by $p_N/q_N=\sum_{n\le N}10^{-n!}$ with $q_N=10^{N!}$ and $|\alpha-p_N/q_N|<10^{-(N+1)!}<1/q_N^N$.

Conclusion. For any $d$, this violates Liouville's bound with $d=N$, so $\alpha$ is transcendental.