$\newcommand{\Z}{\mathbb{Z}}$I want to prove that $\alpha := \sum_{i\geq0}p^{i!} \in \Z_p$ is a $p$-adic Liouville number. $p$-adic Liouville number is defined as follows.
$\alpha \in \Z_p$ is called a $p$-adic Liouvile number if $$\liminf_{n\to\infty} \sqrt[n]{|n-\alpha|_p} = 0.$$
The following is what I tried. If I put $n = \sum_{i=0}^k p^{i!}$, then $$\sqrt[n]{|n-\alpha|_p} = p^{-(k+1)!/n} = p^{-(k+1)!/\sum_{i=0}^k p^{i!}}.$$ I want $\text{power index} \to -\infty$, but I can't prove.
I read this and its references as I can, but I can't find the proof. Although this article says the proof is easy, it seems rather difficult for me.