@Lord_Farin small question: do you know a version of this lemma where $\alpha$ is not a constant and depends on some variable $m$ ?
let $\alpha \in \mathbb{R}^+$ and $p_1,p_2,\dots, q_1, q_2, \ldots \in \mathbb{N}$ such that $\left|\alpha q_n - p_n \right| \neq 0$ for all $n \in \mathbb{N}$ and $ \displaystyle \lim_{n \rightarrow \infty} p_n = \lim_{n \rightarrow \infty} q_n = \infty\,$ if $\displaystyle\lim_{n \rightarrow \infty} \left| \alpha q_n - p_n \right| = 0.\,$ Then $α$ is irrational.