criterion for irrationality Given the following theorem:

Let $x\in\mathbb{R}.$ Is there a $\delta>0$ and a rational sequence $\{p_n/q_n\}_n$ with $p_n/q_n\neq x$ and $$\left|x-\frac{p_n}{q_n}\right|<\frac{1}{q_n^{1+\delta}}$$ for all $n$, then x is irrational.

Who did find this statement and when? Is there a proof somewhere?

I'd suspect this to be known to Liouville (as it "smells" a lot like his approximation theorem of Comptes Rendue 18 (1844)), but this specific observation may be even older.
The proof is simple: Assume $x=\frac ab$ is rational.
Then for any rational $\frac pq\ne x$ of a sequence as given, we have
$$\frac1{q^{1+\delta}}>\left|x-\frac pq\right|=\frac{|aq-bp|}{bq}\ge \frac 1{bq}, $$
hence
$$ q<b^{1/\delta}.$$
But with bounded denominator there are even only finitely many rationals with the weaker condition $\left|x-\frac pq\right|<1$.
