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?

The true theorem is
Theorem (Liouville).-For every real algebraic number $x$ of degree  $n≥2$ exists a positive number $c (x)$ such that $| x-p / q |>\frac{ c (α)}{ q ^ n}$ holds for all rational 
$\frac pq  (q> 0)$.
By definition,  a Liouville number $x$ is such that for all $n$ large enough there is a rational
$\frac pq$ with $q\gt 1$ such that $|x- \frac pq|\le \frac{1}{q^n}$ and this such a  number $x$ is necessarily transcendental.
The theorem of Roth says that for every irrational $x$ algebraic and all $\epsilon> 0$ the inequality  $$|x-\frac pq | <\frac{1}{ q ^{2 + ε}}$$ has only a finite number of solutions in irreducible rational $\frac pq$  which implies that for all $\epsilon\gt  0$ there exists a positive constant $C (x, ε)$ such that the inequality  $| x-\frac pq |> \frac{C (x, ε)}{ q^{2 + ε}}$ is holds for all rational $\frac pq$ with $q>0$.
This is a profound theorem by which Roth won the Fields Medal (International Congress of Mathematicians, Edinburgh, 1958).
