Approximation of real numbers with rational number sequences

Let $\alpha\in\mathbb{R}$, and let $C>\max\{1,\alpha\}$. It is shown that if $\{p_n/q_n\}$ is a sequence formed out of all rational numbers $p/q$ such that

$$\left\vert\alpha -\frac{p}{q} \right\vert \leq \frac{1}{Cq^2},$$

where $p\in\mathbb{Z}$ and $q\in\mathbb{N}$ are relatively prime numbers, then either $\{p_n/q_n\}$ has finitely many elements or

$$\limsup_{n\to\infty}\frac{\log\log q_n}{\log n}\geq 1,$$

where the points $\{q_n\}_{n\in\mathbb{N}}$ are ordered by increasing modulus. This implies that the sequence of denominators $\{q_n\}_{n\in\mathbb{N}}$ grows exponentially as a function of $n$, and so the density of rational numbers which approximate $\alpha$ well in the above sense is relatively low.