How could we see that $\{n_k\}_k$ converges $\infty$? Let $x \in \Bbb R\setminus \Bbb Q$ and the sequence $\{\frac {m_k} {n_k}\}_k$  concerges to $x$. The question is from this comment by Ilya:

How could we see that $\{n_k\}_k$ converges $\infty$?

Thanks for your help.

Let $M$ be fixed. Show that there exists such $k_0$ that
$$n_k>M, k\geq k_0.$$
By assuming the contradiction, we'll get such subsequance $\{n_{k_j}\}_j$ that
$n_{k_j} \leq M$ for all $j\geq 1$. Note that
$$\frac{m_{k_j}}{n_{k_j}}\to x.$$ Since such fractions can written as
$$\frac{m_{k_j}}{n_{k_j}}=\frac{A_{k_j}}{M!},$$
where $A_{k_j}$ are integers, then $\frac{m_{k_j}}{n_{k_j}}$ cannot tends to an irrational number. So there is no such subsequance $\{n_{k_j}\}_j$.
