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.

If it is bounded, say by $M$, then all the quotients can be written as $A_k/M!$ where $A_k$ is an integer, which clearly does not converge to an irrational.
