Let \(\displaystyle \alpha \in \mathbb{R}-\mathbb{Q}\) and \(\displaystyle (x_n)_{n\ge 1},(y_n)_{n \ge 1}\subset \mathbb{Q}^*\) such that \(\displaystyle \lim_{n\to \infty} \frac{x_n}{y_n}=\alpha\). Prove that \(\displaystyle \lim_{n\to\infty}x_n=\lim_{n\to\infty}y_n=\infty\).

I guess if \(\displaystyle \lim_{n\to\infty}x_n=\lim_{n\to\infty}y_n\le\infty\) then \(\displaystyle \lim_{n\to \infty} \frac{x_n}{y_n}=\alpha \in \mathbb{Q}\) which is impossible.