# Limit

#### yo79

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 $$\displaystyle \lim_{n\to\infty}x_n=\lim_{n\to\infty}y_n=\infty$$.

#### Hoempa

Math Team
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.

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