Is $\bigcap_{n=1}^\infty \left( \mathbb{R} \setminus A_n \right) $ countable set? Let $A_1, A_2 , \ldots \subseteq \mathbb{R}$ such that $\exists k \quad \mathbb{R} \setminus A_k$ is countable set. Is it true that  $$\bigcap_{n=1}^\infty \left( \mathbb{R} \setminus A_n \right) $$ is also is countable set

$$\bigcap_{n=1}^{\infty} \left( \mathbb{R} \setminus A_n \right) \subseteq \mathbb{R}\setminus A_k$$
But it is an elementary fact that a subset of a countable set is countable.
