Find $\bigcap \mathcal A^c$ and $\bigcup \mathcal A^c$ with the proof.

Question

Let $A_n =\{x \in \Bbb R :-\frac {1}{n} \lt x \lt \frac{1}{n}\}$,$n \in \Bbb N$ and define the indexed family $\mathcal A^c = \{ A_{n}^{c} :n \in \Bbb N \}$. Find $\bigcap \mathcal A^c$ and $\bigcup \mathcal A^c$ with the proof.

I have no idea how to do.

Answer 1

$\cap_n \mathcal A_n =\cap_n (-\frac 1 n,\frac 1 n)^{c}=(\cup_n (-1,1))^{c}=(-1,1)^{c}=\mathbb R \setminus (-1,1)$. I leave it to you to show that $\cup_n \mathcal A_n =\mathbb R \setminus \{0\}$.