Work out \(\bigcup_{n=1}^{\infty} A_{n}\) and \(\bigcap_{n=1}^{\infty} A_{n}\), where \(A_{n}\) is defined as follows for \(n \in \mathbb{N}\) :

(a) \(A_{n}=\{x \in \mathbb{R} \mid x>n\}\).

(b) \(A_{n}=\left\{x \in \mathbb{R} \left\lvert\, \frac{1}{n}

(c) \(A_{n}=\left\{x \in \mathbb{R} \left\lvert\,-n

(d) \(A_{n}=\left\{x \in \mathbb{Q} \left\lvert\, \sqrt{2}-\frac{1}{n} \leq x \leq \sqrt{2}+\frac{1}{n}\right.\right\}\).