Limit superior and limit inferior of a sequence of sets

Let \((A_n)_{n\geq 1}\) be a sequence of subsets of a set \(X\). Its limit superior and limit inferior are \[ \begin{aligned} \limsup_{n\to\infty} A_n &\coloneqq \bigcap_{n\geq 1} \bigcup_{j\geq n} A_j = \theset{x\in X \suchthat x\in A_n \text{ for infinitely many } n}, \\ \liminf_{n\to\infty} A_n &\coloneqq \bigcup_{n\geq 1} \bigcap_{j\geq n} A_j = \theset{x\in X \suchthat x\in A_n \text{ for all but finitely many } n}. \end{aligned} \]