Limit superior and limit inferior of a sequence of sets

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