Let \((A_n)_{n\geq 1}\) be a sequence of subsets of a set \(X\). Its limit inferior and limit superior are \[ \liminf _{n \to \infty} A_n=\bigcup_{n \geq 1} \bigcap_{j \geq n} A_j, \qquad \limsup _{n \to \infty} A_n=\bigcap_{n \geq 1} \bigcup_{j \geq n} A_j. \] A point \(x\in X\) lies in \(\liminf_n A_n\) if and only if \(x\in A_j\) for all but finitely many \(j\), and in \(\limsup_n A_n\) if and only if \(x\in A_j\) for infinitely many \(j\). In particular, \(\liminf _{n \to \infty} A_n \subseteq \limsup _{n \to \infty} A_n\).
A point lies in \(\bigcap_{j\geq n}A_j\) for some \(n\) exactly when it lies in every \(A_j\) with \(j\geq n\) for some \(n\), that is, in all but finitely many \(A_j\). A point lies in \(\bigcup_{j\geq n}A_j\) for every \(n\) exactly when for every \(n\) it lies in some \(A_j\) with \(j\geq n\), that is, in infinitely many \(A_j\). A point in all but finitely many \(A_j\) lies in infinitely many of them.