14.5 Limits of Sets
Definition 14.5.1 (Limit of Sets). Let $X$ be a set and $\seq{E_n}\subset 2^{X}$, then the limit superior of $\seq{E_n}$ is
\[\limsup_{n \to \infty}E_{n} = \bigcap_{n \in \natp}\bigcup_{k \ge n}E_{k}\]
and the limit inferior of $\seq{E_n}$ is \[\liminf_{n \to \infty}E_{n} = \bigcup_{n \in \natp}\bigcap_{k \ge n}E_{k}\]
Lemma 14.5.2. Let $X$ be a set and $\seq{X_n}\subset 2^{X}$, then
\[\limsup_{n \to \infty}E_{n} = \bracs{x \in X|x \in E_n \text{for infinitely many }n}\]
and \[\liminf_{n \to \infty}E_{n} = \bracs{x \in X|x \in E_n \text{for all but finitely many }n}\]
Proof. For any $x \in X$, $x \in \limsup_{n \to \infty}E_{n}$ if and only if for any $n \in \natp$, there exists $k \ge n$ such that $x \in E_{k}$, if and only if $x \in E_{n}$ for infinitely many $n \in \natp$.
For any $x \in X$, $x \in \liminf_{n \to \infty}$ if and only if there exists $n \in \natp$ such that $x \in E_{k}$ for all $k \ge n$, if and only if $x \in E_{n}$ for all but finitely many $n \in \natp$.$\square$