---- > [!definition] Definition. ([[limit inferior and limit superior]] of *sequences*) > The **limit inferior** of a [[sequence]] $(x_{n})$ (say, in $[-\infty, \infty]$) is defined $\liminf_{n \to \infty} x_{n}:= \lim _{n \to \infty} ( \inf_{m \geq n} x_{m}).$ The **limit superior** of a [[sequence]] $(x_{n})$ (say, in $[-\infty, \infty]$) is defined $\limsup_{n \to \infty} x_{n}:= \lim_{n \to \infty} (\sup_{m \geq n} x_{n}).$ ![[CleanShot 2025-10-11 at [email protected]|500]] Image from Wikipedia. ^definition > [!equivalence] > > Equivalently, since the sequences $(\inf_{m \geq n}x_{m})_{n \in \mathbb{N}}$ and $(\sup_{m \geq n} x_{m})_{n \in \mathbb{N}}$ are manifestly [[monotonic map|monotonically increasing and decreasing]] respectively, by the [[monotone convergence theorem for sequences]] we have $\liminf_{n \to \infty} x_{n}= \sup_{n \in \mathbb{N}} \inf_{m \geq n}x_{m}=\sup \{ \{ \inf x_{m}: m \geq n \} : n \in \mathbb{N} \}$ > and $\limsup_{n \to \infty} x_{n}=\inf_{n \in \mathbb{N}} \sup_{m \geq n} x_{m}= \inf\{ \{ \sup x_{m}: m \geq n \} : n \in \mathbb{N}\}.$ > > > [!definition] Definition. ([[limit inferior and limit superior]] of *sets*) > The **lim sup of a sequence $(A_{n})$ of sets** in a universe $\Omega$ is $\bigcap_{n=1}^{\infty}\bigcup_{k \geq n} A_{k}=\{ x \in \Omega : x \in A_{n} \text{ for infinitely many }n \}.$ ^definition ---- #### ---- #### References > [!backlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM [[]] > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ``` > [!frontlink] > ```dataview > TABLE rows.file.link as "Further Reading" > FROM outgoing([[]]) > FLATTEN file.tags as Tag > WHERE Tag = "#definition" OR Tag = "#theorem" OR Tag = "#MOC" OR Tag = "#proposition" OR Tag = "#axiom" > GROUP BY Tag > ```