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Limit of a Monotonic Set Sequence

Last updated Nov 1, 2022

# Definition

Let $({A}{n}){n=1}^{\infty}$ be a Sequence

of Sets. Then if

(A_{n})

is Non-Decreasing Function

\lim\limits_{n \to \infty} A_{n} = \bigcup\limits_{n \in \mathbb{N}} A_{n},

an if

(A_{n})

is Non-Increasing Function then

\lim\limits_{n \to \infty} A_{n} = \bigcap\limits_{n \in \mathbb{N}} A_{n}.

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Liminf of Set Sequence

Liminf of Set Sequence

...\infty} A{n}$ there exists an $n \in \mathbb{N}$ so that $\forall k \geq n$ , $x \in A{k}$ by definition of and . Thus $$\liminf\limits{n \to \infty} A{n} = \{x \in \bigcup\limits{n \in \mathbb{N}} A{n} : x \not\in A{n}......

11/7/2022
Limsup of Set Sequence

Limsup of Set Sequence

...\to \infty} A{n}$ for every $n \in \mathbb{N}$ there exists $k \geq n$ , s.t. $x \in A{k}$ by definition of and . Thus $$\limsup\limits{n \to \infty} A{n} = \{x \in \bigcup\limits{n \in \mathbb{N}} A{n} : x \in A{n}......

11/7/2022

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