Complement of Limit Inferior is Limit Superior of Complements
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Theorem
Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence of sets.
Then:
- $\ds \map \complement {\liminf_{n \mathop \to \infty} \ E_n} = \limsup_{n \mathop \to \infty} \ \map \complement {E_n}$
where $\liminf$ and $\limsup$ denote the limit inferior and limit superior, respectively.
Proof
\(\ds \map \complement {\liminf_{n \mathop \to \infty} \ E_n}\) | \(=\) | \(\ds \map \complement {\bigcup_{n \mathop = 0}^\infty \bigcap_{i \mathop = n}^\infty E_n}\) | Definition 1 of Limit Inferior of Sequence of Sets | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcap_{n \mathop = 0}^\infty \map \complement {\bigcap_{i \mathop = n}^\infty E_n}\) | De Morgan's Laws | |||||||||||
\(\ds \) | \(=\) | \(\ds \bigcap_{n \mathop = 0}^\infty \bigcup_{i \mathop = n}^\infty \map \complement {E_n}\) | De Morgan's Laws | |||||||||||
\(\ds \) | \(=\) | \(\ds \limsup_{n \mathop \to \infty} \ \map \complement {E_n}\) | Definition 1 of Limit Superior of Sequence of Sets |
$\blacksquare$
Sources
- 1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras: Problem $1.1.1$