Complement of Limit Superior is Limit Inferior of Complements

Theorem

Let $\sequence {E_n}_{n \mathop \in \N}$ be a sequence of sets.

Then:

$\ds \map \complement {\limsup_{n \mathop \to \infty} \ E_n} = \liminf_{n \mathop \to \infty} \ \map \complement {E_n}$

where $\limsup$ and $\liminf$ denote the limit superior and limit inferior, respectively.

$\ds \map \complement {\limsup_{n \mathop \to \infty} \ E_n}$

$=$

$\ds \map \complement {\bigcap_{n \mathop = 0}^\infty \bigcup_{i \mathop = n}^\infty E_n}$

$\ds $

$=$

$\ds \bigcup_{n \mathop = 0}^\infty \map \complement {\bigcup_{i \mathop = n}^\infty E_n}$

$\ds $

$=$

$\ds \bigcup_{n \mathop = 0}^\infty \bigcap_{i \mathop = n}^\infty \map \complement {E_n}$

$\ds $

$=$

$\ds \liminf_{n \mathop \to \infty} \ \map \complement {E_n}$

$\blacksquare$

1970: Avner Friedman: Foundations of Modern Analysis ... (previous) ... (next): $\S 1.1$: Rings and Algebras: Problem $1.1.1$