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