Complement of Limit Superior is Limit Inferior of Complements

Theorem

Let $\left\{{E_n : n \in \N}\right\}$ be a sequence of sets.

Then:

$\displaystyle \complement \left({\limsup_{n \to \infty} \ E_n}\right) = \liminf_{n \to \infty} \ \complement \left({E_n}\right)$

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

Proof

 $\displaystyle \complement \left({\limsup_{n \to \infty} \ E_n}\right)$ $=$ $\displaystyle \complement \left({\bigcap_{n \mathop = 0}^\infty \bigcup_{i \mathop = n}^\infty E_n}\right)$ Definition of limit superior $\displaystyle$ $=$ $\displaystyle \bigcup_{n \mathop = 0}^\infty \complement \left({\bigcup_{i \mathop = n}^\infty E_n}\right)$ De Morgan's Laws $\displaystyle$ $=$ $\displaystyle \bigcup_{n \mathop = 0}^\infty \bigcap_{i \mathop = n}^\infty \complement \left({E_n}\right)$ De Morgan's Laws $\displaystyle$ $=$ $\displaystyle \liminf_{n \to \infty} \ \complement \left({E_n}\right)$ Definition of limit inferior

$\blacksquare$