lim sup & lim inf of Sets

The concept of lim sup and lim inf can be applied to sets too. Here is a nice characterisation of lim sup and lim inf of sets:

For a sequence of sets \{E_k\}, \limsup E_k consists of those points that belong to infinitely many E_k, and \liminf E_k consists of those points that belong to all E_k from some k on (i.e. belong to all but finitely many E_k).

Proof:
Note that
\begin{aligned}  x\in\limsup E_k&\iff x\in\bigcup_{k=j}^\infty E_k\ \text{for all}\ j\in\mathbb{N}\\  &\iff\text{For all}\ j\in\mathbb{N}, \text{there exists}\ i\geq j\ \text{such that}\ x\in E_i\\  &\iff x\ \text{belongs to infinitely many}\ E_k.  \end{aligned}
\begin{aligned}  x\in\liminf E_k&\iff x\in\bigcap_{k=j}^\infty E_k\ \text{for some}\ j\in\mathbb{N}\\  &\iff x\in E_k\ \text{for all}\ k\geq j.  \end{aligned}

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