Limit Superior includes Limit Inferior

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Theorem

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


Then:

$\ds \liminf_{n \mathop \to \infty} E_n \subseteq \limsup_{n \mathop \to \infty} E_n$


Proof

Select any $\ds x \in \liminf_{n \mathop \to \infty} E_n$.

Then by definition 2 of Limit Inferior of Sequence of Sets, $x$ belongs to $E_i$ for all but finitely many values of $i$.

Therefore $x$ belongs to $E_i$ for infinitely many values of $i$.

So by the definition 2 of Limit Superior of Sequence of Sets:

$\ds x \in \limsup_{n \mathop \to \infty} E_n$

$\blacksquare$


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