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:
- $\ds x \in \limsup_{n \mathop \to \infty} E_n$
$\blacksquare$
Sources
- 1951: J.C. Burkill: The Lebesgue Integral ... (previous) ... (next): Chapter $\text {I}$: Sets of Points: $1 \cdot 1$. The algebra of sets