Inclusions regarding the limsup and liminf of sets: $ \liminf E_n \subset \limsup E_n $

Question

Let $\{ E_n \}_{n \in \mathbb{N} }$ be a sequence of sets in some ambient set $\Omega $. I want to show that

$$ \liminf E_n \subset \limsup E_n $$

My attempt: IF $x \in \liminf E_n = \bigcup_{k=1}^{\infty} \bigcap_{n \geq k} E_n $, then there is some $k_0 \in \mathbb{N}$ so that $x \in \bigcap_{n \geq k_0} E_n $. How can I show that $x \in \bigcap_{k =1}^{\infty} \bigcup_{n \geq k} E_n = \limsup E_n $ ??

Answer 1

Assume $x \in \cap_{n \geq k_0} E_n$. Then

For $1 \leq k \leq k_0$, we have $x \in E_{k_0}$ and hence $x \in \cup_{n \geq k} E_n$.

For $k > k_0$, we have $x \in E_k $ and hence $x \in \cup_{n \geq k} E_n$.

Therefore $x \in \cap_{k \geq 1} \cup_{n \geq k} E_n$.