$\liminf E_k \subset \limsup E_k $

$\liminf E_k \subset \limsup E_k $

I am wondering if my proof is correct? Thank you for whoever willing to
take a look at it for me.
Proof $\liminf E_k \subset \limsup E_k $
Consider $x \in \liminf E_k$, then $x \in E_1 \cup E_2 \cup \cdots$, and
$x \in E_2 \cup E_3 \cup \cdots$ and so on, that is to say, $x \in
\bigcap_{m=1}^\infty E_m \cup E_{m+1} \cup \dots$. Since $x \in E_r \cup
E_{r+1} \cup \dots$, we showed $x \in \limsup E_k$.