Finding the $\limsup$ of a sequence of sets Is my proof if this equivalence correct?

$$\limsup_{n\to\infty} A_n=\bigcap_{i=1}^\infty \bigcup_{j=i}^\infty A_i=\{\text{ elements that belong to infinitely mane } A_i \text{'s }\}$$

Pf.
Let $B_i=\bigcup_{j\ge i} A_j$. Let $x \in \limsup$, then:
($\rightarrow$) $x\in \limsup \implies x \in B_i$ for all $i\ge 1 \implies x \in $ infinitely many $A_i$'s.
($\leftarrow$) Suppose $x \in$ infinitely many $A_i$'s. Then $x \in A_i \implies x \in B_i$ for all $i\ge 1 \implies x \in \bigcap_{j\ge 1} B_j=\limsup$.

I don't think your proof is correct. Just note that
$$
x \in \limsup_{n \to \infty} A_n \quad \iff \quad \forall i \geq 1, \exists j \geq i : x \in A_j \quad \iff \quad x \text{ belongs to infinitely many $A_i$'s}
$$
