elementary set theory - Disjoint union with limsup

For any sets $A_n,n\in\mathbb{N}$ consider
$$A^+:=\limsup_{n\to\infty}A_n:=\bigcap_{n=1}^{\infty}\bigcup_{k=n}^{\infty}A_k,~~~~~E_m:=\bigcup_{n\geq m}A_n.$$
Show that the sets $E_m, m\geq 1$ can be written as a disjoint union
$$E_m=A^+\uplus\biguplus_{n\geq m}(E_n\setminus E_{n+1}).$$

I do not have a working idea. I started with writing $E_m$ as a disjoint union, i.e.
$$E_m=A_m\uplus\biguplus_{i=m+1}^{\infty}A_i\setminus\bigcup_{j=m}^{i-1}A_j=A_m\uplus\biguplus_{i=m+1}^{\infty}\bigcap_{j=m}^{i-1}A_i\setminus A_j$$

and additionally I see that
$$A^+=\bigcap_{n=1}^{\infty}E_n.$$
But I do not know if this is helpful...

Would be great to get a help resp. answer.

With kind regards,

math12

Answer

Hint: For $x \in E_m$ consider
$$\sup_{A_k \ni x} k.$$
Show that if this supremum is infinite, then $x \in A^{+}$. Otherwise, denote this supremum by $n$ and show that $x \in E_n \setminus E_{n+1}$.