Closure Operator and Set Operations In Engelking, General Topology stand the following exercise:
Show that for any sequence $A_1, A_2, \ldots$ of subsets of a topological space we have
$$
 \overline{\bigcup_{i=1}^{\infty} A_i} = \bigcup_{i=1}^{\infty} \overline{A_i} \cup \bigcap_{i=1}^{\infty} \overline{\bigcup_{j=0}^{\infty} A_{i+j}}.
$$
I have no idea how to show this, I can't even show that the right side yields a closed set, because when I build the complement I got infinite intersections, which aren't necessarily open. Any hints how to solve this exersice?

The inclusion (RHS) $\subseteq$ (LHS) is trivial, so I will give you a hint for the reverse inclusion:
Assume that $x$ is not in the right hand side. Then there is a $j\ge1$ such that $x\notin\overline{\bigcup_{i=j}^\infty A_j}$, i.e. there is a neighborhood $V$ disjoint from this union starting at $j$. But for each $i\ge0$ there is also a neighborhoods $U_i$ disjoint from $A_i$. Now intersect $W:=V\cap U_1\cap...\cap U_j$. Does this $W$ intersect the union of all $A_i$?
