Complement of the Cantor Set

I am looking at the complement of the cantor set in $[0,1]$ as the union of open intervals of decreasing length $\displaystyle\bigcup_{i=1}^{\infty}A_i$ where $A_1 = (\frac{1}{3},\frac{2}{3}), A_2 = (\frac{1}{9},\frac{2}{9})\cup(\frac{7}{9},\frac{8}{9})$ and so on.

I am trying to prove that $\overline{\displaystyle\bigcup_{i=1}^{\infty}A_i} = \displaystyle\bigcup_{i=1}^{\infty}\overline{A_i}$.

I know that this is not true in general and is probably not true in this case, but can someone give a proof or a counterexample

Thank you very much

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• The complement of the Cantor set is dense in $[0,1]$.
• The closure of each individual $A_n$ only has finitely many extra points.
Jonas, as I recall if $X$ is dense in $Y$ then $Y\subseteq\operatorname{cl}(X)$, and if the axiom of countable choice for finite sets is assumed then a countable union of finite sets is countable. I also remember that uncountable-countable = uncountable. Which point am I missing here? – Asaf Karagila Sep 9 '11 at 15:58
Perhaps an explicit number would be good to work on. Show that $1/4$ belongs to the closure of the union, but not to the union of the closures. – GEdgar Sep 9 '11 at 16:29