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