How to proof: Closure of set is equal to intersection of sets

Question

I know that $$\overline{A} = \bigcap_{r>0}[A+S(0,r)].$$ I need to proof that $$\overline{A} = \bigcap^{\infty}_{n=1}[A+S(0,2^{-n})].$$ I already have shown $$\bigcap^{\infty}_{n=1}[A+S(0,2^{-n})] \subset \overline{A}.$$ How to show that $$\overline{A} \subset \bigcap^{\infty}_{n=1}[A+S(0,2^{-n})]?$$