Is it true that $\partial\partial E=\partial E$

Let $E\subset\mathbb{R}^n$ be an arbitrary subset. Is it true that $$\partial\partial E=\partial E$$ The inclusion $\partial\partial E\subset \partial E$ is pretty clear to me since $\partial E$ is closed, however I fail to see the other inclusion. How to prove it?

• What about the rationals in the reals? – Michael Burr Apr 7 '16 at 11:08

Consider $\mathbb Q\subset \mathbb R$ then $\partial \mathbb Q=\mathbb R$ but $\partial\partial \mathbb Q=\emptyset$ .
However you cold prove that $\partial \partial \partial A= \partial\partial A$ for any set A.
$\partial\partial E=\partial E$ iff $\partial E$ has no interior points