Construct a compact set of real numbers whose limit points form a countable set.

I searched and found out that the below is a compact set of real numbers whose limit points form a countable set.I know the set in real number is compact if and only if it is bounded and closed.It's obvious it is bounded since $\,d(1/4, q) < 1\,$ for all $\,q \in E.$

However, I'm not sure how this is closed.

Is there any simpler set that satisfies the above condition?

Thank you!

$$E = \left\{\frac 1{2^m}\left(1 - \frac 1n\right) \mid m,n \in \mathbb N\right\}.$$