Question: What is the closure of the set $\{(-1/2)^n \, : \, n \in \mathbb{N} \} \cup \{0\}$ as a subset of $\mathbb{R}$? Is it compact? Is it connected?
My answer: The closure is the set itself. It is compact as the set is clearly bounded in $\mathbb{R}$ and closed as the closure is itself. No it's not connected in $\mathbb{R}$ as any two open intervals in $\mathbb{R}$ where the union is the subset is a counterexample.
Just checking I'm right?