If $K \subseteq \mathbb{R}^n$ is compact then $\overline{K^{\circ}}$ is compact

Question

I came accross the following statement which I've been trying to prove: $K \subseteq \mathbb{R}^n$ is compact then $\overline{K^{\circ}}$ is compact as well.

Since we are in $\mathbb{R}^n$ I just need to show that $\overline{K^{\circ}}$ is closed and bounded. It is obviously closed and since $K^{\circ} \subseteq K$ and $K$ is bounded we know that $K^{\circ}$ is bounded. But why can we say that $\overline{K^{\circ}}$ is bounded as well? This last part is not so clear to me.

Answer 1

$\overline{K^{\circ}}\subset\overline{K} = K$ bounded.