Why is $\Delta u$ bounded, if $u\in C^2(\overline{\Omega})$ and $\Omega\subseteq\mathbb{R}^n$ is a bounded domain?

Question

Let $\Omega\subseteq\mathbb{R}^n$ be a bounded domain and $u\in C^2(\overline{\Omega})$. Why must $\Delta u$ be bounded?

Answer 1

The closure of a bounded subset of $\mathbb R^n$ is compact. A continuous function on a compact set is bounded.