If $C$ is closed, is $\mathbb{R}^+C=\{kc:k\geq0,c\in C\}$ also closed? Here for simplicity, assume $C\subset\mathbb{E}$ which is a finite dimensional Euclidean space.
It is clear that $kC$ is closed. $\mathbb{R}^+C=\cup_{k\geq 0} kC$ and the union of infinite closed sets may not be closed. Is there a counterexample or a proof?