sum of convex cone with apex at 0 and its closure

Question

Let $K\subseteq\mathbb R^n$ be an open convex cone with apex at 0, and let $\mathrm {cl}K$ be its closure, how to prove that $K+\mathrm {cl}K=K?$

Answer 1

I think the proposition is not true.

If this proposition would be true then for any convex cone $K$ (as $0\in K$) we would have $\mathrm {cl} K= \{0\}+\mathrm {cl}K\subset K+\mathrm {cl}K = K$ (the last equality is the proposition), so $\mathrm {cl} K\subset K$, which generally is not true.