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> Let $A$ be a $m\times n$ matrix and consider the cones $G\_0=\{d\in\mathbb R^n:Ad<0\}$ and $G'=\{d\in\mathbb R^n:Ad\le0\}$
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> Prove that $G'$ is a convex closed cone.
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Lets see that $G'=\overline{ G'}.$ Note that this contention $G'\subset\overline{ G'}$ is always true. Let's see the other contention.

Suppose $d\in\overline{G'}$ **and** $d\not\in G'. $ Thus For every open ball $B$ with $d\in B$ we have that $B\cap G'\neq\emptyset$ **and** $ Ad>0$

How can I reach the contradition? Is this a good way to prove it or is there a better way? I don't know

And how to prove convexity?

Please help me please thanks