Given that $C$ and $D$ are closed convex cones. I need to prove that for dual cones $C^*$ and $D^*$ that $C^*+D^*$ is a closed convex cone. I know that because $C$ and $D$ are closed convex cones, then $C^*$ and $D^*$ are also closed convex cones because of dual cone properties.
Because addition is closed then $C^*+D^*$ is closed.
To show its a cone. I take an $x \in C^*+D^*$ and let $\alpha$ be a positive scalar. Then I let $x=x_1+x_2$ for some $x_1 \in C^*$ and $x_2 \in D^*$ since $C^*$ and $D^*$ are cones $\alpha x \in C^*$ and $\alpha x \in D^*$. Since $\alpha x= \alpha(x_1+x_2)=\alpha x_1+ \alpha x_2$, $\alpha x \in C^*+D^*$.
I am just having problems showing it is convex? I was thinking about starting it similar to showing it was a cone but I get stuck?