TITLE: How to proof the intersection of two closed convex cones is a closed convex cone?
QUESTION [1 upvotes]: Let $C$ and $D$ be closed convex cones in $R^n$. I am trying to show that $C\cap D$ is a closed cone. 
I started with 
Take any point $x_1 \in C$  and $x_2 \in D$ with $\theta_1,\theta_2\geq0$ and $\theta_1+\theta_2=1$. 
After that I do not think I am doing it right. I think I am suppose to let $x=\theta_1x_1+\theta_2x_2$ and show that $x\in C\cap D$? Am I on the right track?

REPLY [2 votes]: Start with $x_1,x_2 \in C \cap D.$ Now, this means that $x_1,x_2 \in C$. Because $C$ is a convex cone, we know that for any $\theta_1,\theta_2>0$ that $$\theta_1x_1+\theta_2x_2 \in C.$$ Similarly, $$\theta_1x_1+\theta_2x_2 \in D.$$ Hence, $$\theta_1x_1+\theta_2x_2 \in C\cap D$$ hence $C \cap D$ is convex. Now, just show it must also be a cone. Simply show that $\alpha > 0, x_1 \in C \cap D \Rightarrow\alpha x_1 \in C \cap D.$ This is straightforward.
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