I want to show the following property:
If $C\in\Bbb R^n$ is a convex cone, and $D$ is the set of all directions in $C$, then $$\forall x\in D^c \implies x \in C^c$$
My intuition tells me that I may want to prove this by contradiction. So here goes:
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Proof $\forall x\in D^c \implies x \in C^c$.
Suppose not. I.e. suppose $\forall x\in D^c \implies x \notin C^c$ or in other words $\forall x\in D^c \implies x \in C$. What this says is that for any vector not in any direction of $C$, it is in $C$. But since all vectors in the direction of $C$ were in $D$, and this says it is not, we arrive at a contradiction! Since this is false, the opposite must be true, ie that $\forall x\in D^c \implies x \in C^c$ is true. $\ Q.E.D.$
My fear with this is my "English interpretation" of the statement. I have never proved a theorem or property this informally (if this is even informal). This problem seems so trivial that it is hard to construct something concrete. Would providing an example be a better option, i.e., a graphical example?
The "suppose not" is: there is some $x \in D^c$ such that $x \in C$. To contradict the statement you're trying to prove, you don't need this for all $x \in D^c$, just for one.