Theorem: Let $A$ and $C$ be two matrices. The system of linear inequalities $Ax<0$ and $Cx \leq 0$ has a solution iff the following equation in $\lambda$ and $\mu$ does not have a solution$$A^T \lambda + C^T \mu =0$$ where $\lambda \geq 0$, $\lambda \neq 0$, $\mu \geq 0$.
What is the meaning of the $\lambda \geq 0$ and $\lambda \neq 0$ part? why not write $\lambda > 0$?
I am confused! Is there an elementary proof of this theorem?