Could someone help me with this problem
Let $A$ be an $m \times n$-matrix and let $C = \{Ax : x \in \mathbb{R}^n , x ≥ 0\} \subseteq R^m$.
Prove that if there exists $a \in \mathbb{R}^m$ such that $a · c \leq \alpha$ for all $c \in C$, then $a · c \leq 0$ for all $c \in C$.
Maybe I don't get your notations right, but this is what I understand : $x=(x_1,\dots,x_n)\ge0$ means $(\forall i)\,x_i\ge0$, and $a\cdot c$ is scalar product of vectors $a$ and $c$.
If this is correct, then if there exists $c\in C$ such that $a\cdot c>0$, then $\lambda c\in C$ for all $\lambda>0$ (because $c=Ax$ for a certain $x\ge0$, and as $\lambda x\ge0$, $\lambda c=A(\lambda c)$).
But then, $a\cdot(\lambda c)=\lambda (a\cdot c)$ can take any positive value, therefore can exceed $\alpha$, which is supposed false. So the hypothesis doesn't hold, and there is no $c$ such that $a\cdot c>0$.