Let $M$ be an $n \times m$-matrix of real numbers. Let $\nu \in \mathbb{R}^n$. Define the set $$S = \{ M^T \mu \ \ | \ \ \mu \in \mathbb{R}^m \ \ \text{and} \ \ \mu \cdot \nu \le 0 \ \ \} $$
How does one show that this set is bounded (having already showed it is closed and convex), if we are given that for any $\mu$ with $\mu \cdot v \le 0$, it holds that $M^T \mu$ cannot be positive (where by positive I mean nonnegative with at least one strictly positive element)?
In fact, the proof I want is the one where we assume that the set is unbounded, and then show that there exists a $\mu$ with the property that $\mu \in S$ and $M^T\mu > 0$ (one element strictly positive, all others nonnegative).