Let $P=\{(x,y)\in \mathbb{R}^{n+m} : Ax\ge b \}$ be a polyhedron in $\mathbb {R}^{n+m}$, with $x\in \mathbb {R}^n$ and $y\in \mathbb {R}^m$.
How can I show that the projection $\pi _X(P) = \{x\in \mathbb {R}^n : (x,y)\in P \ \mathrm{for \ some} \ y\in \mathbb{R}^m\}$ is also a polyhedron in $\mathbb{R^n}$?
A polyhedron must be a set of points that satisfy a finite number of linear inequalities, or equivalently, a finite intersection of half-spaces.