I am currently reading into polytope theory, and stumbled upon the following proposition, which I do not really understand:
Let $$ P = \{ x \in \mathbb{R}^{k+l} \mid Ax\leq b\} $$ be a rational polyhedron, i.e., $A$ and $b$ are rational. Then $$\mbox{conv} (P \cap (\mathbb{Z}^k \times \mathbb{R}^l))$$ is also a rational polyhedron.
I do not understand what implications it has that the first k entries of the vectors are whole numbers. If they weren't, I'd think that the convex hull is a polyhedron nevertheless. So what does the first k entries being whole really mean? What does it change in the proof?