I'm searching for a reference for the following simple fact about polytopes.
FACT:
If $\{x\;|\;Ax \leq b\}$ defines a polytope, and if $c\geq 0$ then $\{x\;|\;Ax\leq b+c\}$ also defines a polytope (i.e. a bounded polyhedron).
Although this fact is intuitively obvious, and not difficult to prove, I would like to justify it in at most one or two lines of text. So does this fact follows directly as a special case of some well known theorem?
Obs: It is clear that if $Ax\leq b$ is satisfied, then $Ax\leq b+c$ is satisfied. Therefore, the polytope defined by $Ax \leq b$ is included in the polyhedron defined by $Ax\leq b+c$. The question is about showing that $Ax\leq b+c$ defines a polytope, i.e., a bounded polyhedron.
Just observe that the polyhedra $P=\{x \in \mathbb{R}^n\ |\ Ax\leq b\}$ and $Q=\{x \in \mathbb{R}^n\ |\ Ax\leq b+c\}$ have the same recession cone $$rec(P)=rec(Q)=\{x \in \mathbb{R}^n\ |\ Ax\leq 0\}$$. By hypothesis $rec(P)=\{\emptyset\}$ because $P$ is a polythope. The thesis follows.