- The cofficient of linear constraints must be rational. otherwise, your lemma will not be true.
- Since your polytope $P \subset \mathbb{R}^n$ is unbouded and defined by rational linear constraints, it has a rational recession direction, i.e. there exists $ 0 \neq d \in \mathbb{Q}^n$ such that $x+\lambda d \in P $, for all $x \in P$ and $ \lambda \geq 0 $. Therefore, if $x_0\in P \cap \mathbb{Z}^n $ then $x_0 +\lambda d \in P \cap \mathbb{Z}^n $ for infinite many values of $\lambda$.
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