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Edit: the following argument is true when the polyhedron $P$ is the intersection of finitely many linear constraints: $a_1 x_1 + \ldots + a_n x_n \leq b$.

1. The coefficient of linear constraints must be rational. Otherwise, your lemma will not be true (e.g. consider $\sqrt{2}x_1-\sqrt{3}x_2=0$).
2. Since your polyhedron $P \subset \mathbb{R}^n$ is unbounded 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 $\lambda$.

Edit: the following argument is true when the polyhedron $P$ is the intersection of finitely many linear constraints: $a_1 x_1 + \ldots + a_n x_n \leq b$.

1. The coefficient of linear constraints must be rational. Otherwise, your lemma will not be true (e.g. consider $\sqrt{2}x_1-\sqrt{3}x_2=0$).
2. Since your polyhedron $P \subset \mathbb{R}^n$ is unbounded 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 infinitely many values of $\lambda$.

Edit: the following argument is true when the polytope $P$ is the intersection of finite many linear constraints: $a_1 x_1 + \ldots + a_n x_n \leq b$.

1. The cofficient of linear constraints must be rational. Otherwise, your lemma will not be true.
2. 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 $\lambda$.

Edit: the following argument is true when the polyhedron $P$ is the intersection of finitely many linear constraints: $a_1 x_1 + \ldots + a_n x_n \leq b$.

1. The coefficient of linear constraints must be rational. Otherwise, your lemma will not be true (e.g. consider $\sqrt{2}x_1-\sqrt{3}x_2=0$).
2. Since your polyhedron $P \subset \mathbb{R}^n$ is unbounded 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 $\lambda$.

1. The cofficient of linear constraints must be rational. Otherwise, your lemma will not be true.
2. 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 $\lambda$.

Edit: the following argument is true when the polytope $P$ is the intersection of finite many linear constraints: $a_1 x_1 + \ldots + a_n x_n \leq b$.

1. The cofficient of linear constraints must be rational. Otherwise, your lemma will not be true.
2. 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 $\lambda$.