Corollary

A point $x$ lies in the closure of a subgroup $G$ of $\mathbf{R}^n$ if and only if $(u|x)$ is an integer for all $u \in \mathbf{R}^n$ such that $(u|y)$ is an integer for all $y \in G$.

Let us apply this characterization of points lying in the closure of a subgroup $G$ to the case of the subgroup $G$ generated by the $n$ vectors $e_j$ of the canonical basis $(1 \leq j \leq n)$ and by an arbitrary number $m$ of points $a_i$ $(1 \leq i \leq m)$ of $\mathbf{R}^n$. To say that $(u|e_j)$ is an integer for $1 \leq j \leq n$ means that the $n$ coordinates of $u$ are integers; therefore: