Exercise 9, § 1

Let $G$ be a subgroup of $\mathbf{R}^n$, not necessarily closed, of rank $p$, and let $V$ be the largest vector subspace contained in $\overline{G}$. If $V$ has dimension $q \leq p$, show that $G$ is the direct sum of $V \cap G$ and a discrete subgroup of rank $p - q$ contained in a vector subspace complementary to $V$ [note that, for each $x \in \overline{G}$, $(x + V) \cap G$ is dense in the linear variety $x + V$].