Let $\Lambda \subset \mathbb{R}^n$ be an Euclidean lattice with generator matrix $B$. Define the center density $\delta(\Lambda)$ in the usual way as $\delta(\Lambda) = \rho^n/|\det{B}|$, where $\rho$ is the packing radius of $\Lambda$ (i.e., $\rho = (1/2) \displaystyle \min_{0 \neq x \in \Lambda} \left\| x \right\|$). Let $\delta_n$ be the supremum of $\delta(\Lambda)$ over all lattices in $\mathbb{R}^n$. Mahler's Selection Theorem implies that there is a lattice whose density achieves $\delta_n$ (see e.g. [Lekkerkeker and Gruber, Geometry of Numbers], pp. 260).

My question is: what is the most elementary way of proving the fact that there exists a lattice achieving the best packing density, without using any Selection Theorem (is there any)?