For each natural number $n$, we may define the (optimal) sphere packing density in $\Bbb R^n$ to be the number $$ D_n = \limsup_{r\to\infty} D_n(r), $$ where $$\begin{align} D_n(r) = \sup \Big\{ \frac{k\omega_n}{(2r)^n} :& \text{there exist $a_1,\dots,a_k$ such that $B(a_j)\subset [-r,r]^n$} \\ &\ \ \text{and $B(a_j)$'s are pairwise disjoint} \Big\}. \end{align}$$
Of course, $B(a)$ denotes the open unit ball centered at $a$, and $\omega_n$ is the $n$-volume of the unit ball in $\Bbb R^n$. I have no idea if this is how the density is normally defined or not, but it should be equivalent to the standard definition nonetheless. My $D_n(r)$ is the maximal density of packing unit balls into an $n$-cube of side length $2r$, and $D_n$ is the limit as the side length of this cube goes to infinity.
Is there a relatively simple argument to show that $D_n <1$ for all $n\ge 2$?
I know that calculating $D_n$ precisely is a hard task (except for some very special values of $n$), for example the Kepler conjecture, which says that $D_3 = \frac{\pi}{3\sqrt{2}} \simeq 0.7405$, has only been proved in the last ten year, after it had been an open problem for several centuries. I don't need to know that value of $D_n$, I just want to know if it's always bounded away from $1$. While it seems intuitive that we should have $D_n < 1$ always (except for the trivial case $D_1$, where $1$-dimensional balls are just lines), I can't think of a quick proof of it off the top of my head.
It doesn't seem too hard to show that $D_n(r) < 1$ for each fixed $r$ since there is definitely some wasted space between the unit balls, however, I can't think of any a priori reason why there wouldn't be a better and better possible configuration as $r$ grows large that may make $D_n(r)\to 1$ as $r\to\infty$. I am thinking that adapting the proof of Besicovitch covering theorem could do the trick, but I haven't tried writing down the details yet.
I know essentially nothing regarding this topic, so perhaps the answer to my question is already well-known or even a standard result in sphere packing. If so, I would be delighted if anyone could point me to a good resource that discusses this kind of question.