The $\varepsilon$-packing number of the unit cube $[0,1]^d$ with respect to the infinity norm is the biggest number of $\varepsilon$-strictly-separated points, i.e., the biggest cardinality of a set of points $E \subset [0,1]^d$ such that each two distinct $x,y\in E$ satisfy $\|x-y\|_\infty > \varepsilon$.
It seems geometrically intuitive to me that this is at most $\left( \bigl\lfloor \frac{1}{\varepsilon}\bigr\rfloor + 1 \right)^d$, where $\lfloor x \rfloor$ is the biggest integer lower or equal to $x$ (I am very naively taking a uniform grid of $(\varepsilon + \varepsilon')$-spaced points, for a small $\varepsilon'$).
Is this true? If so, how is it proven?
I can only found bounds like $(2/\varepsilon + 1)^d$ that upper bound the packing number with the related notion of covering number and proceed to find a covering.