A simple proof of a reverse Minkowski theorem for integral lattices

We prove that for any integral lattice $\mathcal{L}\subset {\mathbb{R}}^n$ (that is, a lattice $\mathcal{L}$ such that the inner product $⟨{\mathbf{y}}_1,{\mathbf{y}}_2⟩$ is an integer for all ${\mathbf{y}}_1,{\mathbf{y}}_2\in \mathcal{L}$) and any positive integer $k$, $$\mathrm{\mid }\{\mathbf{y}\in \mathcal{L}:\mathrm{\parallel }\mathbf{y}{\mathrm{\parallel }}^2=k\}\mathrm{\mid }\le 2\left(\genfrac{}{}{0px}{}{n+2k-2}{2k-1}\right),$$ giving a nearly tight reverse Minkowski theorem for integral lattices.