  In this paper we study sequences of lattices which are, up to similarity,
projections of $\mathbb{Z}^{n+1}$ onto a hyperplane $\bm{v}^{\perp}$, with
$\bm{v} \in \mathbb{Z}^{n+1}$ and converge to a target lattice $\Lambda$ which
is equivalent to an integer lattice. We show a sufficient condition to
construct sequences converging at rate $O(1/ |\bm{v}|^{2/n})$ and exhibit
explicit constructions for some important families of lattices.
