Note on a theorem of Davenport

Let $\Lambda$ be a $n$-dimensional lattice, and $c_1,\ldots,c_{n-1}$ be any $n-1$ vectors in $n$-dimensional real Euclidean space. We show that there exists a basis $\alpha_1,\ldots,\alpha_n$ of $\mathsf\Lambda$ such that
$$ |\alpha_i-Nc_i|=O(\log^2N),\leqslant (1\leqslant i\leqslant n-1) $$
holds for any real number $N\ge 2$, where the constant implied by the $O$ symbol depends only on $\Lambda$ and $c_1,\ldots,c_{n-1}$.