The product of linear nonhomogeneous forms

We show that for an arbitrary unimodular lattice $\Lambda$ of dimension $n$ and an arbitrary point $C=(c_1,c_2,\dots,c_n)\in R^n$ a point $Y=(y_1,y_2,\dots,y_n)\in\Lambda$ can be found and also a number h, satisfying the condition $1\le h\le2^{-n/2}\theta^{-1}+1$ ($0<\theta\le2^{-n/2}$), such that the inequality
$$ \prod_{i=1}^n|y_i+hc_i|<\theta $$
will be satisfied.