The Relevant Domain of the Hilbert Function of a Finite Multiprojective Scheme

Let X be a zero-dimensional scheme contained in a multiprojective space. Let $s_i$ be the length of the projection of X onto the i-th component of the multiprojective space. A result of Van Tuyl states that the Hilbert function of X, in the case when X is reduced, is completely determined by its restriction to the product of the intervals $[0, s_i - 1]$. We prove that the same is also true for non-reduced schemes X.