Abstract: For multi-dimensional integer-valued sequence, a new proof of the upper-bound and lower-bound estimations on the product of all its components is given in this paper. Those estimations are very important to characterize the Kondratiev space of random test function, in which space it is convenient to study random distribution, random partial differential equation and other problems.

Key words: Multidimensional integer value sequence, Product inequality, The upper-bound and lower-bound estimates