The Cartesian product of an infinite number of non-empty sets can't be non-empty. It may sometimes be ill-defined (or even undefined) for an uncountable number of sets, but it can't be empty. The proof is simple: if every set has one element, then the Cartesian product has one element.
Increase the number of elements in any set, the number in the Cartesian product increases as well. If the number is already infinity, the cardinality may remain unchanged, but it can't decrease.