Let $X$ be a smooth projective algebraic variety over an algebraically closed field and let $A,B$ be closed irreducible subvarieties of complementary codimension in $X$. Let $n$ denote their intersection product, and let $Z$ denote the union of the zero dimensional components of the intersection (with multiplicities). It seems likely that $n$ is then an upper bound for the size of $Z$, but how to show this?

(I asked this question on Math.SE but did not get any answers)