Zero dimensional components of an intersection

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)

• I am indeed assuming that $n$ is positive. If there are no zero dimensional components in the intersection then the claim is automatically true. What I am thinking is that if the intersection product is positive, and the intersection is not proper, then there could not be too many isolated points in the intersection; indeed if we move the varieties a bit the number of isolated points can only grow.
– Xzz
Nov 8, 2016 at 7:05

It's not true scheme-theoretically, at least. Let $X=\mathbb P^4$ with coordinates $w,x,y,z,\Omega$, let $A$ be the $xy\Omega$-plane, and $B$ the union of the $wx\Omega$- and $yz\Omega$-planes. Then $A\cap B$ is a fat point of length $3$, whereas the intersection number is $2$.