Zero dimensional component of an intersection

Question

Let $X$ be a smooth projective variety and let $A,B$ be closed irreducible subvarieties of complementary dimension in $X$ and with intersection product equal to $n$. The varieties $A$ and $B$ may not intersect properly; the intersection is a union of components. Let $Z$ be the union of the zero dimensional components (counted with multiplicities). It seems likely that $n$ is an upper bound for the multiplicity of $Z$, but how to show it?