Only finite many multiplicities of Cartier divisor are non zero I am trying to understand the multiplicities of Cartier divisors in Liu’s book on the relevant chapter. Actually I cannot understand the following conclusion of this remark.
Now let $ $ be an open everywhere dense subset of $$ such that $_{}=0$.
Then: Any $∈$ of codimension 1 such that $mult()≠0$ is a generic point of $−. $
THEN: The above implies that in an open affine subset of $X$, there are only finite many codimension 1 points such that the corresponding multiplicity are non-zero. I’ve seen
this question but I really need to understand the full conclusion, namely the last statement.

Actually i found out. We restrict to an open affine $V$ and the closure of the intersection $U\cap V$ in $V$ is $V$. Hence we can apply the first claim, since the affine $V$ is noetherian and hence has finitely many irreducible components.
