A confusion regarding the concept of principal divisor

Question

Let $M$ be a complex manifold. Then a Weil-divisor $D$ on $M$ is given by the formal sum $\sum_{V\in\nu}{\eta}_{V}.V$, where $\nu$ is a locally finite collection of irreducible analytic hypersurfaces in $M$ and ${\eta}_{V}\in\mathbb{Z}$ for each $V\in\nu$. Now, consider a meromorphic function $f$ on $M$. For each irreducible analytic hypersurface $V\subset M$ one can define an integer $ord_{V}(f)$ called the order of $f$ along $V$. Lastly, one defines a Weil -divisor corresponds to $f$ as $(f)=\sum_{V} ord_{V}(f).V$, where $V$ runs over all irreducible analytic hypersurfaces in $M$ [See Griffiths-Harris "Principles of Algebraic Geometry", page 130-131 for details]. Clearly, this collection is not locally finite. So, to conclude that $(f)$ is a Weil-divisor one needs to show $\{V:ord_{V}(f)\neq 0\}$ is locally finite. But I am quite unsure how to establish this. Any help is appreciated.