Support of a section of a coherent module on a complex manifold

Question

Let $X$ be a complex manifold of dimension $n$, and $M$ be a coherent $\mathcal{H}_X$-modules. Let $s \in M$ be a section, and $h \in \mathcal{H}_X$ be a holomorphic function on $X$. I want to prove that if $s \vert_{X_h} = 0$, then there exists $N$ such that $h^N s = 0$, where $X_h = \{ x \in X : h(x) \neq 0\}$. I think I need some sort of Hilbert's Nullstellensatz but I don't know how to proceed. Any hint?