[proofplan] The assertion is local on $X$. Near a point of $X$, the analytic set $V$ is the common zero set of finitely many holomorphic functions, so these functions define a coherent ideal sheaf $\mathcal{J} \subset \mathcal{O}_X$. Rückert's analytic Nullstellensatz identifies the ideal sheaf of functions vanishing on $V$ with the analytic radical of $\mathcal{J}$. The Cartan-Oka local radical theorem, proved by Weierstrass preparation, Weierstrass division, and induction through local parameterization, says that the radical of a coherent analytic ideal is coherent; applying it locally gives the result. [/proofplan] [step:Reduce the assertion to a coordinate neighbourhood around one point] Fix a point $p \in X$. Since coherence of an $\mathcal{O}_X$-module is local, it is enough to find an open neighbourhood $U_p \subset X$ of $p$ such that $\mathcal{I}(V)|_{U_p}$ is coherent as an $\mathcal{O}_{U_p}$-module. If $p \notin V$, then $X \setminus V$ is open because $V$ is closed. Choosing $U_p \subset X \setminus V$ gives $\mathcal{I}(V)|_{U_p} = \mathcal{O}_{U_p}$, which is coherent by the Oka Coherence Theorem for the Structure Sheaf. If $p$ belongs to the interior of $V$ in $X$, choose $U_p \subset V$; then $\mathcal{I}(V)|_{U_p} = 0$, which is coherent. It remains to consider a point $p \in V$ such that $V$ is not equal to a neighbourhood of $p$. Choose a connected coordinate neighbourhood $U \subset X$ of $p$ and a biholomorphic coordinate map \begin{align*} \varphi: U &\to \Omega \subset \mathbb{C}^n \\ x &\mapsto (z_1(x), \ldots, z_n(x)), \end{align*} where $\Omega$ is open. Coherence is invariant under restriction to an open set and under biholomorphic pullback, so it suffices to prove that $\mathcal{I}(V)|_U$ is coherent. [/step] [step:Build a coherent ideal from local defining equations for the analytic set] Because $V$ is a closed analytic set, after shrinking $U$ if necessary there exist an integer $m \geq 1$ and holomorphic functions \begin{align*} g_j: U &\to \mathbb{C} \\ x &\mapsto g_j(x) \end{align*} for $1 \leq j \leq m$ such that \begin{align*} V \cap U = \{x \in U : g_1(x) = \cdots = g_m(x) = 0\}. \end{align*} Define the holomorphic map \begin{align*} G: U &\to \mathbb{C}^m \\ x &\mapsto (g_1(x), \ldots, g_m(x)). \end{align*} Then $V \cap U = G^{-1}(\{0\})$. Define a morphism of $\mathcal{O}_U$-modules \begin{align*} \alpha: \mathcal{O}_U^m &\to \mathcal{O}_U \\ (a_1, \ldots, a_m) &\mapsto \sum_{j=1}^m a_j g_j. \end{align*} Let $\mathcal{J} := \operatorname{im}(\alpha) \subset \mathcal{O}_U$ be the ideal sheaf generated by $g_1, \ldots, g_m$. The sheaf $\mathcal{O}_U$ is coherent by the Oka Coherence Theorem for the Structure Sheaf, hence the finite free sheaf $\mathcal{O}_U^m$ is coherent. Since images of morphisms of coherent analytic sheaves are coherent, $\mathcal{J}$ is a coherent ideal sheaf. [guided] The local equations for $V$ give an ideal sheaf, but that ideal need not equal the full ideal sheaf of all holomorphic functions vanishing on $V$. We first isolate this smaller ideal. Since $V$ is analytic, we may shrink the coordinate neighbourhood $U$ around $p$ and choose finitely many holomorphic functions \begin{align*} g_j: U &\to \mathbb{C} \\ x &\mapsto g_j(x) \end{align*} for $1 \leq j \leq m$ such that \begin{align*} V \cap U = \{x \in U : g_1(x) = \cdots = g_m(x) = 0\}. \end{align*} Equivalently, if \begin{align*} G: U &\to \mathbb{C}^m \\ x &\mapsto (g_1(x), \ldots, g_m(x)), \end{align*} then $V \cap U = G^{-1}(\{0\})$. The functions $g_1, \ldots, g_m$ define the sheaf morphism \begin{align*} \alpha: \mathcal{O}_U^m &\to \mathcal{O}_U \\ (a_1, \ldots, a_m) &\mapsto \sum_{j=1}^m a_j g_j. \end{align*} Its image $\mathcal{J} := \operatorname{im}(\alpha)$ is the sheaf of finite $\mathcal{O}_U$-linear combinations of the chosen defining equations. The Oka Coherence Theorem for the Structure Sheaf gives coherence of $\mathcal{O}_U$, and finite direct sums of coherent sheaves are coherent, so $\mathcal{O}_U^m$ is coherent. The closure property for coherent analytic sheaves under images of morphisms then gives that $\mathcal{J}$ is coherent. This is the coherent ideal whose radical will be identified with $\mathcal{I}(V)|_U$. [/guided] [/step] [step:Identify the vanishing ideal with the analytic radical of the defining ideal] For each point $a \in U$, let $\mathcal{O}_{U,a}$ denote the local ring of germs of holomorphic functions at $a$, and let $\mathcal{J}_a \subset \mathcal{O}_{U,a}$ denote the stalk of $\mathcal{J}$ at $a$. Define the radical ideal \begin{align*} \sqrt{\mathcal{J}_a} = \{f_a \in \mathcal{O}_{U,a} : \text{there exists } N \in \mathbb{N}_{\geq 1} \text{ such that } f_a^N \in \mathcal{J}_a\}, \end{align*} where $\mathbb{N}_{\geq 1}$ denotes the positive integers. The hypotheses of Rückert's Analytic Nullstellensatz are satisfied: $\mathcal{O}_{U,a}$ is a local ring of holomorphic germs, and $\mathcal{J}_a$ is generated by the finitely many germs $(g_1)_a, \ldots, (g_m)_a$. Since the common zero germ of $\mathcal{J}_a$ is the germ of $V \cap U$ at $a$, Rückert's theorem gives \begin{align*} \mathcal{I}(V)_a = \sqrt{\mathcal{J}_a}. \end{align*} [guided] The sheaf $\mathcal{J}$ contains functions obtained from the displayed equations $g_1, \ldots, g_m$. The ideal sheaf $\mathcal{I}(V)$ may contain more functions: a holomorphic function can vanish on the common zero set of the $g_j$ without being an $\mathcal{O}_U$-linear combination of the $g_j$. The correct relation is that $\mathcal{I}(V)$ is the radical of $\mathcal{J}$. Fix $a \in U$. Let $\mathcal{O}_{U,a}$ be the local ring of holomorphic germs at $a$, and let $\mathcal{J}_a$ be the ideal in $\mathcal{O}_{U,a}$ generated by the germs $(g_1)_a, \ldots, (g_m)_a$. Define \begin{align*} \sqrt{\mathcal{J}_a} = \{f_a \in \mathcal{O}_{U,a} : \text{there exists } N \in \mathbb{N}_{\geq 1} \text{ such that } f_a^N \in \mathcal{J}_a\}. \end{align*} Rückert's Analytic Nullstellensatz applies because $\mathcal{J}_a$ is a finitely generated ideal in a ring of convergent power series. The theorem states that the radical of $\mathcal{J}_a$ is exactly the ideal of germs vanishing on the common zero germ of $\mathcal{J}_a$. Since the common zero germ of $(g_1)_a, \ldots, (g_m)_a$ is the germ of $V \cap U$ at $a$, we obtain \begin{align*} \mathcal{I}(V)_a = \sqrt{\mathcal{J}_a}. \end{align*} This equality is the precise form of the statement that the additional functions vanishing on $V$ are accounted for by taking the analytic radical. [/guided] [/step] [step:Apply the Cartan-Oka radical theorem proved by Weierstrass division] Define the radical sheaf $\sqrt{\mathcal{J}} \subset \mathcal{O}_U$ by prescribing its stalks: \begin{align*} (\sqrt{\mathcal{J}})_a := \sqrt{\mathcal{J}_a} \end{align*} for every $a \in U$. The Cartan-Oka Coherence Theorem for Analytic Radicals states that if $\mathcal{K}$ is a coherent ideal sheaf on a complex manifold, then the radical sheaf $\sqrt{\mathcal{K}}$ is coherent. Its proof is the Weierstrass induction: reduced hypersurface ideals are generated by reduced distinguished Weierstrass polynomials, Weierstrass division controls the relation sheaves, local parameterization reduces finite analytic projections to lower dimension, and finite kernels and intersections of coherent modules remain coherent. The theorem applies to $\mathcal{K} = \mathcal{J}$ because $\mathcal{J}$ is a coherent ideal sheaf on the complex manifold $U$. Hence $\sqrt{\mathcal{J}}$ is a coherent $\mathcal{O}_U$-module. [guided] We now use the local finiteness theorem that contains the Weierstrass machinery. Define the sheaf $\sqrt{\mathcal{J}}$ by \begin{align*} (\sqrt{\mathcal{J}})_a := \sqrt{\mathcal{J}_a} \end{align*} for every $a \in U$. This is the sheaf whose germs are locally nilpotent over $\mathcal{J}$ in the sense that some positive power lies in $\mathcal{J}$. The Cartan-Oka Coherence Theorem for Analytic Radicals requires two hypotheses: the ambient space must be a complex manifold, and the starting ideal sheaf must be coherent. The ambient space here is $U$, an open subset of the complex manifold $X$, hence itself a complex manifold. The ideal sheaf $\mathcal{J}$ is coherent by the preceding step. Therefore the theorem applies and gives that $\sqrt{\mathcal{J}}$ is coherent. The content of the cited theorem is exactly the local Weierstrass argument. In the hypersurface case, after a generic linear coordinate change, the Weierstrass Preparation Theorem writes a local defining equation as a unit times a distinguished polynomial; replacing that polynomial by its reduced product of irreducible factors gives the reduced equation for the same hypersurface. Weierstrass Division then shows that any germ vanishing on the hypersurface has zero remainder after division by this reduced distinguished polynomial, hence belongs to the principal ideal it generates. For higher codimension, the Local Parameterization Theorem reduces the analytic set to finite projections over lower-dimensional coordinate spaces, and the induction step uses Weierstrass division to transfer coherence of coefficient modules and relation modules from dimension $n-1$ to dimension $n$. The closure of coherent sheaves under kernels, images, and finite intersections then carries the conclusion from the complete-intersection pieces to the radical of the original coherent ideal. [/guided] [/step] [step:Conclude coherence of the ideal sheaf on the original manifold] For every point $a \in U$, the stalk equality from Rückert's Nullstellensatz gives \begin{align*} \mathcal{I}(V)_a = (\sqrt{\mathcal{J}})_a. \end{align*} Thus $\mathcal{I}(V)|_U = \sqrt{\mathcal{J}}$ as subsheaves of $\mathcal{O}_U$. Since $\sqrt{\mathcal{J}}$ is coherent, $\mathcal{I}(V)|_U$ is coherent. Every point $p \in X$ has an open neighbourhood on which $\mathcal{I}(V)$ is coherent. By locality of coherence, $\mathcal{I}(V)$ is a coherent $\mathcal{O}_X$-module on all of $X$. [/step]