[proofplan] The radical is identified with the ideal sheaf of the analytic zero set of $\mathfrak a$. Coherence of $\mathfrak a$ makes the zero set analytic, and Cartan's coherence theorem for analytic sets says its vanishing ideal is coherent. The analytic Nullstellensatz identifies that ideal with the radical. [/proofplan] [step:Associate an analytic set to $\mathfrak a$] Because $\mathfrak a$ is coherent, it is locally generated by finitely many holomorphic functions $f_1,\dots,f_r$. The common zero locus \begin{align*} V(\mathfrak a)=\{x\in X: f_1(x)=\cdots=f_r(x)=0\} \end{align*} is therefore a closed analytic subset of $X$, independent of the chosen local generators. [/step] [step:Use Cartan coherence for the vanishing ideal] Let $\mathcal{I}(V(\mathfrak a))$ be the sheaf of holomorphic germs vanishing on this analytic set. Cartan's coherence theorem for ideal sheaves of analytic subsets states that $\mathcal{I}(V(\mathfrak a))$ is coherent. This is the main analytic input. [/step] [step:Apply the analytic Nullstellensatz] The analytic Nullstellensatz gives equality on each stalk: \begin{align*} \mathcal{I}(V(\mathfrak a))_p=\sqrt{\mathfrak a_p}. \end{align*} Thus the sheaf $\mathcal{I}(V(\mathfrak a))$ is exactly $\sqrt{\mathfrak a}$. Since the left-hand side is coherent, $\sqrt{\mathfrak a}$ is coherent. [/step]