Computing a support of a sheaf

Let $$Z=\{1/n:n\in\mathbb{Z}-\{0\}\}$$. Now define the sheaf $$\mathcal{J}_Z$$ as $$J_Z(U)=\{f:f\text{ is a holomorphic function on U with vanishing on }Z\}.$$ Find $$\text{Supp}(\mathcal{J}_Z).$$

My attempt: for each $$x\in\mathbb{C}$$, if $$x=0$$, then by identity theorem, the only holomorphic function near zero & vanishing on $$Z$$ is the zero function, so $$0\notin \text{Supp}(\mathcal{J}_Z)$$. If $$x\neq 0$$, clearly the function $$\sin(\pi/z)$$ is a holomorphic function vanishing at $$Z$$, and defined everywhere but zero. Therefore, each stalk at $$x\neq 0$$ is nonzero, so $$\text{Supp}(\mathcal{J}_Z)=\mathbb{C}-\{0\}.$$

Am I correct? Please indicate if I have any mistakes.

• Yes you are correct. – Roland Mar 19 at 15:36