Hi--


Where can I find a proof of this theorem: For each $r \in \mathbb{Z}\_{+}$, 
there exists a complex entire function $f(z)$ such that $f(r) \neq 0$ but
$f(r+1)=f(r+2)=\cdots =0$, 
i.e. $f(z) \in I\_{r+1}$ but $f(z) \neq I\_{r}$, where
$I\_{r}= \{ f \in R \ | f(r)=f(r+1)= \cdots =0\}$
where $R$ is the ring of complex entire functions



Weierstrass Theorem
Why don't you start with a function with zeros at the integers,
for instance $\sin\pi z$, and then somehow eliminate the zero at $r$?


