I need to prove that no two distinct holomorphic functions agree on all of $\frac{1}{n}$ where $n$ is an integer.
So the identity theorem says that two functions $f,g$ are identical iff the set of all complex numbers in a domain having $f(z) = g(z)$ have a limit point in that domain.
So $\frac{1}{n}$ does have a limit point in the complex plane right? It may not belong to the set, but it does have a limit point $z=0$.
What is incorrect in this argument? What is the correct argument?
Best Answer
Just look at $f(x)=\sin(\frac{\pi}{x})$ and $g(x)=0$, they agree on all points $x_n=\frac{1}{n}$ but are not the same...