Equivalent conditions for a holomorphic function to vanish identically

Let \(\Omega\subseteq\CC\) be a nonempty connected open set and let \(f\) be holomorphic on \(\Omega\). The following are equivalent.

1. \(f(z)=0\) for all \(z\in\Omega\).

2. There are \(z_0\in\Omega\) and a sequence \((z_k)_{k\geq1}\) of distinct points of \(\Omega\) with \(z_k\to z_0\) and \(f(z_k)=0\) for all \(k\).

3. There is \(z_0\in\Omega\) with \(f^{(n)}(z_0)=0\) for all \(n\geq0\).