I am reviewing Stein's complex analysis , and there's one theorem I listed follows:
Suppose $ f$ is a holomorphic function on a region $\Omega$ that vanishes on a sequence of distinct points with a limit point in $\Omega$ , then $f$ is identically $0$ . (Here region means both open and connect)
My question is : If we do not assume the limit point is in $\Omega$ , the limit point might in the boundary of $\Omega$ , can we get the similar conclusion ? Since the orginal proof using the power series of the limit point $z_0$ , I have no idea how to deal with this .