A nonzero holomorphic function cannot have accumulating zeros inside its domain.Suggest a slogan
Let \(f\colon D_r(a)\to\CC\) be holomorphic and not identically zero. Then there exists \(0<\rho<r\) such that \(f(z)\neq0\) for all \(z\in D_\rho(a)\sm\ts{a}\).
If \(f(a)\neq0\), continuity of \(f\) gives a disc about \(a\) on which \(f\) does not vanish. Otherwise \(f(a)=0\). Since \(f\) is represented by its Taylor series on \(D_r(a)\) and is not identically zero, some Taylor coefficient is nonzero, so \(a\) is a zero of some finite order \(N\ge1\): \(f(z)=(z-a)^Ng(z)\) with \(g\) holomorphic on \(D_r(a)\) and \(g(a)\neq0\). By continuity, \(g\) does not vanish on some disc \(D_\rho(a)\), and then \(f\) does not vanish on \(D_\rho(a)\sm\ts{a}\).