Suppose \(f(z)\) is analytic in some neighborhood of the point \(z_0\) without the point \(z_0\) itself; let \(z_0\) be an essential singularity of \(f(z)\).
Show that for any complex number \(C\), there is a sequence of points \(z_n\) (\(n=0,1,2,\ldots \)) such that \[ z_n\to z_0\qquad \text {and}\qquad f(z_n)\to C. \] (In other words, in every neighborhood of an essential singularity, the function \(f(z)\) is arbitrary close to every complex number.)