Accumulating zeros force an essential singularity

Let \(f ( z )\) be a non-constant analytic function in \(| z | > 0\) such that \(f ( z _ { n } ) = 0\) for infinite many points \(z _ { n }\) with \(\operatorname* { l i m } _ { n \to \infty } z _ { n } = 0\) . Show that \(z = 0\) is an essential singularity for \(f ( z )\) . (An example of such a function is \(f ( z ) = \sin ( 1 / z ) . )\)