= Hyperbolic zero sequences on a punctured disk
{title2=$z_n=e^{-n^2},\qquad e^{-\rho(e^{-1},z_n)}=n^{-2}$}
The curvature-minus-one <hyperbolic metric on the punctured disk> assigns distance $2\log n$ along the radial geodesic from $e^{-1}$ to $e^{-n^2}$. Its exponential distances therefore sum to a finite value; the <triangle inequality> gives the same conclusion for any fixed basepoint. Nevertheless no nonzero bounded <holomorphic function> has these zeros: the <Riemann removable singularity theorem> extends it at zero, and the <identity theorem> then makes it identically zero. Simple connectivity is necessary in the hyperbolic characterization of bounded analytic zero sets.
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