= Hyperbolic tube around a radial segment
{title2=$\operatorname{Im}w/|w|\geq\operatorname{sech}k$}
Under the <Cayley transform between the half-plane and disk>, a radial segment ending at a boundary point becomes a vertical half-ray. If a point is within <hyperbolic distance> $k$ of that half-ray, it lies in a fixed angular sector. Indeed, in the <complex upper half-plane>, $\cosh\rho(w,iy)=(|w|^2+y^2)/(2y\operatorname{Im}w)\geq|w|/\operatorname{Im}w$. This converts bounded hyperbolic approach to a boundary point into nontangential approach, allowing <normal-family proof of angular boundary convergence> to transfer radial limits of bounded <holomorphic functions>.
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