= Normal-family proof of angular boundary convergence
If a bounded <holomorphic function> $g$ on the <complex upper half-plane> has $g(iy)\to\ell$ as $y\downarrow0$, the rescaled functions $g(z/n)$ form a <normal family>. Every subsequential limit equals $\ell$ on the imaginary axis, hence everywhere by the <identity theorem for holomorphic functions>. The entire sequence converges locally uniformly to $\ell$. For a nontangential sequence $z_j=x_j+iy_j\to0$, put $n_j=\lfloor1/y_j\rfloor$; then $n_jz_j$ remains in a fixed compact subset of the half-plane. This gives $g(z_j)\to\ell$. The bounded example $e^{-i/z}$ shows that unrestricted boundary convergence can fail.
// Destination: analysis.bigb
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