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 $y\downarrow 0$, 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 $z_j=x_j+i{y}_j\to 0$, put $n_j=\lfloor 1/y_j\rfloor$; then $n_j{z}_j$ 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