= Pointwise anchoring of a holomorphic normal family
{title2=$\sup_{f\in\mathcal F}|f(z_0)|<\infty$}
For a holomorphic <normal family>, limits are holomorphic or identically infinity. A uniform bound at one interior point excludes infinity on its <connected> domain. Every sequence therefore has a finite locally uniform subsequential limit. The <Cauchy integral formula> makes the <derivatives> converge locally uniformly too, so the <derivative> family is normal. Without the point bound this can fail: $n(1+z^2/2)$ tends uniformly to infinity on the unit disk, whereas its <derivatives> $nz$ are not normal at zero.
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