A family of holomorphic or meromorphic maps is normal when every sequence has a subsequence converging locally uniformly in the spherical metric, possibly to infinity.
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: tends uniformly to infinity on the unit disk, whereas its derivatives are not normal at zero.
Choose a countable cover by connected disks on which a family is normal. From any sequence extract successive subsequences converging on each disk and use the diagonal sequence. The limits agree on overlaps; a finite holomorphic limit and infinity cannot agree on a nonempty overlap. Connectedness therefore gives one global type of limit, and finite compact subcovers give locally uniform convergence. This proves normality of the family on the whole domain from normality near every point.
If a bounded holomorphic function on the complex upper half-plane has as , the rescaled functions form a normal family. Every subsequential limit equals on the imaginary axis, hence everywhere by the identity theorem for holomorphic functions. The entire sequence converges locally uniformly to . For a nontangential sequence , put ; then remains in a fixed compact subset of the half-plane. This gives . The bounded example shows that unrestricted boundary convergence can fail.
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A family of meromorphic functions on a domain that omits three fixed points of the Riemann sphere is normal. For holomorphic functions with values in the complex plane, omitting two fixed values suffices.
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