= Normality is a local property
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.
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