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.