Normality is a local property

Suppose a family of meromorphic functions is a normal family on a neighbourhood of every point of a planar domain. Choose a countable cover by open discs with that property. From any sequence extract successive subsequences converging locally uniformly in the chordal metric on the first disc, second disc, and so on; take the diagonal subsequence argument.

The local Riemann sphere valued limits agree on overlaps, since the same subsequence converges there. They therefore define one continuous global limit. Locally it is meromorphic or identically infinity, so on a connected domain it is globally meromorphic or identically infinity. A finite cover of each compact subset transfers the local uniform estimates to that subset. Thus the original family is a normal family on the entire domain. The argument handles poles using the chordal metric; it does not require every local limit to be a finite holomorphic function.