= 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>.
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