Prove $F$ is normal in $H(\Omega) \iff$ for every $z\in H(\Omega)$ there is a neighborhood $U$ of $z$ such that $F$ is normal in $U$.
The direct implication is immediate because $U\subset \Omega$.
The reciprocal seem obvious but not very immediate. Maybe with this argument? If $F$ is not normal in any neighborhood $U$ of $z_0$ then for functions $f_n$ that take only values in $U$ is not true they converge uniformly to $f$ then it makes no sense to say that $F$ is normal in $H(\Omega)$