Theorem. Uniform limits of polynomial [rmsf-1405][edit]

$\gdef\CC{\mathbb{C}}$

Suppose $f_1, f_2, \cdots$ is a sequence of holomorphic functions $U \to \CC$ with pointwise limit $f: U \to \CC$. Suppose also the following stronger property: for every closed ball $\overline D \subset U$, the value

$$ \sup_{z \in \overline D} |f(z) - f_i(z)| $$

tends to zero as $i$ tend to $\infty$ (uniform convergence on closed balls.) Then $f$ is holomorphic.