Vitali's Convergence Theorem

Let $f_n(z)$ be a sequence of functions, each regular in a region $D$, let $\vert f_n(z)\vert\leq M$ for every $n$ and $z$ in $D$, and let $f_n(z)$ tend to a limit as $n\to\infty$ at a set of points having a Limit Point inside $D$. Then $f_n(z)$ tends uniformly to a limit in any region bounded by a contour interior to $D$, the limit therefore being an analytic function of $z$.

See also Montel's Theorem

References

Titchmarsh, E. C. The Theory of Functions, 2nd ed. Oxford, England: Oxford University Press, p. 168, 1960.