Let $\{f_n(z)\}$ be a sequence of analytic functions converging uniformly to a function $f(z)$ on all compact subsets of a domain $D$. Then $f(z)$ is analytic in $D$.
Suppose we proceed as follows:
It is enough to prove that $f(z)$ is analytic at a point $z_0\in D$. Let $D_0$ be disk with center $z_0$ and contained in $D$. Clearly $f(z)$ is continuous on $D_0$. Moreover because of uniform convergence $$\lim_{n\to\infty}\left( \int_C f_n(z)dz\right)=\int_C f(z)dz$$ for every closed contour $C$ in $D_0$ and hence using Cauchy's theorem, we see that the $\int_C f_n(z)dz=0$ for every closed contour $C$ in $D_0$. Now Morera's theorem finishes the proof.
Question: Where is uniform convergence on compact subsets used?
Best Answer
You need uniform convergence (or something similar) to be able to conclude that $$ \lim_{n\to\infty} \int_C f_n(z)\,dz = \int_C f(z)\,dz $$ In paricular, pointwise convergence is not enough for this.