Let $f_n:B \to \mathbb{R}$ be a sequence of continuous function on $\bar{B}$(closed unit ball) such that all $f_n$ are analytic on open unit ball $B$ . And $f_n$ converges to f in $L^2$ sense in $\bar{B}$. To show $f$ is analytic on $B$.
I am not getting how to use the $L^2$ convergence. As in order to use Cauchy integral theorem on certain compact sets and weistrass theorem to show the limit in analytic , how can we use $L^2$ convergence. Any help is much appreciated.