= Solution
The <uniform limit theorem> first gives <continuity> of $f$ on $D$. For any triangle $\Gamma$ whose closed interior is in $D$, the <Cauchy integral theorem> gives $\int_\Gamma f_n(z)\,dz=0$. Moreover,
$$
\left|\int_\Gamma(f-f_n)(z)\,dz\right|
\le\operatorname{length}(\Gamma)\sup_D|f-f_n|\longrightarrow0.
$$
Hence $\int_\Gamma f=0$, and <Morera's theorem> proves that \b[$f$ is analytic on $D$]. In fact, the proof only needs <locally uniform convergence> on compact subsets, since each triangle is compact; the stated <uniform convergence> on the entire domain is stronger.
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