= Solution
Let the common domain be an <open set> $U$. The <uniform limit theorem> makes the limit $f$ <continuous>. For every triangle whose closed interior lies in $U$, <Cauchy's integral theorem> gives $\int_{\partial T}f_n=0$. Part (ii) passes this identity to the limit, giving $\int_{\partial T}f=0$. <Morera's theorem> now proves that \b[$f$ is <holomorphic>].
The same reasoning works for <locally uniform convergence>, since each triangle boundary is <compact>. No passage of derivatives through a limit has been assumed.
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