= Solution
<Morera's theorem> states that a continuous function on a domain is holomorphic if its contour integral around every triangle whose filled-in interior lies in the domain is zero.
The <uniform limit> $f$ is a <continuous function>, by the <uniform limit theorem>. For any such triangle $T$, the <estimation lemma> gives
$$
\left|\int_{\partial T}(f-f_n)(z)\,dz\right|\leq\operatorname{length}(\partial T)\sup_{z\in\partial T}|f(z)-f_n(z)|\longrightarrow0.
$$
Each integral of $f_n$ vanishes by the <Cauchy integral theorem>. Hence the integral of $f$ vanishes too, and <Morera's theorem> proves \b[$f$ is holomorphic on $D$]. This is the basic <locally uniform convergence of holomorphic functions> argument; the given uniform convergence on all of $D$ is more than needed.
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