= Solution
For the usual piecewise smooth complex <path> $\gamma$, let $\ell(\gamma)$ denote its finite length. The <estimation lemma> gives
$$
\left|\int_\gamma f\,dz-\int_\gamma f_n\,dz\right|
\leq\ell(\gamma)\sup_{z\in\gamma}|f(z)-f_n(z)|.
$$
By <uniform convergence>, the supremum tends to zero, so
$$
\boxed{\int_\gamma f\,dz=\lim_{n\to\infty}\int_\gamma f_n\,dz.}
$$
The uniform limit is <continuous> on the <path>, so its integral exists. This is <uniform convergence and contour integration>; the <path> length is fixed while the functions vary.
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