= Solution
For the preliminary definition, put $S_N(z)=\sum_{k=1}^Nf_k(z)$. The series converges <uniform convergence>[uniformly] to $f$ on $U$ when, for every $\epsilon>0$, some $N_0$ satisfies $|S_N(z)-f(z)|<\epsilon$ for all $z\in U$ and all $N\geq N_0$. Equivalently, $\sup_{z\in U}|S_N(z)-f(z)|\to0$.
Let $L(\gamma)$ be the finite length of the piecewise-smooth path. Finite linearity of the <contour integral> and the integral bound give
$$
\left|\sum_{k=1}^N\int_\gamma f_k(z)\,dz-\int_\gamma f(z)\,dz\right|
=\left|\int_\gamma(S_N-f)\,dz\right|
\leq L(\gamma)\sup_U|S_N-f|\longrightarrow0.
$$
The uniform limit is continuous by the stated allowed result, so its integral is defined. Consequently
$$
\boxed{\sum_{k=1}^{\infty}\int_\gamma f_k(z)\,dz=\int_\gamma f(z)\,dz.}
$$
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