= Solution
{c}
Assume, as required for the displayed integrals, that the $f_n$ are <Riemann integrable> on $[a,b]$. Their <uniform limit> $f$ is also <Riemann integrable>: if $\sup|f-f_N|<\varepsilon$, the upper and lower <Darboux sums> for a <partition of an interval> differ for $f$ by at most the corresponding difference for $f_N$ plus $2\varepsilon(b-a)$. Choosing $N$ and then a sufficiently fine <partition of an interval> proves the <Riemann integrability criterion> for $f$.
The integral estimate then gives
$$
\left|\int_a^b f_n(x)\,dx-\int_a^b f(x)\,dx\right|
\leq(b-a)\sup_{x\in[a,b]}|f_n(x)-f(x)|\longrightarrow0.
$$
This proves that <uniform convergence preserves integrals on a compact interval>. Some integrability hypothesis is necessary: <uniform convergence> alone cannot make the integral of an arbitrary starting <function> exist.
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