 Solution

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 $\sup |f-f_N|<\epsilon$, 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 $2\epsilon (b-a)$. Choosing $N$ and then a sufficiently fine partition of an interval proves the Riemann integrability criterion for $f$.

The integral estimate then givesThis 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.