Past exam of the mathematics course of the University of Cambridge / 2006 / ib / Paper 2 / 3F / Solution

Uniform convergence of $f_n$ to $f$ on an interval $I$ means that for every $ϵ>0$ there is $N$ such that $|f_n(x)-f(x)|<ϵ$ for all $n\ge N$ and every $x\in I$. Equivalently ${\sup }_{x\in I}|f_n(x)-f(x)|\to 0$. The same $N$ must work throughout the interval, unlike pointwise convergence.

For completeness, continuity is preserved: choose $N$ with uniform error below $ϵ/3$, and use continuity of $f_N$ at a given $x$ to make $|f_N(y)-f_N(x)|<ϵ/3$. The triangle inequality then gives $|f(y)-f(x)|<ϵ$. Thus each function involved in the integral is continuous on the closed interval and Riemann integrable. The estimateproves convergence of the integrals by uniform convergence. When $a=b$ both integrals are zero.