Past exam of the mathematics course of the University of Cambridge / 2004 / ib / Paper 1 / 4G / Solution

Uniform convergence means that for every $\epsilon >0$ there is an integer $N$, independent of $x$, such that $|F_n(x)-F(x)|<\epsilon$ for all $n\ge N$ and every $x\in (0,1)$. Equivalently ${\sup }_{0<x<1}|F_n(x)-F(x)|\to 0$. In pointwise convergence, the required $N$ may depend on the chosen $x$.