Past exam of the mathematics course of the University of Cambridge / 2018 / ib / Paper 3 / 12F / a / Solution

The sequence $(f_n)$ converges uniformly to $f$ on $A$ when, for every $\epsilon >0$, there is $N$ such thatfor every $x\in A$.

The function $f$ is uniformly continuous when, for every $\epsilon >0$, there is $\delta >0$ such that $x,y\in A$ and $\Vert x-y\Vert <\delta$ imply $|f(x)-f(y)|<\epsilon$. The same $\delta$ works throughout the domain.