Past exam of the mathematics course of the University of Oxford / 2023 / Prelims / Mathematics II / 5 / a / i / Solution

The sequence of functions $(f_n)$ converges uniformly to $f$ on $E$ ifThe same $N$ must work for every $x$. Equivalently, the supremum norm of $f_n-f$ tends to zero. This is stronger than pointwise convergence, where $N$ may depend on $x$.