Let \(E\) be a set, and let \(f_n\colon E\to\RR\) for \(n\geq 1\) and \(f\colon E\to\RR\) be functions. The sequence \((f_n)\) converges uniformly to \(f\) on \(E\) if for every \(\varepsilon>0\) there exists \(N=N(\varepsilon)\) such that \(\abs{f_n(x) - f(x)} < \varepsilon\) for all \(n\geq N\) and all \(x\in E\).
With \(\norm{g}_\infty \coloneqq \sup_{x\in E}\abs{g(x)}\) for \(g\colon E\to\RR\), the sequence \((f_n)\) converges uniformly to \(f\) on \(E\) if and only if \(\norm{f_n - f}_\infty\to 0\) as \(n\to\infty\).
What does it mean for $f_n$ to converge uniformly to $f$ on $E$?