$x^n$ converges pointwise but not uniformly on $[0,1]$

For \(n\geq 1\) let \(f_n\colon[0,1]\to\RR\), \(f_n(x)\coloneqq x^n\). Then \(f_n\to f\coloneqq\chi_{\theset{1}}\) pointwise on \([0,1]\), since \(x^n\to 0\) for \(0\leq x<1\) and \(1^n = 1\). The convergence is not uniform: for every \(n\), \(\sup_{x\in[0,1]}\abs{f_n(x) - f(x)} = \sup_{0\leq x<1} x^n = 1\).