Past exam of the mathematics course of the University of Cambridge / 2016 / ib / Paper 3 / 2G / b / i / Solution

For each fixed $0<x<1$, $x^n\to 0$, so the pointwise limit is $f(x)=1/(1-x)$. Howeverfor every fixed $n$. Hence the error has infinite supremum on $(0,1)$ and the convergence is not uniform. This does not contradict uniform convergence on any smaller closed interval $[0,r]$, $r<1$, where the error is at most $r^n/(1-r)$.