Talk:Pointwise convergence

I'm not quited convinced that the series {${\displaystyle x^{n}}$} is not uniformly convergent on the interval [0,1).The limit function f(x) is not continious at x0=1. For every 0<x<1 the following is true: lim (n->inf) [sup {|fn(x)-f(x)|} ]=0
${\displaystyle \sup\{\,|f_{n}(x)-f(x)|:0\leq x\leq 1\,\}}$