Prove $x^n$ is not uniformly convergent
Solution 1:
It all boils down to proving that $$\sup_{x\in [0,1]}|x^n-\chi_{\{1\}}|=\sup_{x\in [0,1)}|x^n-0|=\sup_{x\in[0,1)}x^n=1\not\to 0$$
Solution 2:
I think the best way to see that this function doesn't converge uniformly on $x \in [0,1]$ is to note that the limiting function is discontinuous for $x \in [0,a], a<1$ and $x=1$: $$ \lim_{n \to \infty} f_n(x)= \left\{ \begin{array}{rl} 0 &x \in [0,a]\\ 1 & x=1 \end{array} \right. $$