Hello I am trying to understand why the sequence of functions $f_n=nx(1-x^2)^n$ does not converge uniformly to 0 on the interval $[0,1]$ but does on the interval $[a,1]$ where $a\in (0,1)$.
I know that it does not converge on $[0,1]$ as $\int_0^1 nx(1-x^2)^n=1/2$ which is not equal to $\int_0^1 0=0$.
However if I then consider $\mbox{lim}_{n\rightarrow\infty}\sup|f_n(x)|$ I am confused as to why this does not go to 0 on $[0,1]$ but does on $[a,1]$
(unless in the first case i can choose my $x$ to be something like $\frac{1}{n}$ but in the second case I can't-im a bit confused at this bit though)
Thanks for any help-(I hope my post is clear enough, sorry if it is not I am new here)