I'm aware this question has been posted multiple times before but the proofs given haven't been given via the definition of uniform convergence but using sups and differentiation, and I've only come across the definition of uniform convergence so far.
Let $f_n:[0,1]→\mathbb{R}$ be a function defined by $f_n(x)=x^n(1−x)$
How would you prove that $f_n = x^n(1-x)$ converges to $0$ uniformly using the definition?
I've done the following so far:
Let $\epsilon>0$. Then $N >$_______ implies:
$|x^n(1-x)-0|=$...
And I have no idea where to go from here. I've been given the hint that I may need a different argument for $x$ 'close' to 1 from other values of $x$ but I'm quite confused about this - is it trying to get at the fact that if $x$ is close to $1$, then $(1-x)$ is close to $0$?
Thank you in advance!