Let $f_n=(1-x)^n$. Show the convergence is uniform on $[a,2-a]$ where $a \in (0,1)$
I found the pointwise limit function $f$ to be $1$ is $x=0$ and $0$ on $x\leq 1$.
If $x>1$ the limit function is also $0$.
Now I showed that it's not uniformly convergent on $(0,1)$ because there isn't an $N$ which works for the whole interval, since I can take $\lim x\rightarrow 0$.
Now for the main part:
Clearly the local maximums are attained at $a$ and $2-a$ (in the second, could be negative).
Now I think I need to somehow pick $N$ depending on the $\max \{a,2-a\}$ such that $$|f_n(x)-0|<\epsilon$$
Can somebody help me out?
Hint: if $N$ works for both $a$ and $2-a$, then it works for every number in the range.