I have two question here:

1) Suppose $\displaystyle f_n = x^n$ , $\displaystyle \forall x\in [0,1] , n \in $\mathbb{N}$$ , how to find $\displaystyle A_\varepsilon \subseteq [0,1]$ such that $\displaystyle f_n$ is converge UNIFORMLY on $\displaystyle A_\varepsilon$ ?

2) Suppose $\displaystyle m(A) \nless \infty$ ,why there is no such $\displaystyle A_\varepsilon \subseteq A $ such that when $\displaystyle f_n = 1 _{[n,n+1]} $ , $\displaystyle f_n $ is converges uniformly on $\displaystyle A_\varepsilon$