1. ## Uniform congergence Problem.

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

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

2. ## Re: Uniform congergence Problem.

According to the Dini's Theorem, $f_n=x^n\to 0$ uniformly on any compact $K\subseteq [0,1)$ .

3. ## Re: Uniform congergence Problem.

Originally Posted by FernandoRevilla
According to the Dini's Theorem, $f_n=x^n\to 0$ uniformly on any compact $K\subseteq [0,1)$ .
Is there any other approach ?

4. ## Re: Uniform congergence Problem.

Originally Posted by younhock
Is there any other approach ?
Yes, choose for example $K=[0,a)$ with $0 , then for every $0<\epsilon<1$ and for $n$ positive integer such that $n>\log \epsilon/\log a$ we have $n>\log \epsilon/\log x$ for all $x\in K$ , or equivalently $|x^n|<\epsilon$ for all $x\in K$ .

5. ## Re: Uniform congergence Problem.

Originally Posted by FernandoRevilla
Yes, choose for example $K=[0,a)$ with $0 , then for every $0<\epsilon<1$ and for $n$ positive integer such that $n>\log \epsilon/\log a$ we have $n>\log \epsilon/\log x$ for all $x\in K$ , or equivalently $|x^n|<\epsilon$ for all $x\in K$ .
Oh, This is good. Thanks a lot.
How bout Question 2? I cant show that.

6. ## Re: Uniform congergence Problem.

Originally Posted by younhock
2) Suppose $m(A) \nless \infty$ ,why there is no such $A_\varepsilon \subseteq A$ such that when $f_n = 1 _{[n,n+1]}$ , $f_n$ is converges uniformly on $A_\varepsilon$
It seems it could be true or false depending on $A$ (if I understand the question properly).
It would converge uniformly to 0 on $[0,1]$.
It would not converge uniformly (but pointwise) to 0 on $\bigcup\limits_{n=0}^\infty [n,n+2^{-n}]$.