1. Uniform congergence Problem.

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 $\displaystyle A_\varepsilon$

2. Re: Uniform congergence Problem.

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

3. Re: Uniform congergence Problem.

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

4. Re: Uniform congergence Problem.

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

5. Re: Uniform congergence Problem.

Originally Posted by FernandoRevilla
Yes, choose for example $\displaystyle K=[0,a)$ with $\displaystyle 0<a<1$ , then for every $\displaystyle 0<\epsilon<1$ and for $\displaystyle n$ positive integer such that $\displaystyle n>\log \epsilon/\log a$ we have $\displaystyle n>\log \epsilon/\log x$ for all $\displaystyle x\in K$ , or equivalently $\displaystyle |x^n|<\epsilon$ for all $\displaystyle x\in K$ .
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 $\displaystyle m(A) \nless \infty$ ,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 $\displaystyle A_\varepsilon$
It seems it could be true or false depending on $\displaystyle A$ (if I understand the question properly).
It would converge uniformly to 0 on $\displaystyle [0,1]$.
It would not converge uniformly (but pointwise) to 0 on $\displaystyle \bigcup\limits_{n=0}^\infty [n,n+2^{-n}]$.