1. ## Uniform convergence

I thought I completely understood uniform convergence until I ran into this statement that I cannot prove please help!

Suppose $\{f_n\}$ converges point wise on $[a,b]$ and for all $x\in [a,b]$ theres an open interval $I_x$ that contains $x$ such that $\{f_n\}$ uniformly converges on $I_x\cap [a,b]$. Show $\{f_n\}$ uniformly converges on $[a,b]$.

Can anyone walk me through this? Thank you!

2. Originally Posted by MatthewD
Suppose $\{f_n\}$ converges point wise on $[a,b]$ and for all $x\in [a,b]$ theres an open interval $I_x$ that contains $x$ such that $\{f_n\}$ uniformly converges on $I_x\cap [a,b]$. Show $\{f_n\}$ uniformly converges on $[a,b]$.
Consider the collection $\left\{I_x:x\in [a,b]\right\}$ that is an open covering of $[a,b]$.
By compactness there is a finite subcover from that cover.
Whenever we have a finite collection of integers, we have a largest one.
Can that be used for uniformly converges?

3. Well, I haven't learned anything like that yet... is that topology? This is for a basic real analysis course

4. Originally Posted by MatthewD
Well, I haven't learned anything like that yet... is that topology? This is for a basic real analysis course
That is basic real analysis. Even in the most basic analysis course at my college they discuss the fact that closed intervals may always be covered with finitely many open sets (sparing you the technical language)

5. Then how would that relate to uniform convergence? I think I understand, the language of subcovers/covers just hasn't been used yet... but I don't at all see the relation to convergence