# Uniform convergence and equicontinuity

• May 21st 2008, 11:42 AM
spoon737
Uniform convergence and equicontinuity
This problem is giving me a headache.

Suppose $\{f_n\}$ is a sequence of functions defined on [a,b] which converges uniformly to f. Prove that $\{f_n\}$ is equicontinuous. That is, show that whenever $\epsilon >0$, there is a $\delta >0$ such that if n is a positive integer and $x,y \in [a,b]$ with $|x-y|< \delta$, then $|f_n(x)-f_n(y)|< \epsilon$.

Any help on how to prove this would be greatly appreciated. Bear in mind that this is the first time I've heard of equicontinuity, so if there is some nice theorem that produces this as an immediate result, I probably don't know it and thus can't use it. I need to work with the epsilon-delta definition.
• May 21st 2008, 12:30 PM
TheEmptySet
Quote:

Originally Posted by spoon737
This problem is giving me a headache.

Suppose $\{f_n\}$ is a sequence of functions defined on [a,b] which converges uniformly to f. Prove that $\{f_n\}$ is equicontinuous. That is, show that whenever $\epsilon >0$, there is a $\delta >0$ such that if n is a positive integer and $x,y \in [a,b]$ with $|x-y|< \delta$, then $|f_n(x)-f_n(y)|< \epsilon$.

Any help on how to prove this would be greatly appreciated. Bear in mind that this is the first time I've heard of equicontinuity, so if there is some nice theorem that produces this as an immediate result, I probably don't know it and thus can't use it. I need to work with the epsilon-delta definition.

Do we know if f is continous?

If so consider

$|f_n(x)-f_n(y)|=|f_n(x)-f_n(y)+f(x)-f(x)+f(y)-f(y)|$
$\le |f_n(x)-f(x)|+|f(y)-f_n(y)|+|f(x)-f(y)|$

The reson I need f to be continous is we can make the first two terms small by using uniform convergenece but |f(x)-f(y)| can only be made small if f is continous.

I Hope this helps
• May 21st 2008, 12:39 PM
ThePerfectHacker
Quote:

Originally Posted by TheEmptySet
Do we know if f is continous?

Yes. Because $\{f_n\}$ converge to $f$ uniformly, so a uniform limit of continous functions is continous. Who do we know if $\{f_n\}$ are continous? Well, they have to be, because we conclusion we get to is that they are equicontinous. But the poster should have said that each sequence function is continous, otherwise the conditional is false.
• May 21st 2008, 12:59 PM
spoon737
Oops, I forgot to mention that each $f_n$ is, in fact, continuous. Thanks for the help!