Proving Uniform Boundedness of a Pointwise Bounded Family

[Mosis]

Homework Statement

Let f_n:[a,b] -> R be a pointwise bounded, continuous family. Prove there exists an interval (c,d) < [a,b] on which f_n is uniformly bounded.

Homework Equations

no equations

The Attempt at a Solution

I'm stuck. If we have equicontinuity, then this is easy, so I'm thinking we need to prove we have some kind of local equicontinuity or equicontinuity at one point, but I've not been successful in breaking through with these ideas. I don't have a hook!

 

[mighty2000]

Dear student,

Thank you for your post. It seems like you are on the right track with your thoughts about equicontinuity. Here's a suggestion for how to approach this problem:

First, recall the definition of equicontinuity for a family of functions. This means that for any given ε>0, there exists a δ>0 such that for all x,y in [a,b] with |x-y|<δ, we have |f_n(x)-f_n(y)|<ε for all n. This ensures that the functions in the family do not vary too much over small intervals.

Now, let's consider the given pointwise bounded and continuous family of functions. This means that for each x in [a,b], the values of f_n(x) are bounded, and the functions themselves are continuous. Can you use this information to show that the family is equicontinuous at some point in [a,b]? Hint: think about the extreme value theorem.

Once you have established equicontinuity at a point, you can use the fact that the functions are pointwise bounded to show that they are uniformly bounded on some interval containing that point. This should give you the desired interval (c,d) < [a,b]. I hope this helps. Good luck with your proof!