Proving Uniform Boundedness of a Pointwise Bounded Family

• Mosis
In summary, to prove that there exists an interval (c,d) < [a,b] on which f_n is uniformly bounded for a pointwise bounded, continuous family, we can use the concept of equicontinuity. By showing that the given family is equicontinuous at some point in [a,b], we can then use the pointwise boundedness to establish uniform boundedness on an interval containing that point.
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.

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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!

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!

1. What is uniform boundedness in mathematics?

Uniform boundedness is a concept in mathematics that refers to the property of a set of functions to have a common bound. This means that all the functions in the set cannot exceed a certain value, regardless of the input or domain.

2. How is uniform boundedness different from pointwise boundedness?

Pointwise boundedness refers to the property of a single function to have a bound at each point in its domain. On the other hand, uniform boundedness refers to the property of a set of functions to have a common bound that applies to all points in their respective domains.

3. Why is proving uniform boundedness important?

Proving uniform boundedness is important because it allows us to determine the behavior of a set of functions as a whole, rather than just focusing on individual functions. It also helps in analyzing the convergence of the functions in the set and their relationship to each other.

4. What are the steps to proving uniform boundedness of a pointwise bounded family?

The steps to proving uniform boundedness of a pointwise bounded family are as follows:
1. Show that each function in the family is pointwise bounded.
2. Choose a specific point in the domain and find the supremum of the function values at that point for all functions in the family.
3. Use the supremum value to define a bound for the entire family.
4. Show that this bound holds for all points in the domain, thereby proving uniform boundedness.

5. Can a pointwise bounded family be uniformly unbounded?

No, a pointwise bounded family cannot be uniformly unbounded. This is because if a family of functions is pointwise bounded, it means that each individual function has a bound at each point in its domain. Therefore, there cannot be a common bound for the entire family that applies to all points in their respective domains.

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