# Prove that the closure of a bounded set is bounded.

• dustbin
In summary, to prove that the closure of a bounded subset S of ℝ^n is bounded, we can use the definition of boundedness and closure, as well as the concept of open balls. An alternative proof is also suggested, but cannot be used without proof.
dustbin

## Homework Statement

Prove that if S is a bounded subset of ℝ^n, then the closure of S is bounded.

## Homework Equations

Definitions of bounded, closure, open balls, etc.

## The Attempt at a Solution

See attached pdf.

#### Attachments

• Problem.pdf
73.3 KB · Views: 1,315
That looks OK to me apart from a couple of typos.

There is an easier way to prove this too though. Do you know that the closure of S is the intersection of all closed subsets containing S?

Thanks for your input! We have not proved that the closure of S is the intersection of all closed subsets containing S... so I cannot use this result without proof.

dustbin said:
Thanks for your input! We have not proved that the closure of S is the intersection of all closed subsets containing S... so I cannot use this result without proof.

Your original proof looks fine to me. Except when you pick an epsilon, it should be "there exists epsilon" not "for all epsilon".

## 1. What is the definition of a bounded set?

A bounded set is a set that has a finite limit or bound. In other words, all elements in the set are contained within a certain range or interval.

## 2. How do you prove that the closure of a bounded set is bounded?

To prove that the closure of a bounded set is bounded, we must show that the closure also has a finite limit or bound. This can be done by using the definition of closure, which states that the closure of a set contains all of its limit points. By showing that the limit points of the bounded set are also contained within a certain range, we can prove that the closure is bounded.

## 3. Can you provide an example to illustrate this concept?

Sure, let's consider the set A = {1, 2, 3} which is bounded since all elements are contained within the interval [1, 3]. The closure of A, denoted as cl(A), would then be cl(A) = {1, 2, 3} since all limit points are also contained within the interval [1, 3]. This proves that the closure of a bounded set is also bounded.

## 4. What is the significance of proving that the closure of a bounded set is bounded?

Proving that the closure of a bounded set is bounded is important because it helps us understand the behavior of sets with finite limits. It also allows us to make conclusions about the convergence of sequences and continuity of functions.

## 5. Are there any exceptions to this statement?

Yes, there are exceptions to this statement. In some cases, the closure of a bounded set may not be bounded. This can occur when the set contains infinite elements or when the limit points are not contained within a finite range. However, in most cases, the closure of a bounded set is indeed bounded.

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