- #1
dustbin
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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.
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.
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.
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.
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.
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.
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.