A subset in R^n is bounded if and only if it is totally bounded.

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In summary, the proof for the implication "totally bounded -> bounded" involves choosing a number e and a finite number of open balls with radius e that cover A, and showing that there is a maximum distance from b to any other point in A. For the implication "bounded -> totally bounded," we can show that A is contained in a ball with radius K, and then try to reduce it to a finite number of balls. However, using the idea of covering A with balls of rational radii may not work.
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Homework Statement


Prove that a subset in R^n, where n is a finite number, is bounded if and only if it is totally bounded.

Homework Equations


If A is the subset, A is bounded if there is a point b in R^n such that d(x,b)<= K, for a every x in A.

A is totally bounded if for every e> 0, there is a finite number of ball with radius e, that covers A.

The Attempt at a Solution



totally bounded -> bounded

Chose a number e, there is a finite number P of open balls that cover A, each with radius e.
Chose an element b in A. s=sup{d(b,l): l is the center of the balls} must exist since we only have finite amount of balls. Then the max distance from b to another point in A is s+e, and hence A is bounded.
Is this proof correct for this implication?

bounded -> totally bounded
There is an element in R^n such that for any element x in A d(x,b) <= K, for a real number K.
I do not see how to proceed here?
Since Q^n is dense I guess that if we make balls with radius epsilon around every element in R^n that is rational we have a countable number of balls that cover A?, then the problem is to reduce it to a finite number of balls?
 
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bobby2k said:
bounded -> totally bounded
There is an element in R^n such that for any element x in A d(x,b) <= K, for a real number K.
I do not see how to proceed here?
Since Q^n is dense I guess that if we make balls with radius epsilon around every element in R^n that is rational we have a countable number of balls that cover A?, then the problem is to reduce it to a finite number of balls?
I like your thinking, simply because it shows you're actually thinking about this, but I don't think that will work.

Since ##A## is bounded, can we make ##A\subseteq B_K##, where ##B_K=\{x\in\mathbb{R}^n:d(b,x)\leq K\}##?
 

1. What does it mean for a subset in R^n to be bounded?

For a subset to be bounded in R^n, it means that there exists a finite value that serves as an upper bound for all its elements. In other words, all elements in the subset fall within a certain range, and do not approach infinity.

2. What does it mean for a subset in R^n to be totally bounded?

A subset in R^n is totally bounded if it can be covered by a finite number of smaller subsets, each of which is bounded. Essentially, this means that the subset can be broken down into smaller, finite parts that are bounded.

3. How are bounded and totally bounded subsets related?

In order for a subset in R^n to be totally bounded, it must first be bounded. However, not all bounded subsets are totally bounded. A totally bounded subset is a stronger condition than being bounded.

4. What is the significance of a subset being bounded and totally bounded?

These concepts are important in understanding the properties of subsets in R^n. Bounded and totally bounded subsets help us to determine if a subset is compact, which has important implications in the study of analysis and topology.

5. Can you provide an example of a bounded subset that is not totally bounded?

Yes, the set [0,1] in R is a bounded subset, since all its elements fall within the range of 0 to 1. However, it is not totally bounded, as it cannot be covered by a finite number of smaller subsets that are also bounded.

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