Distance between compact subsets

[Ocifer]

Homework Statement

Let A,B be two disjoint, non-empty, compact subsets of a metric space (X,d).

Show that there exists some r>0 such that d(a,b) > r for all a in A, b in B.

Hint provided was: Assume the opposite, consider a sequence argument.

Homework Equations

N/A

The Attempt at a Solution

I've tried a few different characterizations of compactness, but neither one has led me anywhere particularly useful. I'm missing something such that the teacher's hint isn't helping me much.

If I assume the opposite,

Assume that for all r>0, there exists some pair (a,b) such that d(a,b) <= r

I sense that I am supposed to reach a contradiction regarding the property that any sequence in a compact A or B should have a convergent subsequence. But how do I show that I would reach such a contradiction. Can anyone give me a push in the right direction?

 

[Dick]

Homework Statement

Let A,B be two disjoint, non-empty, compact subsets of a metric space (X,d).

Show that there exists some r>0 such that d(a,b) > r for all a in A, b in B.

Hint provided was: Assume the opposite, consider a sequence argument.

Homework Equations

N/A

The Attempt at a Solution

I've tried a few different characterizations of compactness, but neither one has led me anywhere particularly useful. I'm missing something such that the teacher's hint isn't helping me much.

If I assume the opposite,

Assume that for all r>0, there exists some pair (a,b) such that d(a,b) <= r

I sense that I am supposed to reach a contradiction regarding the property that any sequence in a compact A or B should have a convergent subsequence. But how do I show that I would reach such a contradiction. Can anyone give me a push in the right direction?

Make some sequences. Assume d(a_n,b_n)<=1/n for all n. Now pick subsequences.

 

[Ocifer]

Do you mean:

Assume, by way of contradiction, that for all r>0, there exists some pair (a,b) such that d(a,b) <= r.

Let {a_n}, {b_n} be sequences in A and B respectively.

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I'm unclear on your next assumption. Are you using n to index the sequences as well as to bound the distances with 1/n, or was that an unintentional typo? If not, why may I /should I make that assumption?

 

[Ocifer]

If I assume that for all r>0, there exist some pair such that d(a,b) <= r , that allows me to at least construct two sequences in A and B respectively having the property:

d( a_1, b_1 ) <= 1/1
d( a_2, b_2 ) <= 1/2
...

and so on, is that what you meant? I hope I'm understanding you correctly.

If I am trying to show that this property implies there is no convergent subsequence, how does it help. It doesn't tell me about the distances between each of the elements of the {a_n} sequence and the {b_n} sequence, does it?

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Another intuition I had which I'm trying to formalize, but I thought I would run past the forum is this. If I assume that for all r>0, there exists an (a,b) such that d(a,b) <= r, would this not imply that the two sets share a limit point? Since they are compact they must each contain their limit points, and then they would not be disjoint? Can I reach my desired contradiction in that manner?

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[Dick]

If I assume that for all r>0, there exist some pair such that d(a,b) <= r , that allows me to at least construct two sequences in A and B respectively having the property:

d( a_1, b_1 ) <= 1/1
d( a_2, b_2 ) <= 1/2
...

and so on, is that what you meant? I hope I'm understanding you correctly.

If I am trying to show that this property implies there is no convergent subsequence, how does it help. It doesn't tell me about the distances between each of the elements of the {a_n} sequence and the {b_n} sequence, does it?

-----------------------------

Another intuition I had which I'm trying to formalize, but I thought I would run past the forum is this. If I assume that for all r>0, there exists an (a,b) such that d(a,b) <= r, would this not imply that the two sets share a limit point? Since they are compact they must each contain their limit points, and then they would not be disjoint? Can I reach my desired contradiction in that manner?

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The second route. If a_n has a limit point a and b_n has a limit point b, what's d(a,b)?

 

[Ocifer]

Here goes my full attempt:

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Let (X,d) be a metric space. Let A and B be non-empty, disjoint, compact subsets of (X,d).
Then there exists an r>0 such that d(a,b) > r for all a in A, b in B.

Proof:

Assume by way of contradiction that for all r>0, there exists a pair (a,b) such that d(a,b) <= r.

Under this assumption, we may construct the two sequences in A and B respectively, having the property that:

d(a_n, b_n) <= 1/n , for all n in the natural numbers.

Claim: If we consider this sequence of distances, its limit is 0.

Subproof: For any ε > 0, there is an n in the natural numbers such that 1/n < ε . It follows that for any ε > 0, we have some N(ε) such that if n >= N(ε), | d(a_n, b_n) - 0 | < ε .
And so the sequence of distances has limit 0.

From this it follows that the limit of {a_n} is equal to the limit of {b_n}. So A and B share a limit point, and since both contain their limit points, the sets A and B cannot be disjoint. This is a contradiction, and so it follows that there exists some r>0 such that d(a,b) > r for all a in A, b in B.

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Is this okay?

 

[Dick]

Here goes my full attempt:

---------------------------
Let (X,d) be a metric space. Let A and B be non-empty, disjoint, compact subsets of (X,d).
Then there exists an r>0 such that d(a,b) > r for all a in A, b in B.

Proof:

Assume by way of contradiction that for all r>0, there exists a pair (a,b) such that d(a,b) <= r.

Under this assumption, we may construct the two sequences in A and B respectively, having the property that:

d(a_n, b_n) <= 1/n , for all n in the natural numbers.

Claim: If we consider this sequence of distances, its limit is 0.

Subproof: For any ε > 0, there is an n in the natural numbers such that 1/n < ε . It follows that for any ε > 0, we have some N(ε) such that if n >= N(ε), | d(a_n, b_n) - 0 | < ε .
And so the sequence of distances has limit 0.

From this it follows that the limit of {a_n} is equal to the limit of {b_n}. So A and B share a limit point, and since both contain their limit points, the sets A and B cannot be disjoint. This is a contradiction, and so it follows that there exists some r>0 such that d(a,b) > r for all a in A, b in B.

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Is this okay?

The idea is fine. I'm not sure the proof really measures up. Skip to picking convergent subsequences and suppose d(a_n,b_n)<1/n and a_n->a and b_n->b. Use d(a,b)<=d(a,a_n)+d(a_n,b_n)+d(b_n,b) using the triangle inequality. Can you show the right side converges to 0?

 

[Ocifer]

The right hand side terms should all go to zero in the limit as

d(a, a_n) --> d(a,a) = 0
d(b, b_n) --> d(b,b) = 0

and

d(a_n, b_n) --> 0 using the definition of a limit of a sequence in metric spaces, and the fact that d(a_n, b_n) < 1/n for all n.

I think I have it now, thank you for your help.