# Help with proof (sets)

• Apr 23rd 2008, 01:39 AM
gasbasis
Help with proof (sets)
Just wondering if someone could help me with this problem.

For two disjoint compact subsets of R^n, show that there is a minimum positive distance between the points of each set.

Much appreciated!
• Apr 23rd 2008, 03:47 AM
Plato
Quote:

Originally Posted by gasbasis
Just wondering if someone could help me with this problem.
For two disjoint compact subsets of R^n, show that there is a minimum positive distance between the points of each set.

Here is an outline.
Suppose that the two subsets are A & B and that D(A,B)=0.
Use the definition of distance between sets to get a sequence of distinct points in A that must converge to a point in B. That happens due to compactness. That will give you a contradiction.
• Apr 23rd 2008, 09:58 AM
ThePerfectHacker
We can generalize this. If A is compact and B is closed subsets of R^n then d(A,B) > 0
• Apr 23rd 2008, 04:14 PM
gasbasis
Quote:

Originally Posted by Plato
Here is an outline.
Suppose that the two subsets are A & B and that D(A,B)=0.
Use the definition of distance between sets to get a sequence of distinct points in A that must converge to a point in B. That happens due to compactness. That will give you a contradiction.

Thanks a lot for this. Does it make sense to also say that since A,B are compact, the function that describes a path between a point in A and a point in B must have a minimum and a maximum because it is a function on a compact set?