# Thread: Help with proof (sets)

1. ## 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!

2. 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.

3. We can generalize this. If A is compact and B is closed subsets of R^n then d(A,B) > 0

4. 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?