Hi
I'm trying to prove that the following two staments are equivalent.
Let A and B be closed subsets of Rd, and suppose that A is bounded.
i) A and B are disjoint
ii) There exists a pair of sequences (xn) (subset of A) and (yn) (subset of B) such that
lxn-Ynl->0.
Any help would be appreciated, not too sure here to start.
Condition ii) is wrong. It should say
ii') There does not exist a pair of sequences (subset of A) and (subset of B) such that
With that change in wording, it is true that i) and ii') are equivalent. The key thing to notice is that A is closed and bounded, hence compact. That implies that a sequence in A has a convergent subsequence, say for some s in A. If there also exists a sequence in B with then But B is closed, so that implies Thus , so A and B are not disjoint.
That shows that i) implies the modified condition ii').
The converse implication is easier. In fact, if A and B are not disjoint then there exists a point . By taking for all n, you see that condition ii') does not hold.