# Thread: Prove that the following statements are equivalent

1. ## Prove that the following statements are equivalent

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.

2. ## Re: Prove that the following statements are equivalent

Originally Posted by icedtea
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.
What is $\displaystyle Rd~?$

3. ## Re: Prove that the following statements are equivalent

Sorry. Thats my fault, kind of new to this.

Rd should be the set of reals with dimension d.

4. ## Re: Prove that the following statements are equivalent

Originally Posted by icedtea
Sorry. Thats my fault, kind of new to this.
Rd should be the set of reals with dimension d.
The statement is not true.
Let $\displaystyle d=1$, as in the reals.
Let $\displaystyle A=[0,1]~\&~B=[2,3]$. They are disjoint, but the is no sequences $\displaystyle (x_n)\subset A~\&~(y_n)\subset B$ such that $\displaystyle |x_n-y_n|\to 0.$

5. ## Re: Prove that the following statements are equivalent

What if the sets A and B were unbounded? Would the statements hold?

6. ## Re: Prove that the following statements are equivalent

Originally Posted by icedtea
What if the sets A and B were unbounded? Would the statements hold?
Let $\displaystyle A=(-\infty,-1]~\&~B=[1,\infty)$ they are unbounded, disjoint, closed sets. If $\displaystyle x\in A~\&~y\in B$ then $\displaystyle d(x,y)\ge 2$.

7. ## Re: Prove that the following statements are equivalent

Originally Posted by icedtea
Hi

I'm trying to prove that the following two statements are equivalent.

Let A and B be closed subsets of $\displaystyle \mathbb{R}^d$, and suppose that A is bounded.

i) A and B are disjoint
ii) There exists a pair of sequences $\displaystyle (x_n)$ (subset of A) and $\displaystyle (y_n)$ (subset of B) such that $\displaystyle |x_n-y_n|\to0.$

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 $\displaystyle (x_n)$ (subset of A) and $\displaystyle (y_n)$ (subset of B) such that $\displaystyle |x_n-y_n|\to0.$

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 $\displaystyle (x_n)$ in A has a convergent subsequence, say $\displaystyle x_{n_k}\to s$ for some s in A. If there also exists a sequence $\displaystyle (y_n)$ in B with $\displaystyle |x_n-y_n|\to0$ then $\displaystyle y_{n_k}\to s.$ But B is closed, so that implies $\displaystyle s\in B.$ Thus $\displaystyle s\in A\cap B$, 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 $\displaystyle t\in A\cap B$. By taking $\displaystyle x_n=y_n=t$ for all n, you see that condition ii') does not hold.