# Prove that the following statements are equivalent

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• Nov 24th 2011, 12:49 PM
icedtea
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.
• Nov 24th 2011, 01:00 PM
Plato
Re: Prove that the following statements are equivalent
Quote:

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 $\displaystyle Rd~?$
• Nov 24th 2011, 01:06 PM
icedtea
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.
• Nov 24th 2011, 01:13 PM
Plato
Re: Prove that the following statements are equivalent
Quote:

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 $\displaystyle |x_n-y_n|\to 0.$
• Nov 24th 2011, 02:07 PM
icedtea
Re: Prove that the following statements are equivalent
What if the sets A and B were unbounded? Would the statements hold?
• Nov 24th 2011, 02:21 PM
Plato
Re: Prove that the following statements are equivalent
Quote:

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 $\displaystyle d(x,y)\ge 2$.

Now answer your own question?
• Nov 25th 2011, 05:15 AM
Opalg
Re: Prove that the following statements are equivalent
Quote:

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.