Prove that the following statements are equivalent

**icedtea** · November 24th 2011, 12:49 PM

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.

**Plato** · November 24th 2011, 01:00 PM

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 $Rd~?$

**icedtea** · November 24th 2011, 01:06 PM

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

Rd should be the set of reals with dimension d.

**Plato** · November 24th 2011, 01:13 PM

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 $d=1$ , as in the reals.
Let $A=[0,1]~\&~B=[2,3]$ . They are disjoint, but the is no sequences $(x_n)\subset A~\&~(y_n)\subset B$ such that $|x_n-y_n|\to 0.$

**icedtea** · November 24th 2011, 02:07 PM

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

**Plato** · November 24th 2011, 02:21 PM

Originally Posted by icedtea

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

Let $A=(-\infty,-1]~\&~B=[1,\infty)$ they are unbounded, disjoint, closed sets. If $x\in A~\&~y\in B$ then $d(x,y)\ge 2$ .

Now answer your own question?

**Opalg** · November 25th 2011, 05:15 AM

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 $\mathbb{R}^d$ , and suppose that A is bounded.

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

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

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

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