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Math Help - Prove that the following statements are equivalent

  1. #1
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    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.
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  2. #2
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    Re: Prove that the following statements are equivalent

    Quote Originally Posted by icedtea View Post
    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~?
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    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.
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    Re: Prove that the following statements are equivalent

    Quote Originally Posted by icedtea View Post
    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.
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    Re: Prove that the following statements are equivalent

    What if the sets A and B were unbounded? Would the statements hold?
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    Re: Prove that the following statements are equivalent

    Quote Originally Posted by icedtea View Post
    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?
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  7. #7
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    Re: Prove that the following statements are equivalent

    Quote Originally Posted by icedtea View Post
    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.
    Last edited by Opalg; November 25th 2011 at 05:31 AM.
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