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Thread: minimum distance

  1. #1
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    minimum distance

    Let (X,d) be a metric space and let A be a compact subset of X. Let x be a point in X.

    I want to show that there exists a point a* in A such that d(x,a*) is less than or equal to d(x,a) for all a in A.

    ---------------------------------------------

    Attempt:

    Let $\displaystyle D = inf\{d(x,a): a \in A\}$

    Then there is a sequence $\displaystyle (a_n)$ in A such that $\displaystyle d(x,a_n)\rightarrow D$ as $\displaystyle n \rightarrow \infty$

    Because A is compact there is a subsequence $\displaystyle (a_{n_j})$ in A which converges to a point in A, call it a*

    Then $\displaystyle d(x,a_{n_j}) \rightarrow d(x,a^*)$ as $\displaystyle n \rightarrow \infty$

    Then... something... to give d(x,a*) = D which is equivalent to what I was trying to show

    ----------------------

    I'm not sure... it's all a little cloudy to me... I feel like I'm on the right track but I may not be appreciating some of the subtleties...

    Also, the question came with a hint to construct a sequence in A satisfying $\displaystyle d(x, a_n)$ less than or equal to D + 1/n, then use the compactnes of A... That's even cloudier to me :S


    Thanks for any help
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  2. #2
    Senior Member TheAbstractionist's Avatar
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    Quote Originally Posted by RanDom View Post
    Let (X,d) be a metric space and let A be a compact subset of X. Let x be a point in X.

    I want to show that there exists a point a* in A such that d(x,a*) is less than or equal to d(x,a) for all a in A.
    Hi RanDom.

    If $\displaystyle x\in A$ then we can take $\displaystyle a^*=x$ and the result is proved. So assume $\displaystyle x\notin A.$

    Then for each $\displaystyle a\in A$ let $\displaystyle d(x,a)=\epsilon_a>0.$ Then $\displaystyle \{B_{\epsilon_a}(a):a\in A\}$ is an open cover for $\displaystyle A$ and by compactness there is a finite subcover $\displaystyle \{B_{\epsilon_{a_i}}(a_i):1\le i\le n\}.$ Hence there exists $\displaystyle 1\le j\le n$ such that $\displaystyle d(x,a_j)$ is a minimum. Set $\displaystyle a^*=a_j.$
    Last edited by TheAbstractionist; Apr 24th 2009 at 01:36 PM.
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  3. #3
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    Thanks, that makes sense.

    Can you show me how to do it following the hint that was given as well?
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