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Closest point compact set

  1. Oct 21, 2008 #1
    1. The problem statement, all variables and given/known data

    Let X be a metric space and let K be any non-empty compact subset of X, and let x be an element of X. Prove that there is a point y is an element of K st d(x,y) leq d(x,k) for every k an element of K.

    2. Relevant equations

    triangle inequality


    3. The attempt at a solution

    i tried proof by contradiction with the triangle inequality, and it didn't get me anywhere
    1. The problem statement, all variables and given/known data



    2. Relevant equations



    3. The attempt at a solution
     
  2. jcsd
  3. Oct 21, 2008 #2

    morphism

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    Look at the set {d(x,k) : k in K}. Note in particular that it's bounded below.
     
  4. Oct 21, 2008 #3
    its bounded below as k gets closer to x is that inf K?
     
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