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Distance between point and set

  1. Aug 10, 2013 #1
    \in1. The problem statement, all variables and given/known data
    Denote by d(x,A) = inf |x-y|,y [itex]\in[/itex] A, the distance between a point x [itex]\in[/itex] R^n and a set A [itex]\subseteq[/itex] R^n. Show

    |d(x,A)-d(z,A)| [itex]\leq[/itex] |x-z|

    In particular, x → d(x,A) is continuous

    2. Relevant equations

    3. The attempt at a solution

    I have no idea on how to prove this. I drew a picture and the result seemed intuitive but I don't know how to prove it mathematically.

    Appriciate any help!
  2. jcsd
  3. Aug 10, 2013 #2


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    For closed sets, this is easy to show with the triangle inequality. For general sets, I would try to apply the same argument for a converging series of elements of A.
  4. Aug 10, 2013 #3
    Ah yes! I actually tried the triangle inequality but failed. I am going to try again!

    Could you please elaborate some more on the second part? I have been stuck on similar questions because I do not understand this argument.
  5. Aug 10, 2013 #4


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    If there is no y in A such that d(x,A)=d(x,y), there is a sequence yi such that d(x,yi) converges to d(x,A) (for i->infinity).
  6. Aug 12, 2013 #5
    Thank you! I got it right.
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