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Real analysis

  1. Nov 16, 2013 #1
    1. The problem statement, all variables and given/known data

    Prove that if ##S## is a nonempty closed subset of ##E^n## and ##p_0\in E^n## then ##\min\{d(p_0,p):p\in S\}## exists.

    2. The attempt at a solution

    If ##p_0## was in ##S## why would ##\min\{d(p_0,p):p\in S\} = 0?## Is it just because it is the minimum? How about if ##p_0 \in S## then what will ##\max\{d(p_0,p):p\in S\}## be?
     
  2. jcsd
  3. Nov 16, 2013 #2

    Dick

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    max is not really the issue. The minimum of a set of numbers is the smallest number in the set. If you take the open interval (0,1) it doesn't have a minimum. It does have an infimum which is 0 but that's not in the set. So it doesn't have a minimum. So if you put S to be the open interval (0,1) and ##p_0=0##, then the minimum does not exist. If S were closed why is the situation different?
     
    Last edited: Nov 16, 2013
  4. Nov 16, 2013 #3

    PeroK

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    Don't forget that {d(p_0,p):p\in S\} is a non-empty subset of IR bounded below (by 0).
     
  5. Nov 16, 2013 #4
    Thank you Dick and PeroK! I think I understand now.
     
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