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Sequences in Complex Plane which Converge Absolutely

  1. Nov 7, 2011 #1
    Let A be a non-empty subset of the complex plane and let b ∈ ℂ be an arbitrary point not in A. Now define d(A,b) := inf{|z-b| : z ∈ A}. Show that if A is closed, then there is an a ∈ A such that d(A,b) = |a-b|.

    Ok so basically what I did was begin by choosing some arbitrary element of A and labeling it z_1 and then set |z_1 - b|=x_1. Then I defined a rule such that we find some other element in A, z_2, such that |z_2 - b|=x_2 < x_1. So in this way we have defined a sequence of real numbers which is always positive, decreasing, and bounded below by zero, hence it converges to some real number call it x. Now there is a plurality of z ∈ ℂ such that |z-b| = x, the difficult part for me is showing that one of these z is in A.

    I mean I can see that the sequence (z_n) need not converge, since it may simply continue bouncing around a circle of radius x in the complex plane. What I originally wanted to use to prove that there is a z ∈ A such that |z-b|=x was that convergent sequences in closed sets converge to limits within that set. But since (z_n) need not necessarily converge for (x_n) to converge, that puts a damper on things. I was thinking of maybe finding a subsequence which converges, or assuming by contradiction that if none of the z such that |z-b|=x are in A than that would cause (x_n) not to converge, but I can't find a way to set that up properly either.
     
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  3. Nov 7, 2011 #2

    LCKurtz

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    Say you call K = {z: |z - b| ≤ 1}. Then [itex]K\cap A[/itex] is compact and you can get your sequence in there...
     
  4. Nov 7, 2011 #3
    I'm sorry but this is simply too terse and cryptic for me to comprehend. The distance from the closest point in A to b is unknown so your intersection of K and A may be empty. Are you describing a way to find a convergent subsequence? I just really can't tell, please elaborate.
     
  5. Nov 7, 2011 #4

    Dick

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    Saying K = {z: |z - b| ≤ 1} is probably not what LCKurtz meant to write. How about K = {z: |z - b| ≤ 2*d(A,b)}? You know how having a compact set would solve your 'points bouncing around' problem, yes?
     
  6. Nov 7, 2011 #5

    LCKurtz

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    Yes, careless slip there. Thanks Dick.
     
  7. Nov 8, 2011 #6
    Ah, yes, excellent, thanks.
     
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