Sequences in Complex Plane which Converge Absolutely

In summary, the conversation discusses a proof for the existence of an element in a closed subset of the complex plane such that its distance from an arbitrary point outside the subset is equal to the infimum of the distances between the subset and the point. The proof involves defining a decreasing sequence of real numbers and finding a subsequence that converges within the subset, using the fact that the intersection of a compact set and a closed set is also compact.
  • #1
Poopsilon
294
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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  • #2
Say you call K = {z: |z - b| ≤ 1}. Then [itex]K\cap A[/itex] is compact and you can get your sequence in there...
 
  • #3
Say you call K = {z: |z - b| ≤ 1}. Then K∩A is compact and you can get your sequence in there...

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.
 
  • #4
Poopsilon said:
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.

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?
 
  • #5
Dick said:
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?

Yes, careless slip there. Thanks Dick.
 
  • #6
Ah, yes, excellent, thanks.
 

1. What is the definition of absolute convergence in sequences in the complex plane?

A sequence in the complex plane converges absolutely if the sum of the absolute values of its terms converges to a finite limit.

2. How is absolute convergence different from conditional convergence in sequences in the complex plane?

Absolute convergence in sequences in the complex plane means that the sum of the absolute values of the terms converges, regardless of the order in which the terms are added. Conditional convergence means that the sum of the terms converges, but only when added in a specific order.

3. What is the significance of absolute convergence in sequences in the complex plane?

Absolute convergence in sequences in the complex plane is important because it guarantees that the series will converge to a finite value, regardless of the order in which the terms are added. This allows for easier calculation and analysis of the series.

4. Can a series in the complex plane converge absolutely but not converge?

Yes, it is possible for a series in the complex plane to converge absolutely but not converge. This can happen if the series is conditionally convergent but not absolutely convergent.

5. How can one determine if a sequence in the complex plane converges absolutely?

To determine if a sequence in the complex plane converges absolutely, one can use the ratio or root test, which check for the convergence of the absolute values of the terms in the series. If the limit of the ratio or root is less than 1, then the series converges absolutely.

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