How Does the Boundedness of Im(zn) Aid in Proving Convergence of <e^(i*zn)>?

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Discussion Overview

The discussion revolves around the convergence of the sequence where is a sequence of complex numbers with a bounded imaginary part. Participants explore how the boundedness of Im(zn) may assist in proving the convergence of and consider relevant theorems such as Bolzano-Weierstrass.

Discussion Character

  • Exploratory, Technical explanation, Debate/contested

Main Points Raised

  • One participant questions how the boundedness of Im(zn) contributes to proving the convergence of and seeks relevant theorems.
  • Another participant brings up the Bolzano-Weierstrass theorem, noting its applicability to bounded sequences, while acknowledging that the sequence in question is only bounded below.
  • A participant mentions the geometric behavior of , suggesting it forms a circle in the complex plane.
  • There is a suggestion to demonstrate that the sequence is bounded by finding a constant C such that |e^{iz_n}| ≤ C.
  • A later reply expresses uncertainty about the boundedness of and seeks clarification on its behavior, indicating a possible misunderstanding of the sequence's properties.
  • One participant confirms that circles are indeed bounded, implying a connection to the discussion on the boundedness of the sequence.

Areas of Agreement / Disagreement

Participants express uncertainty regarding the boundedness of and whether the Bolzano-Weierstrass theorem can be applied in this context. Multiple competing views remain about the implications of the boundedness of Im(zn) and the behavior of the sequence.

Contextual Notes

The discussion highlights limitations in understanding the implications of boundedness, particularly regarding the application of the Bolzano-Weierstrass theorem to sequences that are only bounded below. There is also uncertainty about the geometric interpretation of and its boundedness.

Who May Find This Useful

This discussion may be useful for those interested in complex analysis, particularly in understanding convergence properties of complex sequences and the application of theorems related to boundedness.

gestalt
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Let <zn> be a sequence complex numbers for which Im(zn) is bounded below.
Prove <e^(i*zn)> has a convergent subsequence.

My question on this is what possible help could the boundedness of the Im(zn) to this proof and what theorem might be of help?
 
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Do you know anything about compactness or Bolzano-Weierstrass??
 
do you know how e^z behaves geometrically?
 
Yes, I do know the B-W theorem. I looked at it but it applies to bounded sequences. This sequence is only bounded below and only on the I am part. I suppose since you asked you see a way it is still applicable.

It was my understanding e^z forms a circle in the complex. Is this what you mean?
 
Last edited:
gestalt said:
Yes, I do know the B-W theorem. I looked at it but it applies to bounded sequences. This sequence is only bounded below and only on the I am part. I suppose since you asked you see a way it is still applicable.

It was my understanding e^z forms a circle in the complex. I this what you mean?

Try to show that the sequence is bounded. Try to show that there is a constant C such that

[tex]|e^{iz_n}|\leq C[/tex]
 
Edit: The latex was not loading but now I see what you mean. I apologize if this sounds ignorant but I did not think e^(i*zn) was bounded. If you are saying I should show it is then I assume it can be.
When I think of this sequence I must be seeing it all wrong. could you give any explanation of how it behaves. If it helps I see nothing but a circle when I think of it. What attribute am I missing?
 
Last edited:
circles are bounded, right?
 

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