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

[gestalt]

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

My question on this is what possible help could the boundedness of the Im(z_n) to this proof and what theorem might be of help?

 

[micromass]

Do you know anything about compactness or Bolzano-Weierstrass??

 

[mathwonk]

do you know how e^z behaves geometrically?

 

[gestalt]

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?

 

[micromass]

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

|e^{iz_n}|\leq C

 

[gestalt]

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*z_n) 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?

 

[mathwonk]

circles are bounded, right?