Complex Sequence proof

[eigenman]

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.

*Note on notation: z_n is read as z sub n.

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?

 

[Freeman]

Let \(\displaystyle z_n=x_n+iy_n.\ x_n,y_n\in\mathbb{R},y_n\) is bounded below.
\(\displaystyle |e^{iz_n}|=|e^{ix_n-y_n}|=e^{-y_n}|e^{ix_n}|=e^{-y_n}\) is bounded above and below.
Every bounded sequence has a convergent subsequence.