Prove or disprove : The limit $\lim_{n\rightarrow\infty}z^3_n$ exists if and only if the limit $\lim_{n\rightarrow\infty}z_n$ exists

Question

Let $z_n$ be a complex sequence. The limit $\lim_{n\rightarrow\infty}z^3_n$ exists if and only if the limit $\lim_{n\rightarrow\infty}z_n$ exists

I think the statement is true. From the definition, an infinite sequence of complex numbers has a limit $z$ if, for each positive number $ε$, there exists a positive integer $n_0$ such that $|z_n − z|<ε$ whenever $n>n_0$.

How can I use the definition to prove the statement above? Is it true in the first place?

Answer 1

The statement is false. Let $\omega=-\frac12+\frac{\sqrt3}2i=e^{2\pi i/3}$, and let $z_n=\omega^n$. Then $(\forall n\in\Bbb N):z_n^{\,3}=1$, but the sequence $(z_n)_{n\in\Bbb N}$ diverges.