Suppose the sequence $z_{n}$ converges to a nonzero limit $A$ and let $\Phi_{n}$ be any sequence of values of $Arg (z_{n})$ satisfying the inequality $|\Phi_{m}-\Phi_{n}|<\pi$ for $m>N, n>N$. Prove that $\Phi_{n}$ converges to one of the values of $Arg(A)$.
I really couldn't find the way to start this. In my previuos question I asked about a similar proof. Maybe this should have something in commun.
My idea for the case where $A$ is not negative real number: let $Arg(z_{m})=\arg z_{m}+2k\pi$ and $Arg(z_{n})=\arg z_{n}+2t\pi$. Since $|\Phi_{m}-\Phi_{n}|<\pi$ and $\arg z_{m}\to \arg A$ and $\arg z_{n} \to \arg A$, $t=k=C$ so $\lim_{n \to \infty} \Phi_{n} = \arg A + 2C\pi$, which would be the value of some of the arguments of $A$.