I need to prove that $\left(1+\frac{z_{1}}{z_{2}}\right)\left(1+\frac{z_{2}}{z_{3}}\right)...\left(1+\frac{z_{n}}{z_{1}}\right)\in\mathbb R$ where $|z_{1}|=|z_{2}|=...=|z_{n}|=1$.
This can be done relatively easily by induction, but I'm looking for more elegant solution.
Any ideas?
Thanks!