So I found this theorem and proof in my abstract algebra book in the section about cyclic groups. Here it goes:
Theorem: If $z^n=1$, then the nth roots of unity are $$z=\text{cis}(\frac{2k\pi}{n})$$where $k\in \mathbb Z^{+}.$
Proof: DeMoivre's Theorem. $$z^n=\text{cis}(n\frac{2k\pi}{n})=\text{cis}(2k\pi)=1$$
My issue: the assumption is already made in the proof that $|z|=r=1$, because DeMoivre's Theorem states that $z^n=r^n\text{cis}(n\theta)$, so $r^n=1$ for this to work. Am I missing something here?