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In elementary proofs like the one elucidated in Kevin McGerthy's answer, there is usually a step where for a given $n\in \mathbb{N}_0$ and given $\alpha\in \mathbb{C}$ a $z\in \mathbb{C}$ has to be found so that $Re(\alpha z^n)<0$. People usually browse very quickly over this part and implicitly use something like Euler's or De Moivre's formula or use a topological method. This step can however be accomplished in a more 'direct' way. Indeed, for this task it suffices to show that $\forall n \in \mathbb{N}_0:\,\forall \beta \in \{1,i,-1,-i\}$ the polynomial $z^n-\beta$ has a root. This can be shown using the Cartesian formula for the square root $$(x+yi)^{\frac{1}{2}}=\left(\frac{x+(x^2+y^2)^{\frac{1}{2}}}{2}\right)^{\frac{1}{2}}+\text{sgn}(y)\left(\frac{-x+(x^2+y^2)^{\frac{1}{2}}}{2}\right)^{\frac{1}{2}}$$ and an induction argument (on $n$). As I suspect that many people find such a remark 'pedantic', I'll waste no more words on it :)