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We may assume that $n\geq 2$. Let $f:X\to \mathbb P^n$ be a finite étale morphism and $H\subset \mathbb P^n$ be a hyperplane. Then $f^*H$ is an ample divisor on $X$ and hence connected. By induction, then the restriction $f^*H\to H$ is an isomorphism, so $\deg f=1$ and $f$ is an isomorphism.