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Here is a differential geometric point of view, which of course works only over $\mathbb C$.

Since $L\to X$ is ample, then it carries a smooth hermitian metric $h$ of positive definite Chern curvature $i\Theta(L)>0$.

Now let $f\colon Y\to X$ be any finite morphism. Then $f^*L$ inherits a smooth hermitian metric which is just the pull-back oh $h$ and its Chern curvature is given by the pull-back $i f^*\Theta(L)$. Since $f$ is finite, then its differential is (more or less) injective, so that $i f^*\Theta(L)$ is positive definite, too.

But then, by Kodaira's projectivity criterion, $f^*L$ is ample.

1

Here is a differential geometric point of view, which of course works only over $\mathbb C$.

Since $L\to X$ is ample, then it carries a smooth hermitian metric $h$ of positive definite Chern curvature $i\Theta(L)>0$.

Now let $f\colon Y\to X$ be any finite morphism. Then $f^*L$ inherits a smooth hermitian metric which is just the pull-back oh $h$ and its Chern curvature is given by the pull-back $i f^*\Theta(L)$. Since $f$ is finite, then its differential is injective, so that $i f^*\Theta(L)$ is positive definite, too.

But then, by Kodaira's projectivity criterion, $f^*L$ is ample.