Write with quantifiers P and ¬P

Question

Let P: "for all natural $n$, exists an integer $p$ such that $n<p<2n$ "

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P: $\forall\, n \in \mathbb{N}\, ||\, \exists \, p\in\mathbb {Z} \, ||\, n<p<2n $

¬P: $\exists\, n \in \mathbb{N}\, ||\, \forall \, p\in\mathbb {Z} \, ||\, n>p>2n $

Is correct the translation?

Answer 1

Not quite. The negation of $n<p<2n$ should be $p\le n$ or $p \ge 2n$.