I'm working my way through Discrete Mathematics by Levin and tried to prove the "Pigeonhole Principle" on my own, as the proof in the book is entirely in words. I'm still new to proofs so I would really appreciate as much feedback and guidance as possible. I'm also still learning about this site so sorry for any formatting issues.
From the text: The Pigeonhole Principle: If more than n pigeons fly into n pigeon holes, then at least one pigeon hole will contain at least two pigeons. Prove this!
My attempt to a proof:
Let $x>n$, $x\in\mathbb{Z}^+$. Then the number of pigeons per hole is $x/n$. Since $x>n$, $x/n>1$ and $x\,\operatorname{mod}\, n\geq 1$, it follows that at least one hole has at least 2 pigeons.
Thank you!
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