Suppose we arrange finitely many pigeons in infinitely many pigeon holes. How do I use the Infinite Pigeonhole Principle to prove that there are infinitely many pigeonholes that contain no pigeons.


1 Answer 1


Let's not use any theorem, but just prove it.

Suppose not, i.e. that only finitely many pigeonholes do not contain pigeons. This means that there are an infinite number of holes with pigeons. But this means that there are infinitely many pigeons, which is a contradiction.

(Or it means that we've cooked and cut the pigeons into very many tiny pieces and distributed them maniacally)

  • 2
    $\begingroup$ Can we combine the pieces into twice as many pigeons as we had previously? $\endgroup$
    – Brian Tung
    May 4, 2015 at 23:53
  • $\begingroup$ I mean in the problem, it says I have to use that principle to prove it $\endgroup$
    – Ray
    May 4, 2015 at 23:55

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