let [tex]a\in R[/tex] and [tex] Q>=3[/tex] where [tex]Q\in Z[/tex], i need to prove that in the set {a,2a,...,(Q-1)a} there exists a number which its distance from the nearest whole number is smaller than 1/Q.(adsbygoogle = window.adsbygoogle || []).push({});

i got this as an assignment in topic of the pigeon hole, now i dont see how to use it here, what i did so far (havent yet finished) is first for the integers it's trivially correct, so i suppose that a is non integer real number, and first prove it for a between 0 and 1, and afterwards for a>=1, with symmetry we can see that if we prove this then it's trivially valid for the negatives.

so basically my proof is constrcutive, if 0<a<1 then the nearest value is 1 or 0, so either |a|<1/Q or |a-1|<1/Q, if they both arent valid we keep on going to the next number 0<2a<2, because a>=1/Q and 1-a>=1/Q we have that 2a must be between 1 and 2, and so we proceed as before.

my two questions are, is this approach valid, and is there any other proofs which might use the pigeon hole principle?

thanks in advance.

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# A question in combinatorics. (perhaps implementation of the pigeon hole principle).

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