Prove that, among the numbers , , , ..., one of them gets arbitrarily close to an integer.

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- June 13th 2012, 11:36 PMrichard1234Prove that...
Prove that, among the numbers , , , ..., one of them gets arbitrarily close to an integer.

- June 14th 2012, 08:37 AMmfbRe: Prove that...
Here is a hint: If with a small , how is that related to ?

Which property of the rational numbers can you use? - June 14th 2012, 11:06 AMrichard1234Re: Prove that...
Haha I already know the solution :) Hence the "that you can solve yourself" part in the title of this forum.

There's a more elegant solution involving the Pigeonhole principle. - July 5th 2012, 08:04 PMcalcmasterRe: Prove that...
Wait, does it say that n must be an integer? Or do we just infer that from the number sequence given? Because if not we can say that there are more combinations of n times root 2 than there are integers so one must be arbitrarily close to an integer? I'm not a calcmaster as my user name implies but I do like challenging problems! Hope I didn't say something too ridiculous...

- July 5th 2012, 08:53 PMrichard1234Re: Prove that...
According to mfb's notation, n is an integer.

There's a fairly simple solution to this... - July 6th 2012, 06:38 AMmfbRe: Prove that...