# Prove that...

• Jun 13th 2012, 11:36 PM
richard1234
Prove that...
Prove that, among the numbers $\sqrt{2}$, $2\sqrt{2}$, $3\sqrt{2}$, ..., one of them gets arbitrarily close to an integer.
• Jun 14th 2012, 08:37 AM
mfb
Re: Prove that...
Here is a hint: If $n \sqrt{2}=m+\epsilon$ with a small $\epsilon$, how is that related to $\frac{m}{n}$?
Which property of the rational numbers can you use?
• Jun 14th 2012, 11:06 AM
richard1234
Re: 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.
• Jul 5th 2012, 08:04 PM
calcmaster
Re: 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...
• Jul 5th 2012, 08:53 PM
richard1234
Re: Prove that...
According to mfb's notation, n is an integer.

There's a fairly simple solution to this...
• Jul 6th 2012, 06:38 AM
mfb
Re: Prove that...
Quote:

Originally Posted by calcmaster
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?

[0.4 , 0.6] as interval in the real numbers has more numbers than there are integers, but it does not have any number less than 0.4 apart from an integer.