1. ## Prove that...

Prove that, among the numbers $\displaystyle \sqrt{2}$, $\displaystyle 2\sqrt{2}$, $\displaystyle 3\sqrt{2}$, ..., one of them gets arbitrarily close to an integer.

2. ## Re: Prove that...

Here is a hint: If $\displaystyle n \sqrt{2}=m+\epsilon$ with a small $\displaystyle \epsilon$, how is that related to $\displaystyle \frac{m}{n}$?
Which property of the rational numbers can you use?

3. ## 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.

4. ## 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...

5. ## Re: Prove that...

According to mfb's notation, n is an integer.

There's a fairly simple solution to this...

6. ## Re: Prove that...

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.