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.
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...