Thread: Prove that...

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

Here is a hint: If with a small , how is that related to ?
Which property of the rational numbers can you use?

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.

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...

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

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

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.