I saw this one in an analysis text. The author didn't say where this inequality comes from or where it is used. I'd like to know more about it if anyone has information on it. Looks like some kind of number theoretic inequality since with prime is always irrational.

Let be such that is irrational. We want to prove there is some such that for all integers with , we have .

I have no clue on how to prove this one. The hint was to rationalize. So Then but supposing , we have and implies is an integer which is a contradiction. Therefore and is an integer. So .