1. The problem statement, all variables and given/known data Let x be an irrational number. Show that the absolute value of the difference between jx and the nearest integer to jx is less than 1/n for some positive integer j not exceeding n. 2. Relevant equations 3. The attempt at a solution Ok, I know that it should be solved using pigeonhole Principle and there is the fact that for real numbers: 0 <= | jx - [jx] | < 1 specifically for irrational numbers: 0 < | jx - [jx] | < 1 that should make the difference but i cant exactly come up with the correct intervals that form the final conclusion.