Pigeonhole Principle & irrational numbers

[mndt]

Homework Statement

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.

Homework Equations

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 can't exactly come up with the correct intervals that form the final conclusion.

 

[HallsofIvy]

I think you are looking at it backwards. It is "n" that is 'given', not j. Given a positive integer, n, divide the interval from 0 to 1 into n equal sized (1/n of course) intervals.

 

[mndt]

I think you are looking at it backwards. It is "n" that is 'given', not j. Given a positive integer, n, divide the interval from 0 to 1 into n equal sized (1/n of course) intervals.

yes, but considering that 1 <= j <= n we have n different js.

that leaves us with n pigeons (js) and n pigeonholes (intervals). and no useful conclusion.

and why only irrational numbers?!