Pigeonhole Principle & irrational numbers

1. Jan 4, 2009

mndt

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.

Last edited: Jan 4, 2009
2. Jan 4, 2009

HallsofIvy

Staff Emeritus
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.

3. Jan 4, 2009

mndt

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?!

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