(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Letxbe an irrational number. Show that the absolute value of the difference betweenjxand the nearest integer tojxis less than1/nfor some positive integerjnot exceedingn.

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.

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# Homework Help: Pigeonhole Principle & irrational numbers

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