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mndt
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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.
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