the inequality in this form that you wrote is false because, for example, it will fail if you put q = 1 and let p go to infinity. the correct form of the inequality is:

is a special case ofLiouville's approximation theorem: if is a root of a polynomial of degree in then there exists such that for all integers with we'll

have: .

Proof of: let and choose let p and q be integers with q > 0. we consider two cases:

case 1. in this case we'll have

case 2. then we'll get now since we have thus: threrefore: