Exercises
[0C1]Prerequisites:[0BW]. (Dirichlet’s approximation theorem) Given an irrational number \(x\), show that there are infinitely many rationals \(𝛼\) such that we can represent \(𝛼=m/n\) in order to satisfy the relation
\[ \left| x - \frac m n \right| {\lt} \frac 1{n^ 2}\quad . \]Some comments.
Note for every fixed \(n≥ 2\) there is at most an \(m\) for which the previous relation holds; but there may not be one.
Note that if the relation holds for a rational \(𝛼\), there are only finite choices of representations for which it holds,
and certainly it holds for the "canonical" representation with \(n,m\) coprimes.
Solution 1