Exercises
[0BY] Prerequisites:[0BS],[0BW].Given \(x,b ∈ ℝ\) with \(x≠ 0\) irrational, and \(\varepsilon {\gt}0\), prove that there is a natural \(M\) such that \(M x-b\) is at most \(\varepsilon \) from an integer.
Let \(𝜑(x)=x-⌊ x ⌋\) be the fractional part of \(x\), we have \(𝜑(x)∈[0,1)\). The above result implies that the sequence \(𝜑(nx)\) is dense in the interval \([0,1]\).
Note that instead if \(x≠ 0\) is rational i.e. \(x=n/d\) with \(n,d\) coprime integers and \(d{\gt}0\), then the sequence \(𝜑(nx)\) assumes all and only the values \(\{ 0,1/d,2/d,\ldots (d-1)/d\} \).
(This is demonstrated by the Bézout’s lemma [ 26 ] ).
Solution 1[ [0C0]]