1 Introduction

To show a real number \(\xi \) is irrational, we usually construct a non-zero sequence \(\{ a_n + b_n\xi \} \rightarrow 0\), when \(n \rightarrow \infty \), where\(a_n, b_n \in \mathbb {Z}\).

Cause if \(\xi \) is rational number \(\frac{p}{q}\), then the sequence has a low bound \(\frac{1}{q}\).