On a theorem of Borel on diophantine approximation

A theorem of É. Borel’s asserts that one of any three consecutive convergents of a real number a, which we denote \(\frac{p}{q}\) $\frac{p}{q}$ , satisfies the inequality \(\left| a-\frac{p}{q} \right| < \frac{C}{q^2}\) $\left|a,-,\frac{p}{q}\right|<\frac{C}{q^2}$ with \(C=\frac{1}{\sqrt{5}}\) $C=\frac{1}{\sqrt{5}}$ . In this paper we give more precise information about the constant C.