On a Theorem of Legendre on Diophantine Approximation

Legendre’s theorem states that every irreducible fraction $\frac{p}{q}$ which satisfies the inequality $\mid \alpha -\frac{p}{q}\mid <\frac{1}{2q^2}$ is convergent to $\alpha$. Later Barbolosi and Jager improved this theorem. In this paper we refine these results.