An elementary proof of Lelli Chiesa's theorem on constancy of second coordinate of gonality sequence

Let $X$ be a K3 surface and $L$ be an ample line bundle on it. In this article we will give an alternative and elementary proof of Lelli Chiesa’s Theorem in the case of $r=2$. More precisely we will prove that that under certain condition the second co-ordinate of the gonality sequence is constant along the smooth curves in the linear system $\mathrm{\mid }L\mathrm{\mid }$. Using Lelli Chiesa’s theorem for $r\ge 3$ we also extend Lelli Chiesa’s Theorem in the case of $r= 2$ in weaker condition.