Let $\displaystyle b=r_0, r_1, r_2... $ be the successive remainders in the Euclidean Algorithm applied to $\displaystyle a$ and $\displaystyle b$. Show that $\displaystyle r_{i+2} \le r_i/2$ for all $\displaystyle i$.

I'm thinking that this is proven by induction but I am stuck on the inductive step.