 Solution

Consider one iteration with $a_i>b_i>0$ and $r_i=a_i\mathrm{mod}{b}_i$. If $b_i\le a_i/2$, then $r_i<b_i\le a_i/2$. If $b_i>a_i/2$, the quotient is $1$, so $r_i=a_i-b_i<a_i/2$. In either case $r_i<a_i/2$, whenceRepeated application, or mathematical induction, provesThis also holds for the final iteration, whose left side is zero. For zero iterations the correct relation is equality $a_0{b}_0=ab$, rather than a strict inequality.