VMO 1993 Problem 6

Define the sequences $a_0$, $a_1$, $a_2$, ... and $b_0$, $b_1$, $b_2$, ... by $a_0$ = 2, $b_0$ = 1, $a_{n+1}$ = $2a_nb_n$/($a_n$ + $b_n$), $b_{n+1}$ = √($a_{n+1}b_n$). Show that the two sequences converge to the same limit, and find the limit.