Sequences $(a_n), (b_n)$ are defined by $a_1 = 1, b_1 = 2$, $a_{n+1} = \frac{ 1 + a_n + a_nb_n}{b_n}$, $b_{n+1} = \frac{ 1 +b_n+ a_nb_n}{a_n}$ for all positive integers $n$. Prove that $a_{2020} < 5$.