ILL 1976 Problem 34

Let $\{a_n\}^{\infty}_0$ and $\{b_n\}^{\infty}_0$ be two sequences determined by the recursion formulas $$a_{n+1} = a_n + b_n,$$ $$b_{n+1} = 3a_n + b_n, n= 0, 1, 2, \cdots,$$ and the initial values $a_0 = b_0 = 1$ . Prove that there exists a uniquely determined constant $c$ such that $n|ca_n-b_n| < 2$ for all nonnegative integers $n$ .