Upper bound of a sequence Suppose that $\left|a_n-b_n\right| \leq \lambda_n\left|a_{n-1}-b_{n-1}\right|$ for all $n$ with $\lambda_n < 1.$ Find an upper bound on $\left|a_n-b_n\right|$ in terms of $\left|a_0-b_0\right|$ and $\lambda= \max \{\ \lambda_i : 1 \leq i \leq n \}$.
I can't think of anything and not even sure where to start. Some help would be great.

HINT: Show by induction on $n$ that
$$|a_n-b_n|\le|a_0-b_0|\prod_{k=1}^n\lambda_k\;.$$
Then get a simple upper bound on the product.
