Comparation between continued fractions

Question

I'm trying to solve the following problem but I'm having some difficulties..

Let $a_0,a_1,\dots,a_n$ and $b_0,b_1,\dots,b_n,b_{n+1}$ be positive integers. Give conditions that make the following statement true: $$[a_0;a_1,\dots,a_n]<[b_0;b_1,\dots,b_n,b_{n+1}]$$

I thing a have to use convergents of the continued fraction but I don't know how. Thanks!

Answer 1

Hints:

1. Left side is between $a_0$ and $a_0 + 1$.
2. $[a_0; a_1, \ldots ] = a_0 + 1/[a1; a_2, \ldots]$