Solution
= Solution
Set $p_{-2}=0,p_{-1}=1,q_{-2}=1,q_{-1}=0$ and define the <continued fraction convergents> by
$$
p_n=a_np_{n-1}+p_{n-2},\qquad q_n=a_nq_{n-1}+q_{n-2},\qquad
\frac{p_n}{q_n}=[a_0;\ldots,a_n].
$$
The recurrence reverses the sign of $p_nq_{n-1}-p_{n-1}q_n$, whose initial value at $n=0$ is $-1$. Hence $p_nq_{n-1}-p_{n-1}q_n=(-1)^{n-1}$. Every common <divisor> of $p_n,q_n$ divides this number, proving $\boxed{\gcd(p_n,q_n)=1}$.