Question: Let $[a_0; a_1, a_2, ...]$, an infinite simple continued fraction, be a positive irrational number. Show that $a_n > 0$ for all $n\geq 1$ (every integer past $a_0$ in the continued fraction is greater than 0).
My attempt: I was thinking of using the sequences $\left(\frac{p_{2n}}{q_{2n}}\right)_{n\geq 1}$ and $\left(\frac{p_{2n+1}}{q_{2n+1}}\right)_{n\geq 1} $, and showing that they converge to the same limit, which is the irrational number $[a_0; a_1, a_2, ...]$. But I am not sure how to proceed with this.