discrete mathematics – Prove that any rational number is positive $frac{m}{n}$ Can be displayed in the form of a “combined fraction”

“Combined fracture” is an expression of the form

$(a_{0},a_{1},a_{2}…,a_{n})=
> a_{0}+frac{1}{a_{1}+frac{1}{a_{2}+frac{1}{a_{3}+_{frac{1}{…+}}}}}$

$a_{0},a_{1},a_{2}…,a_{n}$ are natural numbers. Example:
$frac{275}{52}=(5,3,2,7)= 5+frac{1}{3+frac{1}{2+frac{1}{7}}}$

Prove that any rational number is positive $frac{m}{n}$ Can be displayed in the form of a “combined fraction” (induction for n)

I think for proving with induction start with base (n=1), we get (1)= 1+1=2

then assuming $(a_{0},a_{1},a_{2}…,a_{n})$ is a rational number, then we need to prove that $(a_{0},a_{1},a_{2}…,a_{n},a_{n+1})$ is rational

But I don’t know how to continue from here