# 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