Let’s assume $a_n$ is a sequence of positive integers, and we have another sequence $A_n$ that $$ A_1 = \frac{1}{a_1} $$ $$ A_2 = \frac{1}{a_1 + \frac{1}{a_2}} $$ $$ ... $$
How should we prove that:
A) $A_n$ converges to an irrational number called $A$ between 0 and 1;
and
B) If $a_n$ is periodic, $A$ is a root of an quadratic equation like $mx^2 + nx + p = 0$ ($m$, $n$ and $p$ are positive integers)
Could we generalize the proof found to find a way to write irrational numbers (like $\sqrt{2}$) in continued fraction format?