Let $\theta\in\Bbb{R}_{\gt0}$.
A) Prove that the convergents for the continued fraction expansion of $\theta$ give us better and better rational approximations to $\theta$.
B) Suppose $\theta\notin \Bbb{Q}$. Prove that the convergents $a_n/b_n$ from the continued fraction expansion of $\theta$ form an alternating series of better and better rational approximations to $\theta$: $$\frac{a_1}{b_1}\lt\frac{a_3}{b_3}\lt...\lt\theta\lt...\lt\frac{a_4}{b_4}\lt\frac{a_2}{b_2}$$
I feel if I could do A then B probably follows somehow, and logically it makes sense, but no idea how to prove either!