Let $C_k = \frac{q_k}{p_k}$ denote the $k^{th}$ convergent of $[a_0;a_1,a_2,\dots]$. Prove that $$p_kq_{k-1}-p_{k-1}q_k = (-1)^{k-1}.$$
I am not sure how to prove this and I am not sure what it even means. To my understanding, a convergent is a rational number given by the Continued Fraction Algorithm. For example, if we apply the continued fraction algorithm to $\sqrt{6}$ we get $$c_0 = 2; c_1 = \frac{5}{2}; c_2 = \frac{22}{9}; c_4 = \frac{49}{20}$$
Take $c_3$ for example, what are $P_3$ and $q_3$? I would assume that $p_3 = 49$ and $q_3=20$, but under this assumption the theorem fails. Can anyone give me an explanation and hint for the proof?