Difference between Adjacent Convergents But One of Simple Continued Fraction

Theorem
Let $F$ be a field, such as the field of real numbers $\R$.

Let $n \in \N \cup \{\infty\}$ be an extended natural number.

Let $\left[{a_0, a_1, a_2, \ldots}\right]$ be a continued fraction in $F$ of length $n$.

Let $p_0, p_1, p_2, \ldots$ and $q_0, q_1, q_2, \ldots$ be its numerators and denominators.

Let $C_0, C_1, C_2, \ldots$ be the convergents of $\left[{a_0, a_1, a_2, \ldots}\right]$.

For $k \geq 2$:
 * $p_k q_{k - 2} - p_{k - 2} q_k = \left({-1}\right)^{k} a_k$

That is:
 * $C_k - C_{k-2} = \dfrac {p_k} {q_k} - \dfrac {p_{k - 2} } {q_{k - 2} } = \dfrac {\left({-1}\right)^{k} a_k} {q_k q_{k - 2} }$

Proof
Let $k \geq 2$.